Algorithms and Heuristics for Wavelength allocation in Optical ...

Algorithms and Heuristics for
Wavelength allocation in Optical
Networks
Godfrey Justo
Department of Computer Science
King’s College London, University of London
A thesis submitted for the degree of
Doctor of Philosophy

Abstract
Optical networks allow end-to-end high-speed connection for signal
transmission through optical fibers using wavelength division multiplexing
(WDM). The WDM technology transmit many light signals
concurrently on an optical fiber through wavelength routing, in which
a network switching node routes signals based on their wavelengths.
The technology has the potential to cater for the ever-growing high
bandwidth need on several emerging bandwidth critical applications,
including video conferencing and real time digital imaging.
In this thesis we present improved algorithms and heuristics for allocation
of network bandwidth (wavelength) for different network topologies.
For ring networks we propose different heuristics for wavelength
allocation for networks with/no use of wavelength conversion. These
heuristics are also extended to tree of rings networks without wavelength
conversion. We experimentally evaluate the heuristics and
show a significant improvement for wavelength utilization compared
to previously known algorithms. Moreover, we study ring networks
that allow use of more than one fiber per link (multi-fiber rings) and
wavelength conversion. We show that if a single network node is
equipped with limited wavelength conversion capabilities of degree
two, or a pair of nodes is equipped with limited wavelength conversion
of degree one (fixed conversion), then multi-fiber rings can optimally
allocate the available bandwidth on any instance.
For tree networks that allow more than one optical fiber per link
(multi-fiber trees) we propose approximation algorithms for minimizing
the number of total optical fibers sufficient to satisfy a given

instance. Among other results we propose a 2-approximation algorithm
for spider tree (a tree such that any path has at most one
node of degree greater than 2), and a 4-approximation algorithm for
a tree such that any path has at most three nodes of degree greater
than 2. We study the problem of combining low rate traffic to share
a high bandwidth wavelength (traffic grooming) for path and ring
networks. We propose different heuristics for grooming arbitrary instances
with unit rate requests for path networks with grooming ratio
2 or more, extending the previous work for this problem on path networks
with grooming ratio 2 and all-to-all instances. Moreover, we
consider path networks with grooming ratio 3 and all-to-all instances
on unit rate requests and present results that imply grooming algorithm
with asymptotic performance ratio 1.07. For ring networks we
propose a traffic grooming heuristic that is experimentally compared
to previously known heuristics. Our heuristic improve upon previously
known algorithms for most grooming instances.
3

Contents
0.1 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.2 Mathematical notations . . . . . . . . . . . . . . . . . . . . . . . 16
1 Introduction 17
1.1 General thesis contribution . . . . . . . . . . . . . . . . . . . . . . 21
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Terminology, models and background work 23
2.1 Graph-theoretic concepts . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Optical network models, routing and wavelength allocation . . . 24
2.2.1 Optical networks . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Routing and wavelength allocation . . . . . . . . . . . . . 26
2.2.2.1 RWA with wavelength continuity constraint . . . 27
2.2.2.2 RWA with wavelength conversion . . . . . . . . . 28
2.2.2.3 RWA with multi-granularity wavelengths . . . . . 30
2.3 Special Network topologies . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Experimental performance comparison . . . . . . . . . . . . . . . 33
2.4.1 Request instances . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.2 Routing decisions . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.3 Implementation platform . . . . . . . . . . . . . . . . . . . 36
2.5 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6 Our contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Undirected Rings 49
3.1 RAP model for structured programs . . . . . . . . . . . . . . . . . 50
3.2 Fat cover algorithm for RAP . . . . . . . . . . . . . . . . . . . . 53
4

CONTENTS
3.3 Wavelength allocation by pivot based algorithms . . . . . . . . . . 54
3.3.1 Finding pivot covers . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Fixed pivot algorithm . . . . . . . . . . . . . . . . . . . . 56
3.4 Demand routing and slotting . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 The min-load two phase algorithm . . . . . . . . . . . . . 58
3.5 Numerical results and performance evaluation . . . . . . . . . . . 59
4 Undirected Tree of Rings 66
4.1 Tree of rings model . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Finding pivot cover on ring of tree of rings . . . . . . . . . . . . . 68
4.3 Long pivot-DFS algorithm . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Approximation algorithm . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Input generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Numerical results and performance evaluation . . . . . . . . . . . 74
5 Undirected Rings with Cluster Wavelength Conversion 81
5.1 Cluster wavelength conversion model . . . . . . . . . . . . . . . . 81
5.1.1 Wavelength allocation with conversion framework . . . . . 82
5.2 Wavelength allocation with pivot method and conversion framework 83
5.2.1 Long pivot with wavelength conversion . . . . . . . . . . . 83
5.2.2 Fixed pivot with wavelength conversion . . . . . . . . . . 83
5.3 Related algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Numerical results and performance evaluation . . . . . . . . . . . 85
6 Undirected Multi-fiber Trees 92
6.1 Tree network model . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 PCM algorithm for spider tree . . . . . . . . . . . . . . . . . . . . 94
6.2.1 Path orientation and grouping . . . . . . . . . . . . . . . . 94
6.2.2 PCM reduction to ECBM . . . . . . . . . . . . . . . . . . 95
6.2.3 Bounding performance of APR1 algorithm . . . . . . . . . 96
6.3 Extension to other tree networks . . . . . . . . . . . . . . . . . . 97
6.3.1 PCM algorithm for tree with s(T ) = 3 . . . . . . . . . . . 98
6.3.2 Bounding performance of APR2 algorithm . . . . . . . . . 99
5

CONTENTS
7 Undirected Multi-fiber Rings with Limited Wavelength Conversion
103
7.1 Single-fiber rings with wavelength conversion . . . . . . . . . . . . 105
7.2 Dual-fiber rings with limited wavelength conversion . . . . . . . . 111
7.3 k-fiber rings with limited wavelength conversion for k > 2 . . . . . 114
8 Traffic Grooming on Path Networks - General Instances 120
8.1 General lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 Algorithms for path networks with g = 2 . . . . . . . . . . . . . . 122
8.2.1 Interval partition algorithm . . . . . . . . . . . . . . . . . 122
8.2.2 IPA variant algorithms . . . . . . . . . . . . . . . . . . . . 123
8.2.3 Edge decomposition algorithm . . . . . . . . . . . . . . . . 124
8.2.3.1 Finding valid circuit subgraphs . . . . . . . . . . 127
8.2.3.2 Merging compatible valid subgraphs . . . . . . . 130
8.3 Numerical results and performance evaluation . . . . . . . . . . . 132
8.4 Extension to path networks with arbitrary grooming ratios . . . . 139
8.4.1 Splitting valid subgraphs . . . . . . . . . . . . . . . . . . . 144
9 Traffic Grooming on Unidirectional Path Networks - All-to-all
Instances 145
9.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9.3 Upper bound for A(P n , 3) . . . . . . . . . . . . . . . . . . . . . . 150
10 Traffic Grooming on Ring Networks - General Instances 169
10.1 General lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.2 Grooming Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.3 Fixed edge with edge decomposition algorithm . . . . . . . . . . . 171
10.3.1 Finding closed circuits . . . . . . . . . . . . . . . . . . . . 172
10.3.2 Finding open circuits . . . . . . . . . . . . . . . . . . . . . 173
10.3.3 Merging circuit subgraphs into valid subgraphs . . . . . . 174
10.3.4 Finding valid subgraphs from blue arcs . . . . . . . . . . . 174
10.3.5 ADMs and wavelengths requirement . . . . . . . . . . . . 174
10.4 Related algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6

CONTENTS
10.4.1 Algorithms IV and VI . . . . . . . . . . . . . . . . . . . . 175
10.4.2 Preprocessed iterative matching algorithm . . . . . . . . . 176
10.5 Numerical results and performance evaluation . . . . . . . . . . . 178
11 Conclusion and Future work 186
A Appendix 190
A.1 Undirected rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A.2 Undirected tree of rings . . . . . . . . . . . . . . . . . . . . . . . 196
A.3 Undirected rings with cluster wavelength conversion capabilities . 199
A.4 Traffic grooming on path networks - General Instances . . . . . . 204
A.5 Traffic Grooming on ring networks - General Instances . . . . . . 208
Bibliography 213
7

List of Figures
2.1 Fixed, degree 2, and full converters each with four inputs and
outputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Path, ring, star and mesh networks. . . . . . . . . . . . . . . . . . 32
2.3 A tree of rings with seven rings. . . . . . . . . . . . . . . . . . . . 32
3.1 A loop program and corresponding interval graph. . . . . . . . . . 50
3.2 A cyclic interval graph and corresponding circular arc graph and
interference graph. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Wavelength performance for instances with Uniform-SP distribution
of requests on 64-nodes ring network. . . . . . . . . . . . . . 64
3.4 Wavelength performance for instances with Uniform-LB distribution
of requests on 64 nodes ring network. . . . . . . . . . . . . . 65
3.5 Wavelength performance for instances with Gaussian-LB distribution
of requests on 64 nodes ring network. . . . . . . . . . . . . . 65
4.1 A tree of rings with a numbering of nodes. The numbers in ’[ ]’
are DFS numbers and the numbers in ’( )’ are surrogate numbers. 68
4.2 Different types of requests belonging to ring c. Rings c 1 , c 2 and c 3
have higher DFS numbers than ring c. . . . . . . . . . . . . . . . 69
4.3 A tree of rings with three rings, r 0 , r 1 and r 2 , sharing node u, showing
a long path {a, b}, short path {u, c}, two special links (u − , u)
and (w − , w), processed (dark) nodes and unprocessed (plain) nodes. 72
4.4 Wavelength performance for instances with Uniform-SP distribution
of requests on tree of rings with interval 16 − 32. . . . . . . . 78
8

LIST OF FIGURES
4.5 Wavelength performance for instances with Uniform-LB distribution
of requests on tree of rings with interval 16 − 32. . . . . . . . 78
4.6 Wavelength performance for instances with Gaussian-SP distribution
of requests on tree of rings with interval 16 − 32. . . . . . . . 79
4.7 Wavelength performance for instances with Gaussian-LB distribution
of requests on tree of rings with interval 16 − 32. . . . . . . . 79
5.1 Wavelength and conversion performance for instances with Uniform-
SP distribution of requests on 64 nodes ring network. (a) Number
of Wavelengths (b) Number of Conversions. . . . . . . . . . . . . 89
5.2 Wavelength and conversion performance for instances with Uniform-
LB distribution of requests on 64 nodes ring network. (a) Number
of Wavelengths (b) Number of Conversions. . . . . . . . . . . . . 90
5.3 Wavelength and conversion performance for instances with Gaussian-
LB distribution of requests on 64 nodes ring network. (a) Number
of Wavelengths (b) Number of Conversions. . . . . . . . . . . . . 91
6.1 A spider tree with branch node v. . . . . . . . . . . . . . . . . . . 93
6.2 A set of paths on the same partition using node v incident with
edges e and e ′ . There are m + n paths traversing the edge e, with
m paths ending at node v and n paths bypassing the node v to
edge e ′ . By Step 4, the edges e and e ′ have p · W and q · W paths,
for some integers p and q. Hence the number m of paths ending
at v is equal to (p − q) · W . . . . . . . . . . . . . . . . . . . . . . 96
6.3 A tree T with s(T ) = 3. . . . . . . . . . . . . . . . . . . . . . . . 97
7.1 A dual-fiber ring with four nodes and total of six wavelengths.
Each fiber has three wavelengths, where the first fiber has wavelengths
labeled w 0 , w 2 and w 4 , and the second fiber has wavelengths
labeled w 1 , w 3 and w 5 . The nodes 0 and 2 have fixed wavelength
conversion capabilities. Dotted lines are the wavelengths. At each
node wavelength shifts may occur between pairs of same wavelengths
from distinct fibers (dotted lines) or pairs of distinct wavelengths
attached by optical converters (solid lines). . . . . . . . . 104
9

LIST OF FIGURES
7.2 A dual-fiber ring with four nodes and total of six wavelengths.
Each fiber has three wavelengths, where the first fiber has wavelengths
labeled w 0 , w 2 and w 4 , and the second fiber has wavelengths
labeled w 1 , w 3 and w 5 . The node 1 has degree two wavelength conversion
capabilities. Dotted lines are the wavelengths. At each
node wavelength shifts may occur between pairs of same wavelengths
from distinct fibers (dotted lines) or pairs of distinct wavelengths
attached by optical converters (solid lines). . . . . . . . . 105
7.3 An n-node single-fiber ring network with four wavelengths labeled
w 0 , w 1 , . . . , w 3 , and degree two limited conversion capabilities at
node 0. At each node wavelength channels at the same wavelength
are attached, and in addition at the conversion node 0 the
wavelength pairs (w i , w i+1 ) and (w i+1 , w i ), for 0 ≤ i ≤ 3, and
i ≡ 0(mod 2), are attached. . . . . . . . . . . . . . . . . . . . . . 108
7.4 An n-node single-fiber ring network with four wavelengths labeled
w 0 , w 1 , . . . , w 3 , and degree two limited conversion capabilities at
node 0. At each node wavelength channels at the same wavelength
are attached, and in addition at the conversion node 0 the wavelength
pairs (w i , w i+1 ), 0 ≤ i ≤ 3, are attached. . . . . . . . . . . 108
7.5 A 7-node ring network with seven request paths, {{0, 3}, {3, 6},
{6, 2}, {2, 5}, {5, 1}, {1, 4}, {4, 0}}, which form a single-MCR. . . 110
7.6 A 5-node ring network with four request paths, {{2, 4}, {1, 4}, {4, 1}, {4, 2}}.
110
7.7 An FDR adaptation to k-fiber ring for k ≡ 1(mod 2), with nodes
0, 1, 2, 3 and 4 as the co-primary, primary, co-secondary, secondary
and tertiary node respectively. (a) A 3-fiber ring, and (b) a 5-fiber
ring, with w = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.8 An LDR adaptation to k-fiber ring for k ≡ 1(mod 2), with nodes
0, 1 and 2 as the primary, tertiary and secondary node respectively.
(a) A 3-fiber ring, and (b) a 5-fiber ring, with w = 2. . . . . . . . 116
8.1 An example of invalid closed circuit with nodes a < f < h < z. . . 127
10

LIST OF FIGURES
8.2 ADMs and Wavelength performance for instances with BL-Uniform
requests on 64-node path network (a) ADMs (b) Wavelengths. . . 140
8.3 ADMs and Wavelength performance for instances with UL-Uniform
distribution of requests on 64-node path network (a) ADMs (b)
Wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.4 ADMs performance for instances with Gaussian distribution of requests
on 64-node path network. . . . . . . . . . . . . . . . . . . . 142
9.1 A construction for the graph on p nodes that satisfies the grooming
ratio constraint of 4 and achieves the upper bound γ(4, p) on
number of edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2 Graphs isomorphic to K 4 − e where (a) a < b < d < c, (b) b < a <
c < d, and (c) b < c < d < a. . . . . . . . . . . . . . . . . . . . . . 153
9.3 Graphs isomorphic to K 2,3 where (a) {{b, d}{a, c, e}}, (b) {{a, e}{b, c, d}},
and (c) {{a, d}{b, c, e}}, for a < b < c < d < e. . . . . . . . . . . . 153
10.1 ADMs and Wavelength performance for instances with BL-Uniform
distribution of requests on 64-node ring network (a) ADMs (b)
Wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.2 ADMs and Wavelength performance for instances with UL-Uniform
distribution of requests on 64-node ring network (a) ADMs (b)
Wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
11

Dedication
To
loving mum Tufuwage and the late father Justo,
beautiful wife Rose and daughter I most cherish Catherine,
godfather Kassian and godmother Otilia.
12

Acknowledgements
I would like to thank the department of computer science at King’s college London
for her generosity to offer me a bursary grant for four years of this research study.
I owe lots of gratitude to Tomasz Radzik for his invaluable contributions and
guidance throughout the course of this research.
It is worthy thanking many others who gave inspiration and encouragement
through some contact or academic literature they had authored in line with my
research. There are a host of them but a few that had personally contacted and
thus highly indebted to include Ralf Klasing, Tomas Erlebach and Aris Pagourtzis
for their many useful comments. My fellow PhD student at King’s were so inspiring
in many instances and became close personal friends without whom would
have made far difficult to resist the pressure of discouragement inevitable during
the course of the PhD study. A few that are worthy mentioning include
Adam Wayner, Arjuna Sathiaseelan, Nadia Nicoulson, Sarah Stein and Manal
Mohamed.
Finally, many thanks to my wife Rose and daughter Catherine for all their
patience and love on all those hours that had to be away from them.
13

Glossary and mathematical
notations
0.1 Glossary
A1-g2 Grooming algorithm using Algorithms IV and VI
A2-g2 Grooming algorithm using Algorithm IV and MWM
ADM Add Drop Multiplexor
ADM-LB ADMs lower bound
APR1 Approximation algorithm for spider tree
APR2 Approximation algorithm for the tree T with s(T ) = 3
APX Approximation algorithm for Tree of rings
BFS Breadth-First Search
BL-Uniform Request instance type with bounded length uniform distribution
of requests
CSP Compatible subgraphs partition algorithm
DFS Depth-First Search
DRSP Demand routing and slotting problem
ECBM Edge coloring of bipartite multigraphs
EDA Edge decomposition grooming algorithm framework
EDA-LF EDA with longest forward arc first
EDA-SF EDA with shortest forward arc first
FCV Fat cover algorithm
FDR Fixed-conversion Dual-fiber Ring
FE-EDA Fixed edge with EDA algorithm
FPA Fixed pivot algorithm framework
FP-WC FPA with wavelength conversion framework
Gaussian-LB Routing instance type with Gaussian distribution of requests
routed using the load balancing routing decisions
14

0.1 Glossary
Gaussian-SP
Gbps
GHZ
I2P
IFPA
IFP-WC
ILP
IPA
IPA-LWM
IPA-MWM
IPA-SGM
LDR
LG
LPA
LP-DFS
LP-WC
LWM
MCC
MCR
maxFPA
min2P
minFPA
MWM
MSS
MVC
PCM
PIM
PIM-g2
PW
RAP
RG
RWA
Routing instance type with Gaussian distribution of requests
routed using the shortest path routing decisions
Gigabit per second
Gigahertz
Iterative min-load two phase algorithm
Iterative FPA
IFPA with wavelength conversion framework
Integer linear programming
Interval partition grooming algorithm framework
IPA with largest weight first matching algorithm
IPA with MWM algorithm
IPA with Simple greedy matching algorithm
Limited-conversion Dual-fiber Ring
First-fit coloring heuristic that considers requests in longest first
order
Long pivot algorithm
LPA with Depth First Search
LPA with wavelength conversion framework
Largest weight first matching algorithm
Multi Cycles of Channels
Multi Cycles of Routes
FPA with maximum load link as pivot link
min-load two phase algorithm
FPA with minimum load link as pivot link
Maximum weight matching
Minimum sufficient set
Minimum vertex cover
Path multi-colouring
Preprocessed iterative matching algorithm
PIM with MWM algorithm
Paired Wavelength
Register allocation problem
First-fit coloring heuristic that considers requests in arbitrary
order
Routing and wavelength allocation
15

0.2 Mathematical notations
SONET
SGM
SPR
TDM
UL-Uniform
Uniform-LB
Uniform-SP
WA
WDM
W-LB
Synchronous Optical Network
Simple greedy matching algorithm
Spiral algorithm
Time Division Multiplexing
Request instance type with unbounded length uniform distribution
of requests
Routing instance type with uniform distribution of requests
routed using the load balancing routing decisions
Routing instance type with uniform distribution of requests
routed using the shortest path routing decisions
Wavelength allocation
Wavelength Division Multiplexing
Wavelength lower bound
0.2 Mathematical notations
A(P n , g)
D = (V, A)
g
G = (V, E)
I
K 4 − e
K n
K n1 ,n 2
L
n
N
P n
R
s(T )
T R
W
Number of ADMs needed to satisfy the all-to-all instance on P n
with grooming ratio g
Digraph D with node set V and arc set A
Grooming ratio
Graph G with node set V and edge set E
Request instance
A graph obtained by removing one edge from complete graph
with four nodes
Complete graph with n nodes
Complete bipartite graph with n 1 nodes in one node partition
and n 2 nodes in the other
Network load
Number of nodes on a network
Network
Path network on n nodes
Routing instance
Maximum number of nodes with degree 3 or more on any path
of the tree T
Tree of rings
Total number of wavelengths
16

Chapter 1
Introduction
In the past decade we have seen a dramatic improvement of efficiency for communication
networks due to variants of fast electronic and optical transmission
networks being deployed. Nevertheless, with the fast pace of growth in Internet
and related applications a huge demand of bandwidth has resulted from new
multimedia applications amongst others, thus an important challenge to network
practitioners is to seek solution for provisioning to such ever growing bandwidth
demands. Fiber optics with WDM transmission technology, in which several signals
may be transmitted concurrently on same link under different carrier frequencies/wavelengths,
promise to cater for such huge bandwidth need, however, the
technology still faces a challenge to exploit the potentially enormous bandwidth
in efficient way. The WDM technology transmits many signals concurrently on an
optical fiber using wavelength routing - a network switching node routes signals
based on their wavelengths. In the near future we will likely see optical networks
being widely deployed in bid to benefit from its several important advantages,
including an increased usable bandwidth on optical fiber, reduced electronic processing
cost, protocol transparency and efficient network component (node/link)
failure handling [Gurusamy and Siva Ram Murthy (2002)].
A WDM Optical network consists of a set of nodes and optical fiber links
such that an optical fiber link connects a pair of nodes. A pair of network nodes
communicate by exchanging messages over a connection path (or simply path),
through a set of optical fiber links. An optical fiber supports a fixed number
of wavelengths. Wavelength routing is used to establish a path, in turn the
17

path is allocated a wavelength. A message to be sent from one node to another
is referred to as a connection request (or simply a request). In WDM optical
networks a request is satisfied by establishing a path between communicating
nodes and allocating a wavelength to the path. Since an optical fiber supports a
fixed number of wavelengths, it is possible that a wavelength may not be available
to satisfy a given request, in which case it is blocked. Transmission for a blocked
request is deferred to the future when a wavelength is made available through a
previously allocated connection request completing its transmission.
It is therefore a crucial issue in WDM optical networks to optimize the use
of the available wavelengths, supported on the optical fibers. In such type of
networks several connection requests can be honored concurrently on same fiber
using the WDM technique, where more than one requests can be transmitted
through the same optical fiber but on distinct wavelengths. WDM optical networks
can be broadly categorized into the following two classes.
• All-optical transmission (transparent) network: each connection request
must be assigned the same wavelength in every fiber on the allocated path
between the transmitter node and the receiver node (i.e., wavelength continuity
constraint must be satisfied).
• Optical transmission (opaque) network: a connection request may be assigned
distinct wavelengths in different optical fibers on the allocated path
between the transmitter node and the receiver node (i.e., no wavelength
continuity constraint)
In opaque optical networks, request transmission is analogous to the standard
store-and-forward transmission, in which a request is converted into electronic
format for storage at each intermediate node before it can be retransmitted in
optical form to the next node. For the case of transparent optical network, a
request is only converted into electronic form at its receiving (destination) node.
This allows for potentially much higher transmission throughput than in the
opaque counterparts as there is no electronic bottleneck, the overhead due to the
opto-electronic conversion at intermediate nodes. Nevertheless, in such networks
the problem of wavelength allocation is NP -hard in general [Gargano and Vaccaro
(Feb. 2000)].
18

In [Yang and Ramamurthy (2002, Nov. 2001)] they propose a translucent
optical network as a hybrid to the two broad categories which partially relaxes
the wavelength continuity constraint. In a translucent optical network a request
uses same wavelength in arbitrarily many fiber links it traverses before conversion
to electronic form, and possibly retransmitted in a different wavelength (if
the converting node is not its destination). Such networks are motivated by
an inherent signal impairment in long-distance wavelength transmission which
may necessitate electronic signal regeneration. Several interesting traffic modelling
problems arises in translucent optical networks, including determining the
percentage number of regenerating nodes (sparse regeneration), which result in
maximum performance efficiency (in terms of request blocking) of the network.
Experimental methods are used by [Yang and Ramamurthy (Nov. 2001)] to
demonstrate the required number of regenerating nodes, and was observed that
using just a small fraction of regenerators (at most 20% of the network nodes)
may lead to significant performance gain. Intuitively, whereas in an opaque network
a request is regenerated at every intermediate node, translucent networks
use sparse regeneration in which a request may be regenerated in rather fewer
designated intermediate nodes.
An optical converter is an optical device that is capable of shifting a wavelength
from one wavelength to another optically. Converters have been proposed
as an alternative solution around the bandwidth utilization inefficiency resulting
from the wavelength continuity constraints in all-optical networks. A routing
node equipped with an optical converter can change a wavelength of a request
optically without need to opto-electronic conversion. Optical converters can be
broadly classified into two categories: optical converters with full conversion capabilities
(full converters) that can change an input wavelength to any other
wavelength, and optical converters with partial conversion capabilities (limited
converters) that can change an input wavelength to only a few fixed wavelengths.
A degree of an optical converter is the maximum number of output wavelength
that any input wavelength can be shifted. In wavelength convertible all-optical
networks, if all nodes are equipped with full optical converters, then the wavelength
allocation problem is rendered simple as in the opaque case, moreover,
without the electronic bottleneck. Unfortunately, optical converters are rather
19

expensive devices, especially if they are to provide high degree of wavelength
conversion. Thus their deployment should be minimized, that is, we should seek
to deploy small number of optical converters and with low degree of conversion
possible.
Recent advances in optical fiber technology have lead to an enormous potential
bandwidth in current state-of-art optic-fibers, which are capable of supporting
tens to hundreds of wavelengths, in turn, each supports huge bandwidth capacity
(in the order of 10 - 40 Gbps), while a great deal of network traffic remain relatively
at low rate. It is imperative to combine (groom) several low rate requests
into one high bandwidth wavelength to efficiently use the available wavelength
bandwidth and enhance network connectivity. When more than one requests are
combined to share wavelength bandwidth a device called Add-Drop Multiplexor
(ADM) is used for opto-electronic conversion at the node where traffic is added or
dropped from the wavelength. At such node an ADM may be shared by requests
being added or dropped from the same wavelength. At each node where ADMs
are used, the node incur the opto-electronic conversion cost for processing the
dropped or added requests at that node, since the current technology does not
support processing requests in optical format. Thus we need to minimize the
number of ADMs used by the network.
In this thesis, we consider the following model for efficient utilization of wavelength
bandwidth. We are given a WDM optical network and a set (possibly a
multiset) of requests as source-destination node pairs, the goal is to determine
paths to route each request (routing problem) and allocate a wavelength to a path
such that paths sharing a network link get distinct wavelengths (wavelength allocation
problem). The routing problem is solved using known routing algorithms.
For the case of all-optical networks without conversion and traffic grooming, a
path is allocated to same wavelength at each link, the goal is to minimize number
of wavelengths used. For the case of all-optical networks with conversion capabilities
but no traffic grooming a path may be allocated distinct wavelengths at
different links. Two categories are considered, a wavelength convertible network
where each node is equipped with wavelength conversion capabilities, the goal is
to minimize number of wavelengths used and number of wavelength conversions
performed, and wavelength convertible networks in which some nodes may have
20

1.1 General thesis contribution
wavelength conversion capabilities, the goal is to minimize number of nodes with
such conversion capabilities and the complexity (degree) of optical converters
used. For the case of networks with traffic grooming, several paths may be allocated
to one wavelength, the goal is to minimize number ADMs and wavelengths
used.
1.1 General thesis contribution
We study the wavelength allocation problem in all-optical networks with different
constraints. We propose new heuristics for wavelength allocation in ring networks
without wavelength conversion, and adapt them to tree of rings networks without
wavelength conversion and ring networks with cluster wavelength conversion
capabilities. We consider ring networks that allow multiple fibers per link with
limited optical converters of degree 1 and 2 respectively and show optimal utilization
of wavelength for such ring networks.
We study tree networks that allow multiple fiber per link to satisfy a given
request instance for wavelength allocation, and the goal is to minimize the overall
number of optical fibers used in the network. We give different approximation
algorithms for different tree network constraints. We consider path and ring
networks that allow combining several requests into a wavelength. We present
heuristics for grooming arbitrary instances with unit rate requests for path and
ring networks, and grooming results for path networks with all-to-all instances
on unit rate requests and grooming ratio 3 that imply an asymptotic ratio 1.07.
1.2 Thesis Outline
In Chapter 2 we define some graph theoretic concepts and different problem models
for routing and wavelength allocation in optical networks, and give a review
to previous work. We present wavelength assignment heuristics for ring networks
without wavelength conversion capabilities in Chapter 3. In Chapter 4 and 5 we
extend the latter heuristics to tree of rings without wavelength conversion capabilities
and ring networks with wavelength conversion capabilities. We study the
optimization problem of minimizing the total number of optical fibers sufficient
21

1.2 Thesis Outline
to satisfy a given instance for tree networks that allow multiple optical fibers per
link in Chapter 6. In Chapter 7 we present ring networks that allow multiple fiber
per link and limited wavelength conversion to satisfy any instance using optimal
number of wavelengths. We study path and ring networks that allow combining
several requests into a wavelength in Chapters 8, 9 and 10. We give conclusion
and future work in Chapter 11. In Appendix A we report additional numerical
results for the different heuristics considered in this thesis.
22

Chapter 2
Terminology, models and
background work
Within the last decade optical networks have received a considerable research
attention to which of recent several other research directions have emerged. In this
chapter we define some graph theoretic concepts and other terminologies related
to optical networks. We also describe different problems on optical networks,
review of previous work and our contributions.
2.1 Graph-theoretic concepts
An undirected graph G = (V, E) consists of a finite set V of vertices (nodes) and
set E of edges where the edge {u, v} ∈ E is incident to a pair of nodes u and
v of G. The degree of the vertex v is the number of edges that are adjacent to
v. The degree of the graph G denoted by ∆(G) is equal to the maximum degree
of its nodes. A directed graph D = (V, A) (digraph) consists of a finite set V
of nodes and set A of arcs such that arc (u, v) ∈ A is incident with the pair of
nodes u and v of D. The arc (u, v) is said to be incident from u, but incident to
v. The in-degree of the node v is the number of arcs incident to v. Analogously,
the out-degree of the node u is the number of arcs incident from u. A node of D
is a sink node if its out-degree is 0 (i.e., has no outgoing arcs but one or more
incoming arcs), and is a source node if its in-degree is 0 (i.e., has no incoming arcs
23

2.2 Optical network models, routing and wavelength allocation
but one or more outgoing arcs). An edge can be modeled by a pair of oppositely
directed arcs.
An undirected path of G (path) is a sequence x 1 , x 2 , . . . , x p , of nodes where
{x 1 , x 2 }, {x 2 , x 3 }, . . . , {x p−1 , x p } are edges of G and the nodes x i , i ∈ {1, . . . , p},
are distinct. A directed path of D (dipath) is a sequence x 1 , x 2 , . . . , x p , of nodes
where (x 1 , x 2 ), (x 2 , x 3 ), . . . , (x p−1 , x p ) are arcs of D and the nodes x i , i ∈ {1, . . . , p},
are distinct. A path or dipath is also a simple path or dipath. A closed path or
dipath is a cycle. A graph G is connected if there is a path between any pair of
vertices of G. A digraph D is connected if its underlying graph is connected (the
underlying graph for D is an undirected graph obtained by replacing the arcs of
D with edges).
Given a graph G. A subset M of edges of G is called a matching of G if
the edges in M are pairwise non-adjacent. Given a matching M. A node of G is
matched if it is incident to an edge of M, otherwise, it is unmatched. A matching
M is a maximum matching if it contains the maximum possible number of edges
for any matching of G (there are possibly several such maximum matching). A
graph G is a weighted graph if each edge of G is associated with some real number
as weight. A maximum weight matching M of a weighted graph G is a matching
with maximum possible sum of edge weights for any matching of G.
2.2 Optical network models, routing and wavelength
allocation
In this section we describe the optical network model used in this thesis and
define the routing and wavelength allocation problems in relation to this optical
network model.
2.2.1 Optical networks
An optical network N is modeled by a graph (digraph) such that the nodes represent
communication points in N, and edges (arcs) represent bidirectional (unidirectional)
optical links in N. A communication point may be passive, where
24

2.2 Optical network models, routing and wavelength allocation
it is capable of sending and receiving messages only, or active, where it is capable
of sending, receiving and redirecting (switching) messages. A switching node
provides the ability to switch (direct) any input signal from some input fiberlinks
(port) to any one of the output ports in optical form or to drop a signal.
Two switching models have been proposed; generalized and elementary switches
[Raghavan and Upfal (1994)]. A generalized switching node, is capable of directing
different signals entering the node through an input link to different output
links of the node, and an elementary switching node on the other hand, cannot
direct different signals of an input link entering into the node along different outgoing
fibers. In particular, an elementary switching node is capable of directing
the signals coming along each of its input links to one or more of the output links,
however, cannot differentiate between the different wavelengths coming along the
same link. The generalized switch is the most realistic model favourable for WDM
optical networks. Furthermore, technical results obtained under the generalized
switch model can easily be adapted to the elementary switch model. In this thesis
we consider optical networks modeled by the generalized switching nodes, where
a network link is realized using one or more optical fibers. A single-fiber network
allows one optical fiber per link and a multi-fiber network allows one or more
optical fibers per link. A k-fiber network is a variant of multi-fiber network in
which each link is realized using k optical fibers, for k > 1.
A request of the network N is given as pair of nodes corresponding to a message
to be sent from one node to another. An unordered request {u, v} refers to a
message to be sent from node u to v, or viceversa. That is, any node can be a
transmitter or a receiver node. An ordered request (u, v) refers to a message to be
sent from node u to node v. That is, the node u is the transmitter and the node
v is the receiver. An instance I of N is a set of requests in N (possibly a multiset)
which is to be satisfied. Different constrains may be used to model the instance I
leading to different communication patterns on N. In most practical applications
it is sufficient to model communication patterns using random distributions, for
example Uniform or Gaussian distributions. An example of special instances is
the symmetric communication pattern where the instance I consists of directed
requests such that there is request (v, u) ∈ I for each request (u, v) ∈ I. Such
instances model communication services that require bidirectional reservation of
25

2.2 Optical network models, routing and wavelength allocation
wavelengths. Others include the all-to-all, one-to-all, all-to-one, one-to-many and
permutation communication [Klasing (August, 1998)].
2.2.2 Routing and wavelength allocation
In optical communication we are given a set of connection requests as transmitterreceiver
node pairs and the goal is to establish an optical channel between each
transmitter-receiver node pair. An optical channel consists of a path between the
transmitter-receiver node pair and a wavelength to be allocated to each path. A
routing R for an instance I for network N is a set of paths {p{u, v} : {u, v} ∈ I}
or dipaths {p(u, v) : (u, v) ∈ I} such that each path (dipath) in R consists of a
sequence of links of N that the request in I corresponding to this path traverses,
that is between the request transmitter and receiver nodes. Wavelength allocation
(WA) allocates wavelength to paths using distinct wavelength for paths sharing
a link of N. The term path (dipath) and request can be used interchangeably.
• The load L of network N is the maximum number of paths sharing any link
of N.
• Let I be an instance in N. The routing problem for (N, I) consists of finding
a set of paths R for the instance I in N. Different routing decision objectives
will be described in Section 2.4.2.
• Let R be a routing for I in N. The wavelength allocation problem for (N, I, R)
consists of finding an assignment of wavelength to each path in R, where
no two paths sharing any link of N are assigned the same wavelength. The
goal is to use minimum possible number of wavelengths.
To satisfy request instances the routing and wavelength allocation problem
may be considered independently or as combined problem - routing and wavelength
allocation (RWA).
• Let I be an instance in N. The routing and wavelength allocation problem for
(N, I) consists of finding a routing R for the instance I and the assignment
of wavelength to each path in R, such that no two paths sharing any link
26

2.2 Optical network models, routing and wavelength allocation
of N are assigned with the same wavelength. The goal is to use minimum
possible number of wavelengths.
The network load L is the natural lower bound for the number of wavelengths
needed to satisfy an instance, since at any link traversed by L requests each should
be assigned with distinct wavelength. For networks where there is a unique path
between any pair of nodes (such as tree networks), routing is not needed. But for
networks where a node pair may have alternate paths an efficient optimal routing
scheme may exist or an approximate solution be used.
Two natural optimization problems can be considered for the wavelength allocation
problem.
• Minimize number of wavelengths needed to satisfy a given instance.
• Maximize number of requests satisfied given a fixed number of wavelengths.
In this thesis we consider the problem of minimizing the number of wavelengths
needed to satisfy a given instance (i.e., requests are given a priori). Given
a set of requests (as node pairs), the instance is routed using some known routing
algorithms. Our goal is to allocate wavelengths to paths using minimum
number of wavelengths. The wavelength allocation problem is NP -hard for general
networks including ring networks [Erlebach and Jansen (1997a,b)]. Thus the
standard approach is to consider specific network topologies.
2.2.2.1 RWA with wavelength continuity constraint
The routing and wavelength allocation problem for optical networks may be subject
to various constraints to efficiently use the available wavelength bandwidth.
For all-optical networks without wavelength conversion the RWA problem is constrained
by the wavelength continuity requirement, where each request should be
assigned with the same wavelength at all links it traverses.
For single-fiber all-optical networks the WA problem is related to the well
known vertex colouring problem of graphs. Given a single-fiber all-optical network
N, an instance I and a routing R for I in N. The conflict graph associated with
(N, I, R), denoted by cfl(N, I, R), is the graph G(V, E) with a vertex in V for
27

2.2 Optical network models, routing and wavelength allocation
every request in I, where a pair of vertices in V are adjacent if and only if
corresponding pair of requests share a link of N. The WA problem for (N, I, R) is
equivalent to vertex colouring problem for the conflict graph cfl(N, I, R). Given
⋃c
the graph G(V, E), a proper c-colouring is a partition of the vertices V = ˙
i=1 V i
such that no pair of vertices in V i is adjacent. The vertices in V i are assigned colour
i, hence adjacent vertices receive different colours. The vertex colouring problem
seek to find χ(G) (chromatic number of G), the smallest possible c for which
there exists a proper c-colouring of G. We use the term wavelength allocation
and colouring interchangeably.
Multi-fiber networks provide an alternative approach for efficiently using the
network bandwidth since more wavelengths are made available by the multiple
occurrence of the same wavelength at links, which also reduces the potential of
requests being blocked. For multi-fiber all-optical networks the WA problem is
related to the problem of colouring hypergraphs - graphs whose edges (hyper-edges)
are adjacent to two or more vertices [Ferreira et al. (May 2002)].
2.2.2.2 RWA with wavelength conversion
The restriction caused by the wavelength continuity constraint on all-optical networks
could be relaxed if switching nodes were equipped with wavelength conversion
capabilities. Allowing wavelength conversion at a switching node may
dramatically minimize wavelength requirement, since wavelength assignment conflict
could be resolved by changing wavelength of some requests at intermediate
switching nodes rather than blocking the requests. In particular, if there is no
restriction on the wavelengths on which a request can be transmitted then it is
possible to satisfy any set of requests optimally (i.e., using L wavelengths for an
instance with load L). A switching node that allows changing a signal wavelength
without need for opto-electronic conversion have been proposed [Gurusamy and
Siva Ram Murthy (2002)]. Such switches employ optical converters which are
devices that can shift wavelength from one to another optically.
An optical converter is modeled by a bipartite graph, a graph whose vertices
can be partitioned into two sets such that only pairs of nodes from distinct partition
sets are adjacent. One of the vertex class represent input wavelengths and
28

2.2 Optical network models, routing and wavelength allocation
the other class represent output wavelengths, and edges between vertex pairs
are the possible wavelength shifts. The degree of a bipartite graph models the
the degree (complexity) of optical converter, where a complete bipartite graph
models a full optical converter, and partial (limited) optical converter, otherwise.
A limited converter with degree 1 is a fixed converter. Figure 2.1, shows fixed,
degree 2 and full converter respectively.
1
1
1
1
2
2
2
2
3
3
3
3
4
4
(i) Fixed converter (ii) Degree 2 converter (iii) Full converter
4
4
Figure 2.1: Fixed, degree 2, and full converters each with four inputs and outputs.
Optical converters are expensive devices especially if the degree of conversion
is relatively high, hence a motivation to two prevalent approaches to the study
of all-optical networks with wavelength conversion capabilities.
• Sparse convertible optical networks.
• Limited convertible optical networks.
In sparse convertible optical networks a subset of switching nodes are equipped
with full wavelength conversion capabilities and the goal is to minimize the number
of such switching nodes. That is, sparse conversion seek to achieve optimal
use of the available wavelength by deploying a minimum number possible of full
optical converter at some switching nodes. This problem can be modeled by
minimum sufficient set (MSS) problem. Let the network N be modeled by the
graph G = (V, E). The MSS problem seek to find a sufficient set S ⊆ V for G of
minimum cardinality, such that any instance of N can be satisfied using L colours
if each node of S is equipped with full wavelength conversion capabilities in N.
In limited convertible optical networks, on the other hand, a subset of switching
node is equipped with partial optical converters to achieve near optimal or
optimal utilization of the network bandwidth.
29

2.2 Optical network models, routing and wavelength allocation
2.2.2.3 RWA with multi-granularity wavelengths
So far we have assumed that each request uses the full bandwidth of a wavelength.
This assumption is not realistic for most network traffic since it became apparent
that current state-of-art optic-fibers are capable of supporting tens to hundreds of
wavelengths in turn each supports enormous bandwidth capacity (in the order of
10 - 40 Gbps), while a great deal of network traffic remain relatively at low rate.
For networks with multi-granularity wavelengths several low rate requests can be
combined to share a wavelength bandwidth, thus efficiently using the available
bandwidth and enhance the network connectivity. In such networks a device
called Add-Drop Multiplexors (ADMs) is needed at each node where traffic is
added or dropped from shared wavelength for opto-electronic conversion. At such
node an ADM device can be shared by several traffic on same wavelength. ADMs
devices are costly to use due to opto-electronic conversion cost. For each used
wavelength on the network an ADM is needed at least at transmitter and receiver
node respectively. Therefore, at least two ADMs are needed for each established
path and wavelength, that is, to add traffic (Multiplex) into the wavelength at
transmitter node and to drop traffic (de-Multiplex) from the wavelength at the
receiver node. The number of ADMs requirement should be minimized to improve
efficiency on networks with multi-granularity wavelengths.
Traffic grooming problem seek to combine several low rate requests into a high
bandwidth wavelength with the goal to minimize number of ADMs and wavelengths
used to satisfy an instance. In traffic grooming we assume that wavelengths
are readily available since several connection requests can share wavelength
bandwidth, therefore minimizing number of ADMs is considered as the
primary objective. We consider multi-granularity wavelength networks with unit
rate requests. The grooming ratio g for N is the maximum number of requests
that can share a wavelength of N. That is, given requests of rate c and wavelength
bandwidth C, g = ⌈ C c ⌉. 30

2.3 Special Network topologies
2.3 Special Network topologies
A network is modeled by a connected graph (digraph), where a node corresponds
to a network switch/communication point and an edge (arc) corresponds to a
network link connecting the pair of adjacent switches/communication points. A
network link is said to be bidirectional if it is capable of transmitting signals in
either direction, otherwise, it is unidirectional, i.e., capable of transmitting signals
only in one direction. An edge models a bidirectional network link whereas an
arc models a unidirectional link. Practical networks are commonly modeled of
the topologies circle (ring), chain (path), star, mesh, and tree.
Definition 1 An undirected path network P n with n nodes has a node set V =
{i : 0 ≤ i ≤ n − 1} and edge set E = {{i, i + 1} : 0 ≤ i ≤ n − 2}. A directed path
network with n nodes has the same node set as the undirected path but with arc
set A = {(i, i + 1) : 0 ≤ i ≤ n − 2}.
Definition 2 An undirected ring network with n nodes has a node set V = {i :
0 ≤ i ≤ n−1} and edge set E = {{i, i+1 mod n} : 0 ≤ i ≤ n−1}. A directed ring
network with n nodes (unidirectional ring) has same node set as the undirected
ring but with arc set A = {(i, i + 1 mod n) : 0 ≤ i ≤ n − 1}. The order of the
nodes and edges of a ring in the direction of increasing node identifiers is referred
to as the clockwise orientation.
Definition 3 A tree network is modeled by a connected graph with no cycles. A
set of disconnected tree components is called a forest. Let T be a tree. A node
of T is called a centre node if its removal yield a forest such that the number of
nodes on each connected component of the forest is at most half the nodes in T .
A tree has one or two centre nodes.
Definition 4 A tree is a spider tree if at most one node has degree greater than
2. Analogously, a tree is a star if at most one node has degree greater than 1.
Definition 5 A k-ary tree is a rooted tree in which every internal vertex has k
children. An 1-ary tree is just a path. A 2-ary tree is also called a binary tree.
31

2.3 Special Network topologies
Definition 6 An n = k d nodes d-dimensional mesh M d , has a set of nodes
{ā = (a 1 , . . . , a i , . . . , a d ) : 1 ≤ a i ≤ k, i = 1, . . . , d}. The edges are {(a 1 , . . . , a i , . . . , a d ),
(a 1 , . . . , a i + 1, . . . , a d )}, such that, 1 ≤ a i ≤ k − 1.
Figures 2.2 show some commonly used network topologies.
(i) 5−nodes path
(ii) 7−nodes ring
(ii) 7−nodes star
(iv) 16−nodes mesh
Figure 2.2: Path, ring, star and mesh networks.
Definition 7 A tree of rings network is modeled by a graph that can be constructed
by starting with a ring and then repeatedly adding a new disjoint ring to
the graph and identifying a node on the new ring with a node of the existing tree
of rings.
A tree of rings is a natural extension of a ring network. Figure 2.3 shows a tree
of rings network.
Figure 2.3:
A tree of rings with seven rings.
32

2.4 Experimental performance comparison
2.4 Experimental performance comparison
Numerical results are used to compare the performance of most heuristics and
algorithms considered in this thesis. We compare experimentally the algorithms
for path, ring and tree of rings networks. In this section we describe request
instances types and routing decisions used for our experiments. Also we describe
our implementation platform.
2.4.1 Request instances
We consider networks with arbitrary instances of random requests of Uniform
and Gaussian distributions. The end-nodes of a request of Uniform distribution
are chosen uniformly at random from among the network nodes. Thus requests
are less likely biased. Such requests best reflect the patterns of requests on stand
alone networks. For path and unidirectional ring networks two sub-type of uniform
requests is considered; bounded length uniform (BL-Uniform) and unbounded
length uniform (UL-Uniform or simply Uniform) distribution of requests. The
end-nodes of a UL-Uniform request are chosen just as for a Uniform request, but
the end-nodes of a BL-Uniform request are chosen as follows.
The request length l is first chosen uniformly at random from a fixed integer
range (with a default minimum of 1 and maximum of half the number of network
links). The first end-node i is selected uniformly at random. A coin is tossed
to decide whether the other end-node j is selected to the left or right (counterclockwise
or clockwise) of node i to obtain a request of length l. Note that such
node j may not always exists on both sides of i for path networks, but for any
given l and i at least one side has such j. Thus for path networks, if such j is
only in one side of i no need to toss a coin, but simply select j from the side that
the node exists. Fixing the length l for all instance requests give a variant of BLuniform
distribution of requests we refer to as fixed length uniform distribution
of requests (FL-Uniform). BL-Uniform requests reflect the request patterns for
networks where requests are localized (span short range) uniformly at random.
For path and ring networks the end-nodes of a Gaussian request are chosen as
follows. A network node is fixed about which the request end-nodes are selected.
The normal distribution random variate is used to model a Gaussian request
33

2.4 Experimental performance comparison
where the mean of the distribution models the fixed node about which end-nodes
are selected. A normal variate models the distance of the next selected end-node
from the fixed node. For ring networks the median node is chosen uniformly at
random, whereas for path networks with n nodes the median node is ⌊ n ⌋. A coin
2
is tossed to select a node from one of the two sides (left or right of the median node
for path network; and clockwise or counter-clockwise direction about the median
node for rings) about the fixed node. The deviation of the normal distribution
models the spread of an end-node about the fixed node. By default, the deviation
parameter is set to the value 2. Gaussian instances best reflect request patterns
on a network where most requests are localized. Using a single fixed node implies
that most requests are localized around this fixed node.
For tree of rings T R the end-nodes of a Gaussian requests are chosen as
follows. A distance between a pair of rings on T R is the distance between the
corresponding pair of nodes in the underlying tree of T R. As for ring networks,
each ring of T R has a fixed node about which a request end-node is selected. The
normal variate is used to model the distance between the pair of rings of T R (is
the same ring if the normal variate value is 0) from which the request end-nodes
are selected. A pair of rings at this distance is chosen uniformly at random from
T R. Within a ring of T R the request end-node is selected as for the ring network.
All-to-all instances are used in path networks. For a path with n nodes labeled
from {0 ≤ i ≤ n − 1} the all-to-all instance has n(n−1)
2
requests, where there is
one request for every distinct pair of node. All-to-all instances can be viewed as
edges of the complete graph with n nodes.
2.4.2 Routing decisions
A routing R for instance I on a network N induces a load over each network
link and the maximum load L is the natural lower bound for the number of
wavelengths needed to satisfy the set I. Note that for path networks a request
is routed in a unique path. Different routing approaches have been proposed
in [Cosares and Saniee (1994); Schrijver et al. (February 1998); Stefanakos and
Erlebach (2004)] for ring networks. The general decision version for ring loading
(routing) problem is known to be NP-complete [Schrijver et al. (February 1998)].
34

2.4 Experimental performance comparison
A routing that minimizes the load L seem to be an obvious consideration, however,
[Stefanakos and Erlebach (2004)] show that it may not always be a good
choice, instead they propose minimizing the clique size (the largest complete subgraph)
of the request interference graph (conflict graph). For undirected rings the
study in [Stefanakos and Erlebach (2004)] showed that a routing that minimizes
load or clique size lead to the same performance of wavelength assignment. We
use two routing approaches: shortest path and greedy load balancing routing for
routing decisions on undirected rings and tree of rings.
• Shortest path (SP)
In shortest path routing a network link is associated with nonnegative
weight. A request is routed such that the sum of weight on network links
it traverses is minimum possible. Intuitively, always selects a route that
traverses few links.
• Greedy Load balancing (LB)
In greedy load balancing each network link maintains the number of previously
routed requests that traverse it called link load. The first request is
routed using shortest path routing decision. Subsequent requests are routed
such that the maximum link load induced by requests thus far considered
is minimum possible. Let {i, j} be the next request to be considered by
load balancing routing. The load balancing routing compute the maximum
link load induced by previously routed requests and the request {i, j} when
routed in clockwise (i.e. as (i, j)) and counter-clockwise (i.e., as (j, i)) direction
respectively on the ring. A route for {i, j} that minimizes the maximum
link load is chosen. Break ties by selecting a route that uses few number of
links.
For tree of rings the load balancing routing decisions are performed by
considering one ring at time for rings that the request traverses.
Using different ordering of input requests the load balancing may result into
different load being induced on the network. In a 5 node ring network with
nodes numbered 0, 1, . . . , 4 and given request instance {{4, 0}, {0, 3}, {3, 4}},
for example, the load balancing with input ordering {4, 0}, {0, 3}, {3, 4} and
35

2.4 Experimental performance comparison
{0, 3}, {3, 4}, {4, 0} induce network load 1 and 2 respectively. In this thesis
the load balancing routing decision consider requests in the given order of
a request instance. The load balancing routing, because of the greedy way
it is computed may lead to the higher network load than the shortest path
routing. This often happens for the uniform distribution of request, but
rarely for the Gaussian distribution.
We denote by Uniform-SP and Uniform-LB the routing instances with uniform
distribution of requests routed by the shortest path and the load balancing routing
decisions. Analogously, Gaussian-SP and Gaussian-LB denote the routing
instances with Gaussian distribution of requests and routed by the shortest path
and the load balancing.
2.4.3 Implementation platform
We implemented the different algorithms and instance generators using LEDA
4.5 platform [Mehlhorn and Näher (1999)] under GNU C + + (GCC) compiler
version 3.3. We used a 1.0GHZ AMD Duron Processor with 112.0MB RAM and
Linux operating system (Neilix - 2002 version) that was freely distributed by the
IT support group at the department of computer science King’s college London.
Random integers with uniform distribution were generated using the random
source implementation of the LEDA platform [Mehlhorn and Näher (1999)] where
the stream of integers generated by a random source is only pseudo-random. It
is generated from a seed supplied automatically from the internal clock. Distributions
of Gaussian deviates, on the other hand, were generated using an implementation
described in [Press et al. (1992)] (see pages 288-290), that generates
normally distributed deviates with zero mean and unit variance using the random
source and Box-Muller transformation.
During an experiment a fixed network and instance size is given and ten
random instances of given distribution are generated and the average for each
performance parameter is determine, i.e., average number of wavelengths, ADMs
and conversions respectively. The choice of ten random instances is based on the
experience obtained after several test of random input samples, where an average
of ten random instance samples gave performance results that was close to each
36

2.5 Previous work
individual instance sample. The source code for our experiments is posted at
http://www.dcs.kcl.ac.uk/staff/radzik/PhDstudents/Justo/thesisweb.html, where
you can download and experiment yourself.
2.5 Previous work
Several previous studies considered all-optical networks with single-fiber model,
and proposed different heuristics and algorithms for the routing and wavelength
allocation problem. The routing and wavelength allocation problem is optimally
solved for path networks using the interval graph algorithm in [Golumbic (1980)].
Undirected ring, tree, tree of rings and mesh networks are studied in [Raghavan
and Upfal (1994)] giving a 2-, 3/2-, 3- and poly-logarithmic approximation algorithm
respectively for the RWA problem. The 2-approximation RWA algorithm
for ring networks transforms the instance of a ring network into that of a path
network by cut-one-link approach (by routing connection requests such that no
request traverses the cut link), and use the algorithm in [Golumbic (1980)] to the
resulting instance of a path network. The best known algorithm for the RWA
problem for rings is due to [Kumar (1998)] with a (1.5+1/2e)+o(1)-approximation
asymptotic factor that uses integer programming formulation with randomized
rounding techniques. In [Cheng (2004)] they present the best known deterministic
algorithm for the RWA problem for rings improving the 2-approximation
algorithm in [Raghavan and Upfal (1994)] to a 2 − 1
⌈ L ⌉-approximation
algorithm.
2
They also show how to combine their algorithm with that of [Kumar (1998)] to
obtain a randomized algorithm with a constant factor of 2 − 1 . For the WA
θ(log n)
problem a 1.5-approximation algorithm is presented by [Karapetian (1980)] for
ring networks where request paths are viewed as a family of circular arcs on a
circle. In [Kumar (1998)] they present the best known WA algorithm for rings
with a (1 + 1/e) + o(1) ∼ 1.37 + o(1)-approximation asymptotic bound, using
multicommodity flow and randomized rounding techniques.
The wavelength allocation problem for tree networks is solved in [Raghavan
and Upfal (1994)] by translating a WA instance on the tree network into an
edge colouring instance for multi-graph ( a graph in which a pair of node may be
adjacent to more than one edge). Colouring the edges of the multi-graph using the
37

2.5 Previous work
edge colouring algorithm in [Shannon (1949)] followed by a recolouring technique
lead to a 3/2-approximation algorithm for the original WA instance of the tree
network. Using the edge colouring algorithm in [Nishizeki and Kashiwagi (August
1990)], the latter algorithm is improved in [Mihail et al. (1995)] with asymptotic
9/8-approximation factor.
Both the RWA and WA problems on rings (thus on tree of rings) are NP-hard
[Erlebach and Jansen (1997a); Garey et al. (1980)]. For tree of rings networks
[Raghavan and Upfal (1994)] present a 3-approximation algorithm for the RWA
problem that transform an instance for tree of ring into an instance for tree network
by cut-one-link approach. That is, one link is removed from each ring in
the tree of rings (re-routing any request traversing this link such that all requests
avoid traversing the cut-link), to obtain a tree network, and the solution for the
WA problem on the tree is used as the solution of the RWA problem on the tree of
rings. Note that the latter approach does not work for the WA problem in tree of
rings since request paths are given a-priori and cannot be re-routed. In [Erlebach
(Springer-Verlag, 2001)] they present a 4-approximation algorithm for wavelength
allocation in arbitrary tree of rings networks that uses at most 4L wavelengths.
The best known algorithm for the wavelength allocation problem in arbitrary
tree of rings networks uses at most 3L wavelengths and is a 2.75-approximation
algorithm [Bian et al. (June 2004)]. This algorithm is based on the request processing
order used in [Erlebach (Springer-Verlag, 2001)] and the application of
edge-colouring algorithm for multi-graphs and the algorithm for wavelength allocation
for tree of rings with degree six in [Bian et al. (January 2004)]. Algorithms
with improved approximation factor are further given for a restricted class of tree
of rings networks, including a 2.5-approximation algorithm for tree of rings with
degree six in [Bian et al. (January 2004)] and a 2-approximation algorithm for tree
of rings with degree four in [Deng et al. (2003)]. However, the 3L upper bound is
best possible since there are instances for the WA problem that require at least 3L
wavelength even on a tree of rings with degree four [Bian et al. (January 2004)].
A probabilistic algorithm with poly-logarithmic approximation factor is proposed
in [Raghavan and Upfal (1994)] for mesh-like networks (d-dimension meshes).
Using network flows techniques [Aumann and Rabani (January, 1995)] give a deterministic
algorithm for the RWA problem in mesh networks that is within poly-
38

2.5 Previous work
logarithmic approximation factor and improve upon the probabilistic algorithm
in [Raghavan and Upfal (1994)]. The best known O(1)-approximation algorithm
for mesh networks is presented by [Rabani (October, 1996)], based on linear programs
and randomized rounding techniques.
Directed ring and tree networks have also been considered and greedy techniques
used to design algorithms for the WA problem. In [Mihail et al. (1995)] a
2-approximation algorithm is presented for wavelength allocation in directed ring
networks. The latter is improved to (2 − 1 )-approximation algorithm by [Wilfong
and Winkler (January, 1998)], where L is the network load. For directed
L
tree
networks [Mihail et al. (1995)] propose an upper bound of 15L/4 wavelengths to
satisfy any instance of load L. Further, they exhibit an instance of a directed tree
network that require 3L/2 wavelengths, hence, marking the lower bound. The
upper bound is improved to 7L/4 in [Kumar and Schwabe (January, 1997)] (also
independently in [Kaklamanis and Persiano (Springer-Verlag, 1996)], where they
also show that 5L/4 wavelengths are enough to satisfy the instance exhibited
in [Mihail et al. (1995)] to require 3L/2 wavelengths, hence, marking the best
known lower bound for number of wavelengths to satisfy an instance of load L
on a directed tree network. The best known upper bound use 5L/3 wavelengths
to satisfy an instance of directed trees [Erlebach et al. (1999)], and it is shown
that the upper bound is optimal among the class of greedy algorithms, that is
no greedy algorithm can in general use less than 5L/3 wavelengths. Note that
there is a gap between the best possible upper bound using 5L/3 wavelengths and
lower bound using 5L/4 wavelengths even for binary trees. Randomized greedy
algorithms are proposed in [Auletta et al. (Springer-Verlag, 2000)] to narrow the
gap for binary trees, with an upper bound of 7L/5 wavelengths. Furthermore, in
[Caragiannis et al. (Springer-Verlag, 2001)] a randomized greedy algorithm with
1+ 5 3e
∼ 1.61-approximation factor is proposed for bounded degree trees - a rather
extensive class of tree networks that include binary trees. For tree networks with
symmetric instances [Caragiannis et al. (2000); Caragiannis et al. (2002)], present
a lower bound using (5/4 − ɛ)L wavelengths (c.f., lower bound in [Kumar and
Schwabe (January, 1997)] for general instances).
Although approximation algorithms give performance guarantee it turns out
that for most instances of practical interest they are inherent to poor performance.
39

2.5 Previous work
For this reason heuristic algorithms are designed to complement such cases, and
often are relatively simpler to implement. The most used heuristic techniques
for the WA problem is the first-fit colouring. Given a set W = {λ 1 , λ 2 , . . . , } of
colours and a set P of paths, a path p ∈ P is assigned a colour λ i , with smallest
index i such that no path of P \ {p} with colour λ i intersects with p (i.e., the first
available colour for p).
Several heuristics are presented in [Carpenter and Cosares (2002)] for the
more general demand routing and slot assignment problem (DRSP) for undirected
ring networks.
The DRSP problem is equivalent to the RWA problem when
demands are restricted to unit size, in which demands are requests and slots are
wavelengths. Integer linear programming (ILP) is considered as an alternative
solution method for the RWA problem, nevertheless, there is a practical limit for
the network and instance size that can be efficiently solved. In [Carpenter and
Cosares (2002)] the ILP formulation and corresponding relaxation for the DRSP
problem in ring networks is also presented. The ILP formulation for the RWA in
ring networks is also considered in [Lee et al. (2000)] where the column generation
technique is used to solve the LP relaxation of the formulation, and the branchand-price
procedure applied to obtain the solution. They experimentally show
that optimal solution is obtained to almost all instances under the given node
limit of the branch-and-bound tree.
The general ILP formulation for the wavelength allocation problem in networks
without wavelength conversion capabilities is described as follows. Let N
be a network, I an instance and R a routing for I in N. The network N is modeled
by optical links with W wavelengths labeled w, 1 ≤ w ≤ W . Denote by K the
set of paths in R such that a path in K is labeled k, 1 ≤ k ≤ |K|. For a link e
in N, we denote by K(e) a set of requests that traverse the link e. The goal is to
satisfy the instance I using a minimum possible number of wavelengths, W ∗ , such
that the wavelength continuity constraint - a request receive same wavelength at
every link it traverses, and the overlap constraint - no two requests sharing a link
receive same wavelength, is satisfied. We define the decision variables as follows.
Let zk
w be a variable set to 1 if a request k is assigned wavelength w and 0 otherwise.
Let z w be a variable set to 1 if wavelength w is assigned to a request, and
0 otherwise. The ILP formulation is
40

2.5 Previous work
Min W ∗ = ∑ w
z w (2.1)
∑
s.t.
k∈K(e)
z w k ≤ z w ∀ e ∈ E, w = 1, . . . , W (2.2)
∑
zk w = 1 ∀k ∈ K (2.3)
w
zk w , z w ∈ {0, 1} ∀k ∈ K, w = 1, . . . , W (2.4)
Equation 2.1 is the objective function. The constraint in Equation 2.2 combined
with the integrality constraint in Equation 2.4 ensure that the overlap
constraint at each link is satisfied. The constraint in Equation 2.3 combined with
the integrality constraint in Equation 2.4 ensure that each request is assigned a
wavelength and the wavelength continuity constraint is satisfied.
For wavelength convertible networks it is shown in [Wilfong and Winkler (January,
1998)] that the decision version for the MSS problem is NP -complete, even
in planar networks. In [Kleinberg and Kumar (January, 1999)] the MSS problem
is related to the minimum vertex cover (MVC) problem (a classical NP -hard
problem) implying a 2-approximation factor algorithm for the MSS problem in
general networks. Nevertheless, the MSS problem is shown to be tractable in
several networks. Particularly, a single node equipped with full conversion capabilities
is sufficient in both undirected rings [Ramaswami and Sasaki (December
1998)] and directed rings [Wilfong and Winkler (January, 1998)], and existence of
a linear time algorithm for the MSS problem in arbitrary undirected graphs and
in directed graphs with bounded tree width is shown in [Erlebach and Stefanakos
(April, 2002)]. In addition, [Erlebach and Stefanakos (April, 2002)] show that for
the k-sufficient set problem (i.e., find a sufficient set S for G of size k or decide
that no such sufficient set exists), there exist a fixed parameter tractable algorithm,
an algorithm that runs in time O(f(k).poly(n)), where f(k) is a function
of the parameter and poly(n) is a polynomial in n ( i.e., size of the input) for
arbitrary directed graphs.
Recall that for directed trees [Erlebach et al. (1999)] proved that no algorithm
in the class of greedy algorithms can guarantee a 100% utilization of the
available bandwidth. Motivated by the latter [Auletta et al. (2002)] study the
41

2.5 Previous work
performance of greedy deterministic algorithms in directed binary tree networks
with wavelength conversion. For sparse wavelength conversion they show that
⌊ 1
( n − 1)⌋ full wavelength converting switches are sufficient for the greedy algorithm
to optimally use the available wavelengths ( n is the number of
2 2
nodes
in the network). Further, three limited wavelength conversion models, all-pairs,
top-down and down, are proposed for directed binary tree networks (differing only
on the number of wavelength converting switches placed at nodes and the pair of
links between which the converting switches are placed), and adapt the greedy
algorithm by [Erlebach et al. (1999)] to each model to establish various bounds
for limited wavelength conversion degree that guarantee optimal or near-optimal
use of available bandwidth [Auletta et al. (2002); Caragiannis et al. (2002)].
Networks with heterogeneous wavelength conversion that allow each node to
have an arbitrary degree of optical conversion capabilities have also been considered.
In particular, in [Cavendish and Sengupta (2002)] different heuristics for
wavelength allocation in undirected ring networks with heterogeneous wavelength
conversion that combine greedy and backtracking techniques are presented. The
ILP formulation for the RWA problem in rings with heterogeneous conversion is
also given. An undirected ring network with special degree 2 wavelength conversion
capabilities at each node (cluster conversion) is used for wavelength allocation
performance comparison for the different approaches and routing strategies.
Note that the RWA problem on ring networks with cluster conversion is NPhard
[Cavendish (Nov 12-16, 2001, Beijing, China)]. In [Ramaswami and Sasaki
(December 1998)] they consider ring networks that employ low degree optical
converters at few nodes to realize near-optimal and optimal use of the available
wavelength bandwidth. Using fixed optical converters it is shown that if a single
node of the ring network is equipped with fixed conversion capabilities any
instance of load at most L − 1 can be satisfied using L wavelengths, and this is
best possible for any number of nodes equipped with fixed conversion capabilities.
Furthermore, using limited optical converters of degree 2 it is shown that if
two nodes of the ring network are equipped with degree 2 conversion capabilities
any instance of load at most L can be satisfied using L wavelengths. For star
networks [Ramaswami and Sasaki (December 1998)] show that if the number of
supported wavelength is even then equipping the hub node with fixed conversion
42

2.5 Previous work
capabilities suffices to realize optimal utilization of the network bandwidth. The
latter is extended to tree networks in which limited optical converters with degree
linear in the number of available wavelengths are deployed on patch nodes - a
minimum subset of tree nodes whose removal partition the tree into a set of star
networks. In addition, each star network in the partition set is equipped with
fixed conversion capabilities at its hub node.
As in single-fiber model the routing and wavelength allocation problem is also
NP-hard in the multi-fiber model [Li and Simha (2000); Margara and Simon
(2000)]. Independently, [Li and Simha (2000)] and [Margara and Simon (2000)]
study the RWA problem for k-fiber networks in which each link is realized using
k > 1 optical-fibers. They show a significant improvement in the per-fiber wavelength
utilization factor for most network topologies. Particularly, for undirected
2-fiber star networks optimal utilization of the network bandwidth is realized,
and a 75% bandwidth utilization is best possible for undirected k-fiber rings for
arbitrary k. However, for other k-fiber networks it remain open as to whether
the number of wavelengths can be reduce by a factor strictly larger than k.
Other optimization problems that have been considered for multi-fiber network
model include that of minimizing the total number of fibers needed to
satisfy an instance (given fixed number of supported wavelength per fiber) and
minimizing the number of wavelengths needed to satisfy an instance (given fixed
number of fibers per link). In [Nomikos et al. (2001)] they study the problem
of minimizing the total number of fibers in a multi-fiber network to satisfy a
given instance and show that can be optimally solved in path networks but it
is NP-hard in general networks. For star and ring networks an algorithm with
2-approximation factor is presented, respectively. In [Erlebach et al. (2003)], a
(log n + 1)-approximation algorithm is presented for arbitrary tree networks (n
is the number of network nodes). Using network flow techniques [Chekuri et al.
(2003)] present a 4-approximation algorithm for the problem of minimizing the
number of wavelengths used in multi-fiber tree networks. It is noted in [Erlebach
et al. (2003)] that the latter result imply a 4-approximation algorithm for the
problem of minimizing number of fibers, thus improving upon the results in [Erlebach
et al. (2003)] for this problem for arbitrary trees.
43

2.5 Previous work
For networks with multi-granularity wavelengths the traffic grooming problem
encompasses the routing and wavelength allocation as subproblems. Hence
traffic grooming is at least as hard as the RWA problem, in particular, the traffic
grooming problem is NP-hard even in path networks [Dutta et al. (June, 2003)],
where the RWA problem is optimally solved. The traffic grooming problem was
first pioneered in [Gerstel et al. (1998b); Gerstel et al. (1998a)] for rings with
g = 1 and g > 1 respectively, and has been extensively studied for different network
topologies and grooming ratio g. For networks with g = 1 the problem was
shown to be NP-complete for rings by [Eilam et al. (January 2002)]. Using a
combination of greedy and graph matching techniques a 3/2-approximation algorithm
is presented in [Călinescu and Wan (2002)] for ring topology. The latter is
improved to approximation ratio 10/7+ɛ and 10/7 in [Shalom and Zaks (October
2004)] and [Epstein and Levin (September 2004)] respectively. For general topology,
[Eilam et al. (January 2002)] describe a 8/5-approximation algorithm and
similarly in [Călinescu et al. (January 2002)] a 3/2 + ɛ-approximation algorithm
is presented.
For networks with g > 1 the traffic grooming problem was shown to be NPcomplete
in [Chiu and Modiano (January 2000)] for rings topology and a general
g, and [Zaks (2005)] gives NP-completeness results for ring and path networks for
a general g and for any fixed value of g. In [Flammini et al. (December 2005)] an
approximation algorithm with 2 ln g + o(ln g)-approximation factor is presented
for ring networks with arbitrary grooming ratio g. The algorithm use a greedy
approach to compute subsets of R of size at most g · k (where k is a parameter
dependent on g) that can share a wavelength together with corresponding
weight, in turn an approximation algorithm for the minimum weight set cover
problem in [Chvátal (1979)] is used to find a set cover for R from the subsets obtained.
The set cover is transformed into a partition by eliminating intersections.
It is also noted that the latter algorithm can be adapted to different network
topologies. In [Zhang and Qiao (2000)] several greedy heuristics are presented
for the traffic grooming problem in ring networks for arbitrary grooming ratio
g. An ILP formulation for traffic grooming in unidirectional rings is given in
[Dutta and Rouskas (January, 2002)] and a frame work of bounds (that can be
used to evaluate the performance of heuristics), both upper and lower, on the
44

2.5 Previous work
optimal values of the amount of traffic electronically routed (requiring ADM) in
the network is presented based on a successive decomposition of a ring network
into few nodes. The decompositions are independently solved efficiently to obtain
partial solutions that are combined using dynamic programming to obtain a
sequence of bounds. For this network [Liu and Tobagi (June, 2004)] also present
an ILP formulation and transform into a non-linear formulation that is solved
by decomposing into several smaller ILP subproblems to reduce the computation
time.
For special instances including all-to-one and all-to-all [Chiu and Modiano
(January 2000)] consider unidirectional rings and give optimal solutions for the
all-to-one case with arbitrary g and for the all-to-all case with g = 1 and g =
2 respectively. For all-to-all and distance dependent traffic they present lower
bounds on the number of ADMs required and propose greedy heuristics for a
general g that perform close to the lower bound. Unidirectional rings with allto-all
instances and g = 4 and g = 16 respectively is considered in [Hu (January,
2002)]. The traffic grooming problem is solved exactly for g = 4 and using
minimum number of wavelengths. For g = 16 an optimal solution is achieved
for rings with at most 15 nodes and a heuristic algorithm is proposed for large
rings. In addition they Conjecture a lower bound for g = 16 which is disproved in
[Bermond and Coudert (May 2003)]. Among other results [Bermond and Coudert
(May 2003)] also define the problem of minimizing the number of ADMs for the
unidirectional rings with n nodes and grooming ratio g as follows:
partition
the edges of the complete graph on n nodes (K n ) into W subgraphs B λ ,λ =
1, 2, . . . , W , having |E(B λ )| edges and |V (B λ )| nodes with |E(B λ )| ≤ g and where
∑ W
λ=1 |V (B λ)| has to be minimized (the edges of K n correspond to the circles
(also circuits), the subgraphs B λ correspond to the wavelengths and a vertex
of B λ corresponds to an ADM), and results from design theory [Colbourn and
Dinitz (1996)] is used to obtain optimal and quasi-optimal results for various
values of n and g, respectively (also in [Bermond et al. (2003)]). Furthermore,
design theory is used in [Bermond and Ceroi (2003)] to solved exactly the case
g = 3 for unidirectional rings, and they obtain partial results for the cases g = 5
and g = 6 (also give a simpler method to solve the case g = 4 also considered in
[Hu (January, 2002)]). In [Bermond et al.] results from K 4 − e-designs are used
45

2.6 Our contributions
to obtain optimal constructions for the case g = 5 (except when n ∈ {5, 7, 8}),
and in addition show that it is possible for the minimum number of ADMs to be
realized by only grooming with more than the minimum number of wavelengths
contrary to a Conjecture given in [Chiu and Modiano (January 2000)]. The cases
g = 1 and g = 2 are also shown to be trivially solvable. In [Bermond et al.
(2005)], the case g = 6 is settled for most values of n and quasi-optimal solution
for the few exceptions. The design theory is also used to solve exactly the traffic
grooming problem for bidirectional rings with all-to-all instances and g = 8 in
[Colbourn and Wan (2001)], and for unidirectional path networks with all-to-all
instances and g = 2 in [Bermond et al. (June 2005)] (the case g = 1 is also solved
exactly).
2.6 Our contributions
In this thesis we present improved algorithms and heuristics for allocation of network
bandwidth (wavelength) for different network topologies. For ring networks
we propose different heuristics for wavelength allocation for networks with/no use
of wavelength conversion. For undirected rings with no wavelength conversion we
proposed various heuristics for the wavelength allocation problem in [Justo and
Radzik (July, 2005b)], and experimentally show to improve upon those in [Carpenter
and Cosares (2002)]. We proposed heuristics for wavelength allocation in
undirected rings with cluster wavelength conversion in [Justo and Radzik (July,
2005a)], where we adapted the heuristics for rings without wavelength conversion
and the wavelength conversion framework used by the algorithm for this problem
given in [Cavendish and Sengupta (2002)]. Our heuristics were compared to
those in [Cavendish and Sengupta (2002)] and showed significant improvement
in terms of the number of wavelengths used and the number of wavelength conversions
performed. In [Justo and Radzik (July, 2005b)] we also extended the
heuristics for rings to tree of rings networks, and experimentally compared to
the approximation algorithms for the WA problem for tree of rings presented in
[Bian et al. (June 2004)]. Our heuristics showed a very good wavelength utilization
compared to the approximation algorithms.
46

2.6 Our contributions
We reviewed the results in [Ramaswami and Sasaki (December 1998)] for
single-fiber ring networks and showed that at least two nodes with degree 2 limited
conversion capabilities are necessary to realize optimal bandwidth utilization
for single-fiber ring [Justo and Radzik (July, 2004)]. That is, we showed the existence
of instances of load at most L that cannot be satisfied using L wavelengths
if the single-fiber ring network has degree two conversion capabilities at only one
node. Furthermore, we extended the results of [Ramaswami and Sasaki (December
1998)] for PW single-fiber rings to k-fiber rings (multi-fiber rings that use
k > 1 fibers per link), using low degree limited wavelength conversion capabilities.
We showed optimal wavelength utilization for any k-fiber ring where one
node has limited wavelength conversion capabilities of degree two. Similarly, we
showed the same results for k-fiber rings where two nodes have fixed conversion
capabilities.
We considered the problem of minimizing the number of total optical fibers
sufficient to satisfy a given instance on a multi-fiber tree network (i.e., a tree
network that allows more than one optical fiber per link) in [Justo (2003)]. We
adapted the techniques used in [Nomikos et al. (2001)], and among other results,
proposed a 2-approximation algorithm for spider-tree (independently also is given
in [Potika et al. (Nicosia, Cyprus, Nov 8–10, 2001)]), and a 4-approximation algorithm
for a tree such that any path has at most three nodes of degrees greater
than 2. We studied the traffic grooming problem (combining low rate traffic to
share a high bandwidth wavelength) for networks with multi-granularity wavelengths.
We considered path networks with grooming ratio 2 in [Justo and Radzik
(March, 2006)] and proposed heuristics for grooming arbitrary instances with unit
rate requests, using edge decomposition and interval partition methods. These
heuristics extend the results in [Bermond et al. (June 2005)], where they considered
the traffic grooming problem for this network but with all-to-all instances.
The grooming heuristics we proposed were extended to path networks with arbitrary
grooming ratios. For path networks with all-to-all instances on unit rate
requests and grooming ratio 3 we present results which imply a grooming algorithm
with asymptotic ratio 1.07.
We studied ring networks with grooming ratio 2 and proposed a traffic grooming
heuristic for arbitrary instances with unit rate requests in [Justo and Radzik
47

2.6 Our contributions
(June, 2006)]. This heuristic also uses the edge decomposition based heuristic in
[Justo and Radzik (March, 2006)]. Using weighted matching techniques we apply
the algorithms in [Zhang and Qiao (2000)] and [Călinescu and Wan (2002)] to
ring networks with grooming ratio 2, and experimentally compare their grooming
performance to our heuristic. Our heuristic showed improved performance for
most instances.
48

Chapter 3
Undirected Rings
We study the wavelength allocation problem for undirected rings and propose
heuristics that improve upon those presented in [Carpenter and Cosares (2002)].
Our algorithms are experimentally compared with the best performance algorithm
in [Carpenter and Cosares (2002)], showing an improvement in minimizing
number of wavelengths requirement of up to 11%. For the routing decisions we
use the shortest path routing and the load balancing routing described in Section
2.4.2. For the wavelength allocation, we propose heuristics which adapt the algorithm
presented in [Hendren et al. (1993)] for register allocation problem (RAP)
for a certain class of structured programs.
The RAP is modeled in [Hendren et al. (1993)] as the problem of colouring
cyclic interval graphs. A minimal colouring for the cyclic interval graphs corresponds
to a minimal register allocation on the program variables. The RAP
algorithm for structured programs in [Hendren et al. (1993)] is called the fat cover
algorithm and seeks to find a colouring of the circular arc graph obtained from a
given instance of RAP with small number of colours.
In Section 3.1 we review in more detail the register allocation problem considered
by [Hendren et al. (1993)], and in Section 3.2 we describe their fat cover
algorithm. In Section 3.3 we describe our adaptation of the algorithm of [Hendren
et al. (1993)] to the wavelength allocation problem for ring networks. In Section
3.4 we review the wavelength allocation algorithms proposed by [Carpenter and
Cosares (2002)]. Finally, in Section 3.5, we present experimental results which
49

3.1 RAP model for structured programs
compare performance of our algorithms with the best of the algorithms proposed
by [Carpenter and Cosares (2002)].
3.1 RAP model for structured programs
[Hendren et al. (1993)] consider a structured program iterating over a loop. Figure
3.1 (a) gives an example of such a program. In this example the loop has four
instructions and uses four scalar variables, w, x, y and z. Each variable has a
live range, which is a sequence of intervals corresponding to different iterations
of the loop. Consider for example variable z. For the first iteration, z is defined
outside the loop and ”dies” at instruction 2 within the loop. This period is the
first interval of the life range of variable z. Subsequently, for each iteration i of
the loop, z is defined in instruction 4 of this iteration, and is live between this
definition and the use of z in instruction 2 of the next iteration. The interval graph
in Figure 3.1 (b) represents the live ranges for the program variables in Figure
3.1 (a) capturing the regular periodic nature of variables. The horizontal axis
represents the instruction numbers of the loop, and the vertical axis represents
the variables in the loop. A solid circle designates a point when the variable is
defined (obtains a new value) and a cross designates a point when the current
value of the variable is used for the last time. For example w is defined in
instruction 1 and is last used in instruction 3.
for i = 1 to n {
w = y * 10; (1)
w
1 period 1 period
X
X
x = z * 20; (2)
y = w + 5; (3)
z = x + y; (4)
}
x
y
z
X
1
X
X
X X
2 3 4
X
X
X
(a) A loop program
(b) An interval graph
Figure 3.1: A loop program and corresponding interval graph.
50

3.1 RAP model for structured programs
In Figure 3.1 (b), the live range of a loop variable is represented as a periodic
interval: a sequence of intervals that are equally spaced in time by some period
(for the loop variables a period corresponds to one iteration). Observe that the
same variable can be re-defined several times within the same iteration. The life
range of such a variable consists of a number of periodic non-overlapping intervals.
If the loop has j instructions, then one periodic interval can be characterized by
one or two intervals contained in the interval [0, j + 1] representing one iteration
of the loop. The numbers 1, 2, . . ., j represent the consecutive instructions within
the loop, while the numbers 0 and j + 1 provide the overlap point for successive
iterations. An interval is denoted by [t, t ′ ] if the program variable is alive at both
time points t and t ′ , and is denoted by [t, t ′ ) if after time t ′ the program variable
is no longer alive. For example, the live ranges of variables w and x do not extend
across the loop boundary between iterations, thus, each can be expressed as one
interval, [1 : 3) and [2 : 4) for w and x respectively. The variables y and z, on the
other hand, are defined in one iteration and used in the next, thus its live range
is represented as a pair of intervals, ([0 : 1), [3 : 5]) and ([0 : 2), [4, 5]) for y and z
respectively. Note that the interval [0 : 1) for y and [0 : 2) for z is viewed as an
extension of the interval [3 : 5] and [4 : 5] respectively, that is wrapped around
to fit in one period. Such a wrapped interval is referred to as a cyclic interval.
Figure 3.2 (a) shows the cyclic interval representation for Figure 3.1 (b).
w
x
y
z
[1,3)
w
x
[2,4)
x x
4
[0,1)
[3,5]
x
y
[0,2)
[4,5]
x
x
z 3
0 1 2 3 4 5
y
5;0
2
z
w
1
y
z
x
w
(a) A cyclic interval graph
(b) A circular arc graph
(c) An interference graph
Figure 3.2: A cyclic interval graph and corresponding circular arc graph and
interference graph.
A graph G(V, E) is a circular-arc graph (an interval graph) if its vertices can
be put in a one-to-one correspondence with a set F (S) of circular arcs (intervals)
51

3.1 RAP model for structured programs
on a unit circle (real line) such that two vertices are adjacent in G if and only
if their corresponding circular-arcs (intervals) have nonempty intersection, see
the interference graph in Figure 3.2 (c), for example. Such a set F (S) is called
a circular-arc (interval) model of the circular-arc graph (interval graph). In the
following the set F (S) of circular-arcs on a unit circle is also referred to as circulararc
graph denoted by G CAG , for example, the circular arcs {w, x, y, z} in Figure
3.2 (b).
A time (point) t on a unit circle of G CAG is covered by an interval [t 1 , t ′ 1),
if t 1 ≤ t < t ′ 1, or by an interval [t 1 , t ′ 1], if t 1 ≤ t ≤ t ′ 1. A pair of intervals of
G CAG overlaps if there exists a point t ∈ G CAG that is covered by both intervals.
A circular-arc graph G CAG is constructed from the program variables such that
an arc in G CAG corresponds to the live range of the value of a program variable
and an overlap between two arcs in G CAG represents interference between the
live ranges of two program variables. Figure 3.2 (b) shows a circular-arc graph
modeled by cyclic intervals of Figure 3.2 (a).
Let t 0 be a point in the constructed G CAG representing the overlap of the
successive iterations (i.e., the entry and exit point for different iterations). Only
a program variable with live range that extends over two successive iterations
leads to an interval that covers t 0 in G CAG , referred to as a cyclic interval. In
Figure 3.2 (b) the arc set {y, z} constitute a set of cyclic intervals where the
point 5; 0 correspond to the point t 0 . Let t be any point in G CAG . The width
of G CAG at a point t, w(G CAG , t), is the number of intervals in G CAG covering
t. The maximum width of G CAG , w max (G CAG ), is the maximum w(G CAG , t) over
all points t in G CAG . Analogously, the minimum width, w min (G CAG ), is the
minimum w(G CAG , t) over all points t in G CAG .
If no cyclic interval exists in a circular arc graph, then the circular arc graph
is isomorphic to an interval graph (i.e., interference graph for a set of intervals on
a real line). Interval graphs can be optimally coloured using a greedy algorithm
that proceeds along the line from left to right and assigns the first non-conflicting
colour to each new interval encountered. The colouring obtained this way shows
that the chromatic number of an interval graph is equal to its maximum width
[Golumbic (1980)]. On the other hand finding a minimum colouring for circulararc
graph is NP-hard, as in general graph colouring problem [Garey et al. (1980)].
52

3.2 Fat cover algorithm for RAP
The chromatic number of a circular arc graph is related to its width as given in
the following theorem.
Theorem 1 (Hendren et al. (1993)) Let G CAG be a circular arc graph. Then
G CAG can be coloured using k colours, such that w max (G CAG ) ≤ k ≤ w max (G CAG )+
w min (G CAG ).
3.2 Fat cover algorithm for RAP
We describe the fat cover algorithm for RAP on structured programs due to
[Hendren et al. (1993)], which colours the constructed circular-arc graph G CAG .
A point t of G CAG is a fat point if w(G CAG , t) = w max (G CAG ). Let fp(G CAG )
be a collection of all fat points in G CAG . A fat cover F (I) for G CAG relative to
an interval I ∈ G CAG is a subset of intervals of G CAG such that the following
properties hold.
1. I ∈ F (I).
2. No two intervals in F (I) cover the same point t ∈ G CAG .
3. Every fat point t ∈ fp(G CAG ) is covered by some interval in F (I).
It is noted in [Hendren et al. (1993)] that a fat cover F (I) can be found greedily
(using left to right traversal of the cyclic interval graph) if one exists. The third
property implies that removing the intervals of any fat cover F (I) from G CAG
decreases the maximum width w max (G CAG ) by 1. The fat cover algorithm takes
as input the graph G CAG . Let the subset of intervals of G CAG covering the point
t 0 (i.e., cyclic intervals) be labeled I c1 , I c2 , . . . , I cm , for m ≥ 0. The intervals in
G CAG are coloured as follows.
1. For 1 ≤ i ≤ m, find a fat cover F (I ci ) with respect to a cyclic interval I ci . If
such cover exists then apply a new colour to colour the intervals in F (I ci ).
2. Colour any remaining cyclic interval using a new colour.
3. The remaining uncolored intervals of G CAG form an interval graph. Apply
the algorithm in [Golumbic (1980)] for interval graph colouring using a new
set of colours (that is, colours not used yet).
53

3.3 Wavelength allocation by pivot based algorithms
3.3 Wavelength allocation by pivot based algorithms
In this subsection we describe our adaptation of the fat cover algorithm for RAP
to the WA problem for rings. The fat cover algorithm needs a distinguished point
t 0 on the circle to define cyclic intervals. We do not have such a point on our
optical ring network. Nevertheless, there is a clear analogy between concepts used
by the fat cover algorithm and that for colouring requests of a ring network. Given
a routing R for an instance I on a ring network N, we can create the following
circular arc graph G CAG . The arcs in G CAG correspond to a set of paths in R
and points in G CAG correlate with the links of N, labeled e 0 , e 1 , . . . , e n−1 , for a
ring network with n links. The maximum width w max (G CAG ) for a circular arc
graph G CAG corresponds to the load L on a ring network N. A fat point on G CAG
corresponds to an edge of N traversed by L requests. Likewise, a collection of
fat points fp(G CAG ) on G CAG correlate to a set of edges of N traversed by L
requests.
In analogy to fat covers in circular arc graphs, we define a pivot cover P (r)
relative to a request r (called pivot request) on a ring network N as a maximal
subset of requests in N that satisfy the following properties.
1. r ∈ P (r).
2. No two requests in P (r) share any link of N.
Note that the notion of a pivot cover of N is weaker than the notion of a fat cover
of G CAG . Unlike in a fat cover where every fat point is covered by an interval
of a fat cover, requests on a pivot cover may not necessarily traverse every edge
crossed by L requests. That is, a pivot cover does not guarantee to decrease by
1 the network load for the remaining requests in N.
3.3.1 Finding pivot covers
We denote an undirected request routing path (i, i + 1, i + 2, . . . , j) (where the
additions are modulo n) by (i, j). We consider pivot covers P (r) found in the
following natural greedy way.
54

3.3 Wavelength allocation by pivot based algorithms
1. Given the pivot request r = (i, j), P (r) ← {(i, j)}.
2. Let (i 1 , j 1 ), (i 2 , j 2 ), . . . , (i k , j k ) be the requests, in clockwise orientation, already
added to P (r). If there is a request (x, y) between j k and i 1 (x can
be j k and y can be i 1 ), then from among such requests select one which x
is closest to j k . Break ties by taking the longest possible request (with y
closest to i 1 ). P (r) ← P (r) ∪ {(x, y)}. Otherwise, return P (r).
3. Repeat Step 2.
The WA algorithms which we propose use the pivot cover algorithm as a subroutine.
They repeatedly find a pivot cover P (r) with respect to a request r from
uncoloured requests and assign requests in P (r) a new colour. Different selection
criteria for a pivot request define different variants of our heuristic algorithm for
wavelength allocation on ring networks. Our proposed heuristics select a longest
uncoloured request as pivot request. Since long requests are difficult to colour,
giving priority to such requests may enhance the colouring performance. Let q be
number of requests in N that remain to be considered. As Step 2 indicates, the
next request to be selected into the pivot cover P (r) requires O(q) operations in
the worst case. Thus the pivot based WA algorithms require O(|R| 2 ) operations
in the worst case, where |R| is the instance size.
We propose a WA algorithm called longest pivot algorithm (LPA) that takes
as input the routing R for an instance I on the ring network N. The algorithm
colours requests in N as follows.
1. While some uncoloured requests remain in N,
(a) Select a longest uncoloured request r from N as pivot request
(b) Find a pivot cover P(r) with respect to r.
(c) Assign requests in P (r) a new colour.
Next we propose a WA algorithm called fixed pivot algorithm (FPA) which
uses a pivot request selection rule that is based on fixing a link, referred to
as the pivot link. That is the FPA algorithm selects a pivot request from among
uncoloured requests traversing the pivot link. Suppose that after a certain number
55

3.3 Wavelength allocation by pivot based algorithms
of colouring iterations no more uncoloured requests traverse the pivot link. If all
requests have been already colored, then we are done. Otherwise, observe that
the remaining uncoloured requests do not traverse the pivot link, so they can be
considered as requests of a path network. We use the interval graph algorithm in
[Golumbic (1980)] to colour these requests with optimal number of colours. The
FPA algorithm is described in detail in the next Section 3.3.2.
3.3.2 Fixed pivot algorithm
The Fixed pivot algorithm (FPA) takes the input as in the LPA in Section ??,
and distinguished link as the pivot link. Requests are coloured in two phases as
follows.
1. While some uncoloured requests traverse the pivot link,
(a) Select a longest request r from among uncoloured requests traversing
the pivot link in N as pivot request.
(b) Find a pivot cover P(r) with respect to r
(c) Assign requests in P (r) a new colour.
2. The remaining requests correspond to requests of a path network. Use the
interval graph algorithm [Golumbic (1980)] using a new set of colours (that
is, colours not used yet).
We consider the fixed pivot algorithm with a link traversed by the minimum
number of requests as the pivot link, and denote it minFPA. The following lemma
gives the lower and upper bounds on the number of colours used by this algorithm.
Lemma 1 For a set R of request paths in a ring network, let L denote the load
of the network and L 0 the minimum number of requests traversing the same link.
The minFPA algorithm colours the requests in R using W colours such that L ≤
W ≤ L + L 0 .
Proof : Clearly, if the minFPA algorithm colours all network requests in Phase
1 then L colours are sufficient since the pivot link is traversed by at most L requests.
On the other hand, by choosing a pivot link with L 0 requests, the minFPA
56

3.4 Demand routing and slotting
algorithm uses L 0 colours during Phase 1, and at most L requests traversing any
network link during Phase 2. The remaining requests can be coloured optimally in
Phase 2 using at most L new colours by the interval graph algorithm in [Golumbic
(1980)].
Lemma 1, implies a 2-approximation worst case performance guarantee for the
minFPA algorithm. In fact, selecting any link as pivot gives a 2-approximation
algorithm. Selecting a link with the minimum load L 0 as the pivot link is based
on the intuition that it should be a good idea to try to reach quickly a point when
the remaining uncolored requests form an interval graph which can be coloured
optimally. This selection, however, is only a heuristic and does not guarantee
the overall optimal coloring. If the ring network size n is small (practical rings
typically contain from 3 to 30 nodes [Carpenter and Cosares (2002)]), then it
may be feasible to try each link as the pivot link, and select the best colouring
achieved. This leads to an iterative variant of the fixed pivot algorithm, referred
to as the iterative fixed pivot algorithm (IFPA). The worst case running time for
the IFPA algorithm increases by a factor of n to that of the none-iterative pivot
based algorithms.
The FPA algorithm described in Section 3.3.2 and the fat cover algorithm
described in Section 3.2 are closely related. Relating the pivot link to the fixed
point t 0 on the circle of G CAG , the requests traversing the pivot link to the cyclic
intervals of G CAG , and the pivot covers for the FPA algorithm to fat covers of
the fat cover algorithm, the FPA algorithm looks very similar to the fat cover
algorithm. The difference is that a fat cover and a pivot cover are not exactly
the same notions.
3.4 Demand routing and slotting
In [Carpenter and Cosares (2002)] the demand routing and slotting problem
(DRSP) on ring network is considered. In the DRSP problem we are given a
ring network, with nodes labeled 1, 2, . . . , n (proceeding clockwise) and edges
= e 1 , e 2 , . . . , e n (e i = (i, i + 1)) each with capacity C, and a set of point-topoint
demands to be served. The units of capacity at each edge are numbered
57

3.4 Demand routing and slotting
1, 2, . . . , C, and the units of capacity with number i around the whole ring form
slot i. A demand is specified with a pair of distinct end-points, node i and node
j, where i < j, and a demand amount d k . We wish to route and slot all units
of demands using as few slots as possible. The constraints are that each demand
must be routed entirely in one direction and that no demand unit can change
slots within its route.
When demands are restricted to unit size, the DRSP is equivalent to the
RWA problem, where demands are connection requests and the edge capacity C
is the number of available wavelengths. As an extension of the RWA problem on
rings, the DRSP is NP-hard [Carpenter and Cosares (1997)]. An integer linear
program formulation and several heuristic algorithms for the DRSP are presented
in [Carpenter and Cosares (2002)]. The heuristic algorithms are categorized into
two types, algorithms that combine routing and colouring decision and those
that perform the routing and colouring decisions independently (i.e., two-phased
algorithms). The experimental evaluation presented in [Carpenter and Cosares
(2002)] shows that the two-phased algorithms perform remarkably well compared
to the other approach. In particular, a two-phased algorithm called min-load two
phase algorithm that adapt the algorithm for colouring circular arc graphs in
[Tucker (1975)] is shown to achieve best performance.
We review the min-load two phase algorithm and experimentally compare its
performance with the pivot cover based algorithms presented in the preceding
sections.
3.4.1 The min-load two phase algorithm
The min-load two phase algorithm (min2P) takes as input a routing R for an
instance I (on unit size demands) on a ring network N. Demands are assigned
colours in the following way.
1. Find a node k of N traversed by minimum number of demands.
2. Assign each of the demands traversing the node k with a new colour, subsequently,
mark the colour unavailable at every link traversed by the demand.
58

3.5 Numerical results and performance evaluation
3. The remaining demands correspond to an interval graph since none traverses
the node k. Adapt the interval graph algorithm [Golumbic (1980)]
to colour the remaining demands as follows. Starting from node i =
k, scan through the network links one at a time in clockwise direction,
e i , e i+1 , . . . , e n , . . . , e i−1 , and process any uncoloured demands traversing
current link e j in the following way.
(a) Identify an uncoloured demand traversing the current link e j . Let e h
be the last link being traversed by this demand.
(b) Assign the demand with the first available colour at both, e j and e h
(this guarantees that a colour is available at all links spanned by the
demand), and mark the colour unavailable at the links spanned by the
demand.
Note that like for the fixed pivot algorithm the wavelength minimization for the
min-load two phase algorithm in [Carpenter and Cosares (2002)] can be enhanced
by iterating over all network nodes, a variant algorithm referred to as the iterative
min-load two phase (I2P). Note that the min2P algorithm differs from the FPA
algorithm in Section 3.3.2 in the following two aspects.
• The FPA algorithm colours a collection of requests (a pivot cover) using the
same colour, whereas the min2P algorithm colours one request at a time.
• The FPA algorithm uses a new set of colours (colours not used yet) on
the interval graph algorithm, whereas the mini-load two phase algorithm
attempt to re-use colours used in the previous steps on the interval graph
algorithm.
3.5 Numerical results and performance evaluation
Tables 3.1 and 3.2 present numerical results for ring networks on different number
of nodes and routing instances. The types of instances which we use in our
experiments are described in Section 2.4.1. The average number of wavelengths
59

3.5 Numerical results and performance evaluation
used by each algorithm and the average load (W-LB) are shown. The averages
are taken over ten random instances. Observe that Gaussian instances are easily
coloured under the shortest path routing : all algorithms use optimal number of
colours in our experiments (see in Table 3.1). This is due to the fact that most
requests are short and are concentrated by the median node, hence possibly leaving
some network links being traversed with few/no requests. We do not consider
this instance type in Table 3.2. Unlike the shortest path routing, however, when
a Gaussian instance is routed by the load balancing routing, then the network
load is significantly reduced (almost halved), leading to a more effective way of
using the available bandwidth. For Uniform instances, the shortest path routing
yields relatively low network load compared to the load balancing routing, especially
as the instance size increases. We remark that for cases where requests
are arbitrary the load balancing routing may be considered as a good choice for
routing requests.
We observe that the pivot based algorithms perform better than the DRSP
based algorithms. Note that the min2P algorithm which we use in our experiments
is shown in [Carpenter and Cosares (2002)] to have best performance
among the algorithms considered there. Tables 3.1 and 3.2 show that the algorithm
minFPA has the worst but relatively good performance in most cases in
comparison with the algorithm I2P (which in turn performs better than algorithm
min2P). Further, observe that algorithm LPA gives the best wavelength performance
among the none-iterative algorithms for all the instances we considered.
For medium networks, for example, algorithm LPA gives a wavelength performance
that is up to 6% and 2.5% better than algorithm min2P and minFPA,
respectively (see Table 3.1). When the network size increases, the performance of
algorithm LPA seems to improve further in comparison with the other algorithms,
and for large networks it is up to 11.5% and 3% better than the performance of
algorithms min2P and minFPA, respectively (see Table 3.2). The wavelength
performance for algorithm LPA is optimal or near-optimal (i.e., equal or close to
the network load) for most instances. Although the iterative algorithm IL2P improves
the wavelength minimization upon the algorithm L2P, it is out-performed
significantly by the algorithm LPA under all instances (as seen in Tables 3.1 and
3.2). In addition, algorithm LPA also has better wavelength performance than
60

3.5 Numerical results and performance evaluation
Distribution n |R| W-LB min2P I2P minFPA IFPA LPA
256 83 83 83 83 83 83
16 512 152 153 152 153 152 153
Uniform-SP 1024 294 297 294 294 294 294
256 79 81 79 79 79 79
32 512 145 149 146 146 145 145
1024 285 291 286 286 285 285
256 81 84 83 82 81 81
16 512 161 166 164 162 161 162
Uniform-LB 1024 316 325 320 317 316 316
256 80 86 82 83 81 81
32 512 159 171 166 164 160 161
1024 312 333 327 317 314 314
256 195 195 195 195 195 195
16 512 371 371 371 371 371 371
Gaussian-SP 1024 742 742 742 742 742 742
256 189 189 189 189 189 189
32 512 374 374 374 374 374 374
1024 749 749 749 749 749 749
256 108 110 108 108 108 108
16 512 215 223 217 217 215 215
Gaussian-LB 1024 431 440 436 432 431 431
256 107 111 108 108 107 107
32 512 209 214 210 211 209 209
1024 424 436 428 426 424 425
Table 3.1: Average number of wavelengths for ring networks with 16 and 32
nodes. The instances have 256, 512 and 1024 requests.
61

3.5 Numerical results and performance evaluation
Distribution n |R| W-LB min2P I2P minFPA IFPA LPA
64 1024 283 293 285 285 283 285
Uniform-SP 2048 550 576 562 559 550 552
96 1024 287 297 290 291 287 291
2048 561 592 574 570 561 565
64 1024 307 335 327 321 315 312
Uniform-LB 2048 613 666 655 633 625 619
96 1024 308 338 332 325 320 316
2048 607 688 655 633 627 617
64 1024 421 430 424 426 422 422
Gaussian-LB 2048 850 864 860 859 851 851
96 1024 427 435 430 429 427 427
2048 851 863 857 859 851 851
Table 3.2: Average number of wavelengths for ring networks with 64 and 96
nodes. The instances have 1024 and 2048 requests.
Distribution LPA minFPA min2P FCV
Uniform-SP 1.24 0.42 0.98 12.72
Uniform-LB 1.82 0.54 1.58 16.34
Gaussian-SP 0.45 0.08 0.68 0.08
Gaussian-LB 2.15 1.94 1.66 31.87
Table 3.3: Average time used (in seconds) for ring network with 64 nodes and
instance of 2048 requests (using 1.0GHZ AMD Duron Processor with 112.0MB
RAM).
62

3.5 Numerical results and performance evaluation
algorithm IFPA (which improves the wavelength minimization upon the algorithm
minFPA) for most instances, especially, Gaussian-LB instances for large
ring networks. Furthermore, the iterative algorithm IFPA has an extra factor n
(the size of the network) in its running time in comparison to the none-iterating
algorithm LPA, which is an additional advantage of the algorithm LPA. Table
3.3 shows the average time used for none-iterating algorithm considered for ring
network with 64 nodes and instance of 2048 requests.
We report additional numerical results for different WA heuristics for ring
networks in Appendix A.1. We considered networks with 8, 16, 32, 64, 96 and
128 nodes and additional heuristics listed in Table 3.4. For ring networks with
only 8 nodes and all types of distribution of requests, the performance of all
algorithms we considered except RG was the same: they all always computed
optimal wavelength allocations, using the number of wavelengths equal to W-LB.
The differences between the performance of different algorithms start showing for
networks with 16 nodes, and grow when the size of the network increases (see
Tables A.1, A.2, A.3 and A.4). We should mention again that the input instances
of the Gaussian-SP type turn out to be very easy. For all input instances of
this type which we have considered, all algorithms have always found optimal
solutions.
We summarize the performance results for the algorithms LPA, minFPA, FCV
and min2P in the plots in Figures 3.3, 3.4 and 3.5, for ring networks with 64 nodes
and the Uniform-SP, Uniform-LB and Gaussian-LB distribution of requests. The
plots show that algorithm LPA gives overall the best performance, followed by
the algorithms minFPA and FCV. The good performance of the pivot based
algorithms, and especially the LPA algorithm, could be attributed to the fact
that a pivot cover potentially decreases the network load by 1 (leading to a
better utilization of the colours), and the way the pivot covers are selected gives
priority to longer requests that are potentially difficult to colour.
63

3.5 Numerical results and performance evaluation
maxFPA
RPA
FCV
IFCV
LG
RG
Variant of the FPA algorithm with a link traversed by maximum
number of requests as the pivot link.
Variant of the LPA algorithm with pivot requests selected arbitrarily.
Fat cover algorithm simulation for wavelength assignment on
ring networks. The fixed point t 0 is a link traversed by minimum
number of requests. The cyclic intervals are the requests
traversing the distinguished link.
Variant of the FCV algorithm that iterates over all network
links.
Variant of the first-fit colouring heuristic that colours requests
in the order of a longest requests first.
Variant of the first-fit colouring heuristic that colours requests
in an arbitrary order of requests.
Table 3.4: Other WA heuristics for undirected ring networks
# WAVELENGTHS/W-LB
1.05
1.04
1.03
1.02
1.01
LPA
minFPA
FCV
min2P
1
100 200 300 400 500 600
W-LB
Figure 3.3: Wavelength performance for instances with Uniform-SP distribution
of requests on 64-nodes ring network.
64

3.5 Numerical results and performance evaluation
# WAVELENGTHS/W-LB
1.1
1.09
1.08
1.07
1.06
1.05
1.04
1.03
1.02
1.01
1
LPA
minFPA
FCV
min2P
100 200 300 400 500 600 700
W-LB
Figure 3.4: Wavelength performance for instances with Uniform-LB distribution
of requests on 64 nodes ring network.
# WAVELENGTHS/W-LB
1.04
1.03
1.02
1.01
LPA
minFPA
FCV
min2P
1
100 200 300 400 500 600 700 800 900
W-LB
Figure 3.5: Wavelength performance for instances with Gaussian-LB distribution
of requests on 64 nodes ring network.
65

Chapter 4
Undirected Tree of Rings
The topological structure for tree of rings consist of several sub-rings. In this
chapter we consider the WA problem for undirected tree of rings with given
set of request paths and propose heuristic algorithms that exploit the sub-ring
structure of such networks. We extend the pivot based algorithms presented for
ring networks in Chapter 3 to the tree of rings network. Since a ring network
is a special case of a tree of rings network, the WA problem for tree of rings is
NP-hard as well.
Recently, [Bian et al. (June 2004)] gave a 2.75-approximation algorithm for
the WA problem in tree of rings, which remains the best known approximation
algorithm for this problem in arbitrary tree of rings networks. For the restricted
class of tree of rings networks with node degrees bounded by 6 and 4, a 2.5-
approximation and 2-approximation algorithms are given in [Bian et al. (January
2004)] and [Deng et al. (2003)], respectively. The wavelength allocation framework
in [Bian et al. (June 2004)] is based on edge colouring algorithm for multigraphs
(graphs in which several edges may be incident to the same pair of nodes).
The authors show that the approximation ratio bound of 2.75 is achieved when the
1.1-approximation edge colouring algorithm for multigraphs due to [Nishizeki and
Kashiwagi (August 1990)] is applied, while the 1.5-approximation edge colouring
algorithm for multigraphs in [Shannon (1949)] gives the approximation ratio
bound of 3. In both cases the upper bound on the number of wavelength used
is 3L. The approximation ratios less than 3 come from the lower bounds higher
66

4.1 Tree of rings model
than L. We compare experimentally the performance of the algorithms presented
in this chapter and the approximation algorithms in [Bian et al. (June 2004)].
4.1 Tree of rings model
The tree of rings networks are defined in Section 2.3. We construct a tree of
rings instance from an underlying tree as described Section 4.5. Given a set of
paths on a tree of rings network, the heuristics we present perform the wavelength
allocation task using an ordering of nodes obtained from the depth first search
(DFS) traversal of the underlying tree.
The first ring (the root ring) is chosen from among centre rings defined as
follows. A centre ring on the tree of rings corresponds to a centre node on the
underlying tree defined Section 2.3. A centre ring is likely to be traversed by most
cross-ring requests since many paths between pairs of nodes on distinct rings are
likely to traverse a link on the centre ring.
The DFS traversal of rings on a tree of rings assigns to each ring a nonnegative
integer – the DFS number, starting from the root ring that is assigned number 0.
A node of a tree of rings is referred to as a hub node if its degree is greater than
2, i.e., a node shared by at least two rings. Let c 1 and c 2 be two distinct rings.
Ring c 1 is the parent of ring c 2 , if c 1 and c 2 share a hub node h and c 1 has the
least DFS number amongst all rings that share the hub node h. This node h is
referred to as the hub for the ring c 2 , and the rings with h as the hub (i.e., with
the ring c 1 as the parent ring) are the children of c 1 . Note that the root ring
does not have the hub since it does not have the parent ring.
Given a DFS ordering of the rings, we define the numbering of the nodes using
the clockwise numbering on each ring in the tree of rings. The nodes on the root
ring are first numbered in clockwise direction starting from an arbitrary node
and this numbering is extended to the nodes on other rings considering the rings
according to the increasing DFS numbers. Let h be the hub for the ring c that
is currently being considered. The node h has been already numbered since it
belongs to the parent ring for c. From node h proceeding in clockwise direction
in c, all other nodes of c are numbered using consecutive integers starting from
i + 1, where i is the last number already assigned to previously considered nodes.
67

4.2 Finding pivot cover on ring of tree of rings
Let i+1, i+2, . . . , j be the numbers assigned to the nodes of the ring c other than
the hub. We define the notion of the surrogate number for the hub of c relative
to the ring c as i. Thus all nodes in c, taking the surrogate number for the hub,
are consecutively numbered in clockwise direction. A request on the tree of rings
is said to belong to a ring c if it traverses at least one link of the ring c and c
has the least DFS number amongst all rings on the tree of rings traversed by this
request. Figure 4.1 shows a DFS numbering of rings and node numbering. The
numbers in ’[ ]’ are DFS numbers of rings and the numbers in ’( )’ are surrogate
numbers of hub nodes.
9
0
[3](7)
3 [0] 1
8 2
(3)
[1] 4
5
7[2]
(5)
6
Figure 4.1: A tree of rings with a numbering of nodes. The numbers in ’[ ]’ are
DFS numbers and the numbers in ’( )’ are surrogate numbers.
4.2 Finding pivot cover on ring of tree of rings
For the ring networks, we defined in Section 3.3 a pivot cover relative to a request
and described a greedy way of finding pivot covers. We now extend the notion
of pivot cover to a ring of a tree of rings network. A pivot cover P (r) relative
to a request r for a ring c of a tree of rings network is a subset of requests that
belong to ring c such that r ∈ P (r) and no two requests in P (r) share any link of
the network. We consider hub nodes in a special way when finding a pivot cover
because it is possible that a pair of requests use a hub node and are link disjoint
on ring c, but share a link on one or more other rings.
Let c be a ring in a tree of rings whose requests that belong to c are currently
being processed. Each request belonging to c is of one of the following three
types.
68

4.2 Finding pivot cover on ring of tree of rings
1. A request which is fully within ring c (for example, request p 1 in Figure
4.2).
2. A request which has one endpoint in c and in addition to traversing one or
more links of c, also traverses at least one link of another ring (for example,
requests p 2 and p 3 in Figure 4.2).
3. A request which traverses one or more links of c whose both endpoints are
outside of ring c (for example, request p 4 in Figure 4.2).
c1
c
p
2
p
1
c2
p
3
p
4
c3
Figure 4.2: Different types of requests belonging to ring c. Rings c 1 , c 2 and c 3
have higher DFS numbers than ring c.
For a request p of type 2, let i be the endpoint of this request in c, and h be
the hub node through which this request leaves c. If in the clockwise order of the
nodes of c starting from any link not traversed by p node i comes before node h,
then we say that p exits c at h (or p is an exit request at node h). Otherwise, if
node h comes before node i, we say that p enters c at h (or p is an entrant request
at node h)
For a request p of type 3, let h 1 and h 2 be the hub nodes on c where p leaves
c, and let h 1 come before h 2 in the clockwise order of the nodes of c starting from
any link not traversed by p. We say that request p enters c at h 1 and exit at h 2
(or is an entrant request at h 1 but an exit request at h 2 ).
69

4.3 Long pivot-DFS algorithm
Note that, if two requests p 1 and p 2 belong to ring c, p 1 exits c at a hub node h
and p 2 enters c at this node h then p 1 and p 2 possibly share a link on a ring with
a higher DFS number than c. We adapt the algorithm for finding pivot covers
for ring networks described in Section 3.3.1 to finding pivot covers for a ring on
the tree of rings network by accounting for different types of request belonging
to the same ring (types 1-3).
We find a pivot cover P (r) for a request r belonging to a ring c in the following
way.
1. P (r) ← {r}.
2. Let p ′ be the last request added to P (r), and i and j be the first and the
last nodes on r and p ′ respectively in c in the clockwise ordering starting
from a link not traversed by r. If there is a request p satisfying the following
conditions:
(a) p belongs to c;
(b) p traverses the edges of c from a node x to a node y for some x and y
such that j ≤ x < y ≤ i (in the clockwise orientation of c).
(c) if j = x and p enters c at j and the last request p ′ added to P (r) exits
c at j, then p and p ′ do not share a link outside of c.
(d) if y = i and p exits c at i and r enters c at i then p and r do not share
a link outside of c;
then from among such requests select request p with x closest to j. P (r) ←
P (r)∪{p}. Break the ties by taking request p with y closest to i. Otherwise,
if no request satisfies (2a)–(2d), then return P (r).
3. Repeat Step 2.
4.3 Long pivot-DFS algorithm
We describe the Long pivot-DFS (LP-DFS) WA algorithm for tree of rings network
that uses the pivot cover algorithm and the first-fit colouring heuristic to
70

4.4 Approximation algorithm
colour requests that belong to each ring starting with the root ring. The LP-DFS
algorithm colours requests in the following way.
1. Select the root ring.
2. Compute the DFS numbers of the rings.
3. While some uncoloured requests belongs to the root ring,
(a) select a longest uncoloured request r as pivot request and find a pivot
cover P (r) using the pivot cover algorithm in Section 4.2;
(b) colour requests in P (r) using a new colour.
4. Consider the rings other than the root ring in the order of increasing DFS
numbers,
(a) select a longest uncoloured request r belonging to the current ring as
pivot request and find a pivot cover P (r) for the requests belonging to
this ring and not coloured yet as in Step 3a;
(b) colour each request in P (r) using the first-fit colouring.
4.4 Approximation algorithm
We give in this section an outline of the approximation algorithm for wavelength
allocation in arbitrary tree of rings networks presented in [Bian et al. (June 2004)].
We use this algorithm for comparison in the experimental evaluation of our heuristic
presented in Section 4.3. The input is a set of paths P on a tree of rings
network T R and the output is a valid colouring from W = {λ 1 , λ 2 , . . . , } to P .
The wavelength allocation framework consist of the following three steps.
1. Fix a DFS ordering of all nodes of T R starting from a node of degree two.
2. Process the root node v.
3. Process the other nodes u in DFS order.
Let r 0 be the ring which contains u and the parent of u
71

4.4 Approximation algorithm
(a) Colour the set P ′ of uncoloured paths using node u and fully within
ring r 0 .
(b) Colour the set P ′′ of other uncoloured paths using node u.
u +
b
w
−
processed nodes
r 2
a
u
r 1
c
u
−
r 0
w
unprocessed nodes
(w
−
,w)
−
Special links
(u ,u)
long path
short path
Figure 4.3: A tree of rings with three rings, r 0 , r 1 and r 2 , sharing node u, showing a
long path {a, b}, short path {u, c}, two special links (u − , u) and (w − , w), processed
(dark) nodes and unprocessed (plain) nodes.
The nodes in a ring of T R are processed in clockwise direction. For a node v
we denote by v − and v + the counter-clockwise and clockwise neighbor for v in the
ring of T R respectively. Processing a node v refers to colouring the uncoloured
paths using edges incident to v. In Step 2, the paths using edge (v − , v) are
allocated colours using the first-fit colouring. In addition, uncoloured paths using
the edge (v, v + ) are coloured by the first-fit colouring using a new set of colours
(i.e., colours of W not used on paths using the edge (v − , v)).
The parent of node u in DFS order is node u − in the current ring r 0 . If
node u is shared by k + 1 rings, the other k rings are denoted by r i , 1 ≤ i ≤ k.
Two links are distinguished as special links, that is, the links (u − , u) and (w − , w),
in which the nodes u − and w are processed and the nodes u and w − are yet
to be processed, as shown in Figure 4.3. In Step 3, any unprocessed node u
is considered in DFS order as follows. Let P 0 be the set of paths using special
links (u − , u) or (w − , w). In Step 3a, the set of paths P ′ is coloured using
the colours of W not used for P 0 by the first-fit colouring. Now, an uncoloured
path using the node u is either on one ring (short path) or two rings (long path)
72

4.4 Approximation algorithm
containing the node u respectively, as Figure 4.3 depicts.
Step 3b, transform
the problem of colouring uncoloured paths using node u to the edge-colouring
problem of a multigraph G u (V, E) with rings r i as nodes in V and short and
long paths as edges in E. To exclude loops in G u , a node s i is included in
V (G u ) for every r i in V (G u ). More formally, V (G u ) = {r i , s i : 0 ≤ i ≤ k}, and
E(G u ) = {e = {r i , r j } : e is a long path on rings r i and r j , 0 ≤ i < j ≤ k }.
∪ {e = {r i , s i } : e is a short path on ring r i , 0 ≤ i ≤ k }.
There is a one-to-one correspondence between the paths using u and the edges in
G u . Let γ(G u ) be a valid edge colouring for G u and C γ = {γ 1 , γ 2 , . . .} the set of
colours used in γ(G u ). A mapping f : C γ → W is defined to colour paths using
u as follows. Let P 1 be the set of coloured paths using u prior to Step 3b, and
W 1 be the set of colours allocated to the paths of P 1 .
• For each γ i allocated to edge e ∈ G u corresponding to a path in P 1 with
colour λ j ∈ W 1 , f(γ i ) = λ j .
• For each unmapped γ i , find a distinct λ j ∈ W \W 1 by the first-fit colouring,
and f(γ i ) = λ j .
The WA framework described in this section leads to a 3-approximation and
2.75-approximation WA algorithm for arbitrary tree of rings, using the edgecolouring
results in Theorem 2 and 3 respectively [Bian et al. (June 2004)].
Theorem 2 (Shannon (1949)) For a multi-graph G, a valid edge-colouring using
at most ⌊ 3∆(G)
2
⌋ colours can be found in O(|E(G)|(∆(G) + |V (G)|)) time.
Theorem 3 (Nishizeki and Kashiwagi (August 1990)) For a multi-graph
G, a valid edge-colouring using at most max{⌊ (11∆(G)+8) ⌋, l(G)} colours can be
10
found in O(|E(G)|(∆(G) + |V (G)|)) time, where
l(G) = max
{
L(H) =
⌊
|E(H)|
⌊|V (H)|/2⌋
⌋
}
: H is a subgraph of G with |V (H)| ≥ 3
is a lower bound on the number of colours for edge-colouring of G.
In the following sections we describe the inputs used for experimental comparison
of performance of the algorithms presented above. In Section 4.5 we describe
the routing instance and network instance input generators, and in Section 4.6
we report the numerical results and compare the performance of the algorithms
that have been considered.
73

4.5 Input generator
4.5 Input generator
For request instances we use randomly generated requests of Uniform and Gaussian
distribution defined in Section 2.4.1. For the routing decisions we use the
shortest path routing and the load balancing routing as described in Section 2.4.2.
Let T be a tree. We consider tree of rings T R underlying the tree T (i.e., there
is a ring in T R corresponding to each node in T and two rings share a node if
the pair of corresponding nodes in T are adjacent), constructed recursively as
follows. Start by constructing a ring of T R corresponding to the chosen centre
node of T . Subsequently, put the centre node into a stack data structure. While
the stack is non-empty pop a stack node v and process its children nodes in the
following way. For each child node of v in T add a new ring c of T R such that c
shares a node with the ring corresponding to the node v. Put the child node into
the stack.
We consider instances of tree of rings whose underlying tree is randomly generated.
The input is the number of rings and an integer range (interval) from
which the number of nodes for each ring is chosen uniformly at random. We
start with a forest of trivial components whose cardinality is the number of rings,
then repeatedly choose a pair of nodes uniformly at random from two distinct
connected components of the forest and find a union of the two disconnected tree
components to form a large tree by adding an edge adjacent to the pair of nodes
chosen. Eventually, the input forest is transformed into a tree T . A centre node
for the tree T is determined and the tree of rings instance is generated from the
tree T as above.
4.6 Numerical results and performance evaluation
In this section we report numerical results and compare the wavelength performance
of the approximation algorithm and the LP-DFS algorithm. Tables 4.1
and 4.2 show the average load (W-LB) and the average number of wavelengths
used by each algorithm on tree of rings with 10 rings, where the number of nodes
on a ring is taken from the intervals 8 − 16 and 16 − 32 respectively. The average
74

4.6 Numerical results and performance evaluation
is taken over 10 random instances. We considered two implementations of the approximation
algorithm framework of Section 4.4. The first implementation used
the 1.5-approximation edge colouring algorithm by [Shannon (1949)] (APX0),
and the second implementation used the first-fit colouring heuristic for edgecolouring
of multigraphs (APX1). We observed that the two implementations
used the same average number of wavelengths for all instances we considered. In
the following we denote the approximation algorithms by APX when referring to
both implementations.
The numerical results in Tables 4.1 - 4.2 show that the algorithm APX uses
significantly many wavelengths to satisfy an instance compared to the algorithm
LP-DFS. The LP-DFS algorithm consistently uses number of wavelengths close
to the lower bound (W-LB). For instances we considered the algorithm LP-DFS
used an average number of wavelengths that is within 6% over the lower bound
W-LB for all instances. The average number of wavelengths used by the algorithm
APX, on the other hand, is up to 50% over the lower bound W-LB. The higher
number of wavelengths requirement by the algorithm APX is likely caused by a
technical requirement for each colouring phase (in Steps 2, 3a and 3b of Section
4.4) to use new set of colours to allocate uncoloured paths that share a node
with paths coloured during previous phases. For example, it is possible that two
consecutive colouring phases process sets of paths that share a node but do not
mutually overlap at any link of the network. In such a case the algorithm APX
may use twice as much the number of colours than would possibly be needed.
The fact that the two implementations of the approximation algorithm framework
of Section 4.4 required the same number of wavelengths (i.e., implementations
using the first-fit colouring and the edge colouring algorithm by [Shannon
(1949)]), make it likely that a multigraph instance generated by Step 3b of the
algorithm APX can be easily coloured. But this implies that implementing the
framework by the edge colouring algorithm of [Nishizeki and Kashiwagi (August
1990)] is not likely to improve the number of wavelengths used by the algorithm
APX. The implementations of the approximation algorithm framework also require
different running time as Table 4.3 shows, where the APX1 implementation
is slightly faster than the APX0 counterpart.
75

4.6 Numerical results and performance evaluation
Distribution |R| W-LB LP-DFS APX
128 44 48 58
256 81 86 109
Uniform-SP 512 170 182 224
1024 339 342 506
128 39 42 52
256 99 109 132
Uniform-LB 512 168 180 223
1024 321 324 475
128 20 23 32
256 45 46 67
Gaussian-SP 512 93 94 137
1024 158 160 261
128 20 20 28
256 42 45 58
Gaussian-LB 512 85 90 121
1024 136 136 233
Table 4.1: Average number of wavelengths for tree of rings with 10 rings, where
number of nodes on each ring is taken uniformly at random from the interval
8 − 16. The instances have 128, 256, 512 and 1024 requests.
76

4.6 Numerical results and performance evaluation
Distribution |R| W-LB LP-DFS APX
1024 214 216 281
Uniform-SP 1536 422 427 541
2048 668 677 856
2560 862 869 1103
1024 340 346 447
Uniform-LB 1536 567 575 738
2048 608 616 798
2560 786 798 1027
1024 136 140 196
Gaussian-SP 1536 214 218 309
2048 266 275 404
2560 329 341 500
1024 118 124 171
Gaussian-LB 1536 176 186 252
2048 237 252 345
2560 292 304 422
Table 4.2: Average number of wavelengths for tree of rings with 10 rings, where
number of nodes on each ring is taken uniformly at random from the interval
16 − 32. The instances have 1024, 1536, 2048 and 2560 requests.
Distribution LP-DFS APX0 APX1
Uniform-SP 10.59 99.73 69.63
Uniform-LB 8.57 88.87 61.67
Gaussian-SP 1.52 31.24 24.77
Gaussian-LB 1.91 23.95 19.82
Table 4.3: Average time used (in seconds) for tree of rings network with 10 rings,
where number of node on each ring is taken uniformly at random from the interval
16−32, and instance of 2048 requests (using 1.0GHZ AMD Duron Processor with
112.0MB RAM).
77

4.6 Numerical results and performance evaluation
1.3
1.25
LP-DFS
APX
# WAVELENGTHS/W-LB
1.2
1.15
1.1
1.05
1
300 450 600 750 900 1050 1200 1350
W-LB
Figure 4.4: Wavelength performance for instances with Uniform-SP distribution
of requests on tree of rings with interval 16 − 32.
1.3
1.25
# WAVELENGTHS/W-LB
1.2
1.15
1.1
1.05
LP-DFS
APX
1
300 450 600 750 900 1050 1200 1350
W-LB
Figure 4.5: Wavelength performance for instances with Uniform-LB distribution
of requests on tree of rings with interval 16 − 32.
78

4.6 Numerical results and performance evaluation
# WAVELENGTHS/W-LB
1.7
1.65
1.6
1.55
1.5
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
LP-DFS
RG
LG
APX
100 200 300 400 500 600 700
W-LB
Figure 4.6: Wavelength performance for instances with Gaussian-SP distribution
of requests on tree of rings with interval 16 − 32.
1.65
1.6
1.55
1.5
# WAVELENGTHS/W-LB
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
100 200 300 400 500 600
W-LB
LP-DFS
RG
LG
APX
Figure 4.7: Wavelength performance for instances with Gaussian-LB distribution
of requests on tree of rings with interval 16 − 32.
79

4.6 Numerical results and performance evaluation
We report additional numerical results for different WA heuristics for tree
of rings networks in Appendix A.2. We considered networks on 10 rings with
ring nodes intervals 8 − 16 and 16 − 32, and additional heuristics LG and RG
listed in Table 3.4. For Uniform distribution of requests, the performance of
all algorithms we considered except APX was too close: they all always used
the number of wavelengths differing only by a margin of 1 or 2 wavelengths.
The differences between the performance of different algorithms start showing for
Gaussian distribution of requests (see Tables A.5 and A.6), where the algorithm
LP-DFS also gives overall best performance.
We summarize the performance results for the algorithms LP-DFS and APX in
the plots in Figures 4.4, 4.5, 4.6 and 4.7, for a network with 10 rings and ring nodes
interval 16 − 32 and Uniform-SP, Uniform-LB, Gaussian-SP and Gaussian-LB
distribution of requests. The plots for Gaussian-SP and Gaussian-LB distribution
of requests include the heuristics LG and RG. The plots show that heuristic
algorithms give overall the best performance.
80

Chapter 5
Undirected Rings with Cluster
Wavelength Conversion
In this chapter we study the WA problem for undirected rings with cluster wavelength
conversion capabilities where request paths are given a priori, and propose
WA algorithms that improve upon those presented in [Cavendish and Sengupta
(2002)] for this problem. We experimentally compare with the best performance
algorithm in [Cavendish and Sengupta (2002)] and show improvement of up to
100% and 2% for number of conversions and wavelengths requirement respectively.
Our algorithms are adaptation of the pivot based methods for ring networks without
wavelength conversion presented in Chapter 3, and the wavelength conversion
framework for algorithms for this problem proposed by [Cavendish and Sengupta
(2002)]. For our experiments we use instances with Uniform and Gaussian distribution
of requests defined in Section 2.4.1, and for the routing decisions we
use the shortest path routing and the load balancing routing described in Section
2.4.2.
5.1 Cluster wavelength conversion model
As described in Section 2.2.2.2, wavelength converters allows shifting a wavelength
from one to another. In [Cavendish and Sengupta (2002)] they considered
ring networks with heterogeneous conversion capabilities in which different nodes
of the ring network may have different wavelength conversion capabilities (i.e.,
81

5.1 Cluster wavelength conversion model
with converters of arbitrary degrees of conversion). Ring networks with cluster
wavelength conversion capabilities are an important class of ring networks with
heterogeneous wavelength conversion capabilities considered in [Cavendish and
Sengupta (2002)], which are defined as follows. Let K be an even integer, and
W = {0, 1, . . . , K−1} a collection of available wavelengths on each link of the ring
network. Wavelengths in W can be partitioned into K 2 subsets, W i, 0 ≤ i ≤ K 2 −1,
such that W 0 = {0, 1}, W 1 = {2, 3} . . ., W K
−1
= {K −2, K −1}. In ring networks
2
with cluster wavelength conversion capabilities each node allows for wavelength
conversion only among wavelengths within a partition (cluster W i ). For example,
for the collection W wavelength conversion 0 → 1 and 1 → 0 is allowed, however,
the wavelength conversion 0 → 2, is not allowed. The WA problem for rings with
cluster wavelength conversion capabilities is NP-complete [Cavendish (Nov 12-16,
2001, Beijing, China)].
5.1.1 Wavelength allocation with conversion framework
In this section we describe a wavelength conversion framework that allocates
wavelength to requests one link at a time using the first-fit colouring. A wavelength
shift is attempted on a successive link if the current wavelength become unavailable
and wavelength conversion being possible at current node. Let r = (i, j)
be a request being allocated wavelength using the wavelength conversion framework.
The first fit colouring is used to find a suitable wavelength w p (available
wavelength of least index) at first link i traversed by the request r. On successive
link span the wavelength w p is used, unless w p is not available beyond a certain
node, v, in which case a wavelength conversion w p → w q is attempted at v,
from wavelength w p to wavelength w q , such that w q is the least index wavelength
among all possible wavelength conversions for w p at node v. Once wavelength w q
is used any successive link span for r will attempt to use wavelength w q as before.
Suppose that at node v no suitable wavelength conversion is available. We
backtrack to the last node where a wavelength choice was made and attempt to
choose an alternate wavelength with next least index among possible wavelength
conversions. Backtracking attempt continue until all possible wavelength choices
are exhausted, in which case the request r is allocated using a new wavelength.
82

5.2 Wavelength allocation with pivot method and conversion
framework
5.2 Wavelength allocation with pivot method
and conversion framework
In this section we describe different WA algorithms for rings with cluster wavelength
conversion that combine the pivot method of the algorithms presented in
Chapter 3 and the wavelength conversion framework described in Section 5.1.1.
The notion of pivot covers and algorithm for finding a pivot cover with respect to
a request of a ring network were described in Section 3.3. Our main strategy is to
first allocate wavelength to as many requests as possible using the pivot method
and use the wavelength conversion framework to colour any requests remaining
uncolored yet. That is, our algorithms attempt to make as few calls possible
to the wavelength conversion framework so as to minimize the number of wavelength
conversion requirements since a wavelength conversion is considered as an
expensive operation. In addition, making more calls to the pivot method gives
advantage of improved wavelength utilization.
5.2.1 Long pivot with wavelength conversion
We describe the long pivot with wavelength conversion (LP-WC) algorithm for
the WA problem for rings with cluster wavelength conversion capabilities. Let
L be the network load. The LP-WC algorithm finds L pivot covers which are
assigned using L colours, subsequently, any remaining uncoloured request is allocated
wavelength through the wavelength conversion framework of Section 5.1.1.
The LP-WC algorithm colours requests in the following way.
1. For 1 ≤ i ≤ L, find a pivot cover P (r) with respect to a longest uncoloured
request r. Assign requests in P (r) a new colour.
2. Assign colours to any remaining requests using the wavelength conversion
framework of Section 5.1.1.
5.2.2 Fixed pivot with wavelength conversion
We describe the fixed pivot with wavelength conversion (FP-WC) algorithm for
the WA problem for rings with cluster wavelength conversion capabilities. Let
83

5.3 Related algorithms
e be distinguished as the pivot link and L e the number of requests traversing
the link e. The FP-WC algorithm finds L e pivot covers that are assigned using
L e colours. Any remaining uncolored requests are allocated colours through the
wavelength conversion framework of Section 5.1.1. The FP-WC algorithm colours
requests in the following way.
1. For 1 ≤ i ≤ L e , find a pivot cover P (r) with respect to a longest uncoloured
request using the pivot link. Assign requests in P (r) a new colour.
2. Assign colours to any remaining requests using the wavelength conversion
framework of Section 5.1.1.
As in Section 3.3.2, we consider FP-WC algorithm with a link traversed by
the minimum number of requests as the pivot link, denoted by minFP-WC. If
the ring network size n is small (practical rings typically contain from 3 to 30
nodes [Carpenter and Cosares (2002)]), then it may be feasible to try each link as
the pivot link, and select the best colouring achieved. This leads to an iterative
variant of the FP-WC algorithm, referred to as the iterative fixed pivot with
wavelength conversion algorithm (IFP-WC). The worst case running time for
the IFP-WC algorithm increases by a factor of n to that of the basic FP-WC
algorithm.
5.3 Related algorithms
In this section we describe the best performance WA algorithm in [Cavendish and
Sengupta (2002)] called Spiral algorithm (SPR) for rings with cluster wavelength
conversion capabilities. We experimentally compare the SPR algorithm with our
algorithms for performance in Section 5.4.
The SPR algorithm uses the first-fit colouring and the wavelength conversion
framework of Section 5.1.1 to allocate wavelength to requests of a ring network.
Starting with an arbitrary request the SPR algorithm colours successive requests
following the ring in clockwise direction as follows.
1. Assign colour to the first request using the first-fit colouring.
84

5.4 Numerical results and performance evaluation
2. Let r = (i, j) be the last request to be coloured. Select the next uncoloured
request p = (k, l) such that k has the shortest clockwise distance from j
in the ring among all uncoloured requests. Ties are broken by selecting an
uncoloured request with shortest length.
3. Assign colour to p using the wavelength conversion framework of Section
5.1.1.
5.4 Numerical results and performance evaluation
In this section we report numerical results and performance comparison for the
algorithms LP-WC, minFP-WC, IFP-WC and SPR. We denote by W and C the
average number of wavelength and number of conversion requirement by each
algorithm. The average is taken over ten random instances. Tables 5.1 and
5.2 show the numerical results for all algorithms considered for instances with
Uniform-SP, Uniform-LB, Gaussian-SP and Gaussian-LB distribution of requests
and networks with 32 and 64 nodes.
As was noted in Section 3.5 observe from Tables 5.1 and 5.2 that Gaussian
instances are easily coloured under the shortest path routing : all algorithms use
optimal number of colours in our experiments. In addition all algorithms except
the SPR algorithm do not perform any wavelength conversion, and the number
of conversions performed by the SPR algorithm are fewer (between 1 and 3).
For other instance types we consider the numerical results show that algorithms
LP-WC and IFP-WC give best wavelength performance with up to 2%
better than the SPR algorithm. The worst performance minFP-WC algorithm
amongst pivot based algorithms also outperforms the SPR algorithm on all instances
except the Uniform-LB instances. For all instances the algorithm LP-WC
does not perform any conversion, whereas the algorithm minFP-WC and IFP-
WC performs only up to 2 and 1 conversions respectively. On the other and, the
algorithm SPR performs a significant number of conversions for most instances we
considered. But the iterative algorithm IFP-WC has an extra factor n (the size
of the network) in its running time in comparison to the none-iterating algorithm
85

5.4 Numerical results and performance evaluation
SPR minFP-WC LP-WC IFP-WC
Distribution n |R| W-LB W C W C W C W C
128 41 42 14 42 0 42 0 41 0
32 256 78 79 35 79 0 79 0 78 0
Uniform 512 156 157 66 156 0 157 0 156 0
with SP 1024 288 291 143 290 1 288 0 288 0
128 42 43 18 43 1 43 0 42 1
64 256 77 78 37 78 1 78 0 77 0
512 153 156 84 155 1 154 0 153 1
1024 282 288 167 286 2 284 0 283 1
128 41 42 20 43 1 42 0 42 0
32 256 79 81 40 82 1 80 0 79 1
Uniform 512 158 161 82 162 1 160 0 159 0
with LB 1024 309 315 173 316 1 311 0 311 0
128 41 43 22 44 0 42 0 42 1
64 256 79 83 42 85 1 83 0 81 1
512 154 161 94 162 2 158 0 159 1
1024 309 319 192 323 2 313 0 316 1
Table 5.1: Average number of wavelengths (W) and conversions (C) for instances
with Uniform distribution of requests. The instances have 128, 256, 512 and 1024
requests.
86

5.4 Numerical results and performance evaluation
SPR minFP-WC LP-WC IFP-WC
Distribution n |R| W-LB W C W C W C W C
128 95 95 1 95 0 95 0 95 0
32 256 187 187 1 187 0 187 0 187 0
Gaussian 512 375 375 3 375 0 375 0 375 0
with SP 1024 749 749 3 749 0 749 0 749 0
128 93 93 2 93 0 93 0 93 0
64 256 189 189 1 189 0 189 0 189 0
512 377 377 1 377 0 377 0 377 0
1024 746 746 2 746 0 746 0 746 0
128 53 54 12 54 0 54 0 54 0
32 256 109 111 26 110 0 110 0 110 0
Gaussian 512 210 213 46 212 0 210 0 210 0
with LB 1024 427 436 106 432 0 428 0 428 0
128 56 57 13 57 0 57 0 56 0
64 256 108 110 26 109 0 109 0 108 0
512 214 218 49 216 0 215 0 215 0
1024 422 428 102 424 0 422 0 422 0
Table 5.2: Average number of wavelengths (W) and conversions (C) for instances
with Gaussian distribution of requests. The instances have 128, 256, 512 and
1024 requests.
Distribution LP-WC minFP-WC SPR
Uniform-SP 1.29 0.69 1.43
Uniform-LB 1.78 0.99 2.02
Gaussian-SP 0.40 1.06 0.58
Gaussian-LB 1.98 2.38 3.35
Table 5.3: Average time used (in seconds) for ring network with 64 nodes and
instance of 2048 requests (using 1.0GHZ AMD Duron Processor with 112.0MB
RAM).
87

5.4 Numerical results and performance evaluation
LPA, which is an additional advantage of the algorithm LPA. Table 5.3 shows
the average time used for none-iterating algorithms considered for ring network
with 64 nodes and instance of 2048 requests.
We report additional numerical results for WA heuristics considered in this
chapter in Appendix A.3 for networks with 16, 32, 64 and 96 nodes. The differences
between the performance grow when the size of the network increases
(see Tables A.7, A.8, A.9 and A.10). For all input instances of Gaussian-SP type
which we have considered, all algorithms have always found optimal solutions on
all networks we considered.
We summarize the performance results for the algorithms LP-WC, minFP-
WC, and SPR in the plots in Figures 5.1, 5.2 and 5.3, for ring networks with 64
nodes and Uniform-SP, Uniform-LB and Gaussian-LB distribution of requests.
The plots show that algorithm LP-WC gives overall the best performance, followed
by the algorithms minFP-WC and SPR. The good performance of the pivot
based algorithms, and especially the LP-WC algorithm, could be attributed to
the fact that a pivot cover potentially decreases the network load by 1 (leading
to a better utilization of the colours), and the way the pivot covers are selected
gives priority to longer requests that are potentially difficult to colour. In addition,
pivot covers lead to fewer number of conversion requirements for the pivot
based algorithms.
88

5.4 Numerical results and performance evaluation
1.03
LP-WC
minFP-WC
SPR
# WAVELENGTHS/W-LB
1.02
1.01
1
150 300 450 600 750 900
W-LB
(a)
# CONVERSIONS/W-LB
0.675
0.6
0.525
0.45
0.375
0.3
0.225
0.15
0.075
0
150 300 450 600 750 900
W-LB
LP-WC
minFP-WC
SPR
(b)
Figure 5.1: Wavelength and conversion performance for instances with Uniform-
SP distribution of requests on 64 nodes ring network. (a) Number of Wavelengths
(b) Number of Conversions.
89

5.4 Numerical results and performance evaluation
1.08
# WAVELENGTHS/W-LB
1.07
1.06
1.05
1.04
1.03
1.02
LP-WC
minFP-WC
SPR
1.01
1
150 300 450 600 750 900 1050
W-LB
(a)
# CONVERSIONS/W-LB
0.675
0.6
0.525
0.45
0.375
0.3
0.225
0.15
0.075
0
LP-WC
minFP-WC
SPR
150 300 450 600 750 900 1050
W-LB
(b)
Figure 5.2: Wavelength and conversion performance for instances with Uniform-
LB distribution of requests on 64 nodes ring network. (a) Number of Wavelengths
(b) Number of Conversions.
90

5.4 Numerical results and performance evaluation
1.03
LP-WC
minFP-WC
SPR
# WAVELENGTHS/W-LB
1.02
1.01
1
150 300 450 600 750 900 1050 1200 1350
W-LB
(a)
0.3
# CONVERSIONS/W-LB
0.225
0.15
0.075
LP-WC
minFP-WC
SPR
0
150 300 450 600 750 900 1050 1200 1350
W-LB
(b)
Figure 5.3: Wavelength and conversion performance for instances with Gaussian-
LB distribution of requests on 64 nodes ring network. (a) Number of Wavelengths
(b) Number of Conversions.
91

Chapter 6
Undirected Multi-fiber Trees
A Multi-fiber network allows more than one optic fibers per link making more
wavelengths available to satisfy a given instance of connection requests. When
more than one fibers per link is used to satisfy an instance a natural optimization
problem is to minimize the overall number of fibers requirement on the network.
This problem was first studied by [Nomikos et al. (2001)] and showed that it
can be optimally solved for path (chain) networks and is NP-hard for general
networks. They presented a 2-approximation algorithm for star and ring networks
respectively. In this chapter we adapt the techniques used in [Nomikos
et al. (2001)] for star and chain networks to multi-fiber tree networks and propose
a 2-approximation algorithm for spider trees (also independently given in
[Potika et al. (Nicosia, Cyprus, Nov 8–10, 2001)]), a 3-approximation and 4-
approximation algorithm for a tree such that any path has at most two and three
nodes of degrees greater than 2 respectively. Using this technique we obtain a
log n-approximation and n-approximation algorithm for k-ary and arbitrary trees
with n nodes respectively. Improved results for arbitrary trees are presented in
[Erlebach et al. (2003)] and later in [Chekuri et al. (2003)] with a (log n + 1)-
approximation and 4-approximation ratio respectively.
6.1 Tree network model
Trees and spider trees are defined in Section 2.3. A tree network is modeled by
a tree T = (V, E) where the nodes in V are network nodes and the edges in E
92

6.1 Tree network model
are network links. A tree T is said to be branched tree if there is a non-empty
subset of nodes in V on degrees of at least 3, referred to as branched nodes. An
internal node of T has degree of at least 2 and a leaf node of T has degree equals
1. For example, a path network is a tree network with no branch node and a star
network is a tree network with one branch node, and is the only internal node.
A spider network is a branched tree network with exactly one branch node and
possibly many internal nodes. Figure 6.1 depicts a spider graph with a branch
node v. A path on a tree T is a sequence of nodes v 1 , v 2 , v 3 , . . . , v p such that
{v 1 , v 2 }, {v 2 , v 3 }, . . . , {v p−1 , v p } are edges of T and the v i are distinct. We denote
by s(T ) the maximum number of branch nodes on any path of T .
v
Figure 6.1: A spider tree with branch node v.
The problem of minimizing the overall number of optical fibers in a multi-fiber
network can be modeled by path multi-colouring problem (PCM) [Nomikos et al.
(2001)]. For the PCM problem we are given a network N, a routing R for an
instance I on N as a set of paths P . A link of N can be realized using one or
more optic fibers each supporting fixed number of wavelengths (colours), labeled
1, . . . , W . Given a PCM instance, we wish to satisfy the set of paths in P such
that at any link a wavelength, w, for 1 ≤ w ≤ W , is assigned to zero or more
paths. The goal is to minimize the multiplicity, λ(w), the number of paths sharing
colour w at any link. Note that minimizing λ(w) also imply a minimum number
of optic fibers is used by the network. For path networks the PCM problem can
be optimally solved and for star and rings there is a 2-approximation algorithm
[Nomikos et al. (2001)].
A path subgraph on a tree with a leaf and branch node as end-nodes is referred
to as a tree chain. Let (T, P, W ) be a PCM instance for a tree network T where P
is a set of request paths on T such that a path p ∈ P connects a pair of nodes {u, v}
93

6.2 PCM algorithm for spider tree
in T , and W is a fixed number of wavelengths supported by fibers on the links
of T . We relate a tree chain to a chain network and present a 2-approximation
algorithm for spider trees and a O(s(T ))-approximation algorithm for arbitrary
trees by adapting the techniques in [Nomikos et al. (2001)] for star and chain
networks that involves a reduction of the PCM problem to edge colouring of
bipartite multigraphs (ECBM) problem. There is a polynomial time reduction
from PCM to ECBM [Nomikos et al. (2001)], and optimal algorithm for the
ECBM problem [Cole et al. (2001)].
6.2 PCM algorithm for spider tree
A spider tree has a single branch node and several internal and leaf nodes. Unlike
a star tree where a tree chain has single edge, a tree chain of a spider tree may
have more than one edge. The PCM algorithm for spider tree networks (APR1)
is an adaptation of the PCM algorithms for chain and star networks proposed in
[Nomikos et al. (2001)]. The algorithm APR1 uses two phases to which paths
are oriented and grouped into W paths at each node where paths begin or end,
followed by a PCM reduction to ECBM.
6.2.1 Path orientation and grouping
Given a P MC instance (T, P, W ) for a spider tree T (V, E), where P is a set
of paths in T and W a fixed number of available wavelengths on a fiber. The
path orientation and grouping phase partitions the paths of P traversing an edge
e ∈ T into two sets P ′ e and P ′′
e , and define groups of W paths with same orientation
beginning or ending at the same node, using the following five steps.
1. At each non-leaf node v ∈ T , if there exists a pair of link disjoint paths
p 1 = {u, v} ∈ P and p 2 = {v, w} ∈ P , then replace the paths p 1 and p 2 of
P with a new path p = {u, w}.
2. Assign orientation to paths of P arbitrarily.
3. At every edge e ∈ T partition paths of P traversing e into the sets P e
′
and P e
′′ based on their orientation. Note that at the edge e one or both
94

6.2 PCM algorithm for spider tree
partition sets may be empty. The subscript e for a partition set is dropped
when referring to a partition spanning a tree chain.
4. Let L ′ e and L ′′
e be the number of paths for the path partition P e ′ and P e
′′
respectively. Add L ′ e mod W and L ′′
e mod W dummy dipaths (directed paths
of length 1) in P e ′ and P e ′′ respectively with corresponding orientation. Thus
we have that the number of paths on each edge partition is a multiple of
W .
5. On each tree chain if there exists a pair of link disjoint dipaths p 1 = (u, v)
and p 2 = (v, w), then replace the paths p 1 and p 2 of P with a new path
p = (u, w). Thus at each node (except possibly a branch node), paths on
the same tree chain partition only begin or end but not both. By Step 4
the number of paths that end or begin at such a node are in multiples of
W . Figure 6.2 depicts a node v incident to links e and e ′ . The dashed lines
are oriented paths on the same partition, such that m paths end at v and
n paths bypass node v.
6. At each node of T group arbitrarily the paths that begin or end at a node
into W paths. By Steps 1 and 5 we have that paths on a group share at
least one link of T and that a path belongs to a group of exactly W paths
at its source node and sink node (except possibly for a path with branch
node as the source or sink node, in which case it may belong to a group of
less than W paths).
6.2.2 PCM reduction to ECBM
A colouring of the original paths in P can be obtained from the edge colouring
of a bipartite multigraph H = (A, B, E) in the following way.
1. Define a bipartite multigraph H = (A, B, E) from the paths of T as follows.
At a node v of T where paths begin there is a node in A for each path group
at v, and at a node w of T where paths end there is a node in B for each
path group at w. For every path of T there is an edge in E incident to a pair
of nodes corresponding to the path groups where the path begins (a node
95

6.2 PCM algorithm for spider tree
} m } n
e
v
e’
Figure 6.2: A set of paths on the same partition using node v incident with edges
e and e ′ . There are m + n paths traversing the edge e, with m paths ending at
node v and n paths bypassing the node v to edge e ′ . By Step 4, the edges e and
e ′ have p · W and q · W paths, for some integers p and q. Hence the number m of
paths ending at v is equal to (p − q) · W .
of A) and ends (a node of B). The multigraph H is W -degree bipartite
multigraph since a path group has at most W paths.
2. Use the bipartite edge colouring algorithm in [Cole et al. (2001)] to optimally
colour the edges of H using W colours. By Step 1, the proper edge
colouring of H corresponds to a legal colouring of the paths of P using W
colours.
6.2.3 Bounding performance of APR1 algorithm
In this section we describe the performance analysis of the algorithm APR1 given
in Section 6.2. Let L e be the number of paths traversing a link e ∈ T . The
optimal algorithm, OPT, requires at least ⌈ L e
W
⌉
number of optic fibers to satisfy
paths traversing the link e, since an optic fiber supports at most W paths. The
overall number of optic fibers C opt required by the algorithm OPT is therefore,
C opt ≥ ∑ ⌈ ⌉
Le
.
W
e∈E
In Step 3 the algorithm APR1 partition the paths of P traversing the link
e ∈ T into two sets P e ′ and P e ′′ . Let L ′ e and L ′′
e be the number of paths traversing
the link e in P e ′ and P e
′′ respectively. The ⌈number ⌉ ⌈of ⌉optic⌈ fibers⌉required by the
algorithm APR1 at the link e is therefore, L ′
e
+ L ′′
e
≤ L ′
e +L ′′
e
+1 = ⌈ L e
⌉
W W
W
W +1.
96

6.3 Extension to other tree networks
Hence, the overall number of optic fibers C apr1
APR1 is,
≤ ∑ e∈E
(⌈ ⌉ )
Le
+ 1 = ∑ W
e∈E
required by the algorithm
⌈ ⌉
Le
+ ∑ 1 ≤ C opt + |E|.
W
e∈E
But then, |E| ≤ C opt , since each active link require an optical fiber. Therefore
C apr1 ≤ 2 · C opt , giving a 2-approximation algorithm APR1.
6.3 Extension to other tree networks
In this section we extend the algorithm described in Section 6.2 for spider tree
networks to trees with two or more branch nodes. Let (T, P, W ) be a PCM
instance for a tree T with s(T ) = h, for h ≥ 2, and let a branch node v be
distinguished as the root node for T . The root node for a tree T is determined
through a transformation of the tree T by identifying all nodes of degree 2 to
the tree T ′ with no nodes of degree 2. A centre node of T ′ (defined in Section
2.3) is designated as the root node of the tree T . We define a labeling for the
branch nodes of T with respect to the root node v as follows. The branch node v
is labeled 0. Let k be number of branch nodes contained in a path between the
root node v and a branch node w (the root node v and the node w also counted).
The branch node w is labeled k − 1. For example, the tree network in Figure
6.3 has s(T ) = 3, and suppose that node v is the root node labeled 0. All other
branch nodes receive label 1.
v
Figure 6.3: A tree T with s(T ) = 3.
A subtree of T rooted at a branch node u labeled k consists of nodes and
edges of the tree chains between the branch node u and the leaf-nodes of T with
97

6.3 Extension to other tree networks
no branch node of lower label than k. We say that a path of P is spanned by
a subtree rooted at a branch node u if all links traversed by the path are in
this subtree. We consider paths of P spanned by a subtree rooted at a branch
node independently, starting with subtrees rooted at branch nodes of highest
labels. The partial colouring of paths spanned by each subtree is combined into
a colouring of paths of P . Given a subset of paths spanned by a subtree rooted
at branch node u, the P MC problem is solved by grouping the paths into several
partitions, followed by a reduction of the PCM problem to the ECBM problem
as in Section 6.2.2. We first consider trees T with s(T ) = 3.
6.3.1 PCM algorithm for tree with s(T ) = 3
In this section we describe the PCM algorithm APR2 for trees with s(T ) = 3.
Figure 6.3 shows a tree T with s(T ) = 3 where any path has at most three
branch nodes. Let node v be distinguished as the root node for such tree T . The
node v is labeled 0 and other branch nodes are labeled 1. The paths of P is a
union of paths spanned by the different subtrees rooted at branch nodes labeled
1 called type 1 paths, and the paths spanned by the subtree rooted at the root
node labeled 0, called type 0 paths. Note that a subtree rooted at a node labeled
1 is homeomorphic to a spider tree. Thus we can apply the spider tree algorithm
of Section 6.2 to paths spanned by each of such subtree. A type 0 path may use
more than one branch nodes. We consider type 0 paths by applying Steps 1, 2,
and 3 of Section 6.2.1, in addition to the following two steps.
1. The nodes of a subtree rooted at a node labeled 0 are visited in postorder
(i.e., process all nodes of a tree by recursively processing all subtrees, then
finally processing the root). Let P ′ and P ′′ be chain partitions which maintains
sets of path groups with at most W paths for paths that begin and
end at a node of T respectively. At the current node add a path that begin
or end into a path group with less than W paths of the corresponding path
partition. If no path group with less than W paths exists, create a new
path group and add any remaining paths. If the current node is a branch
node other than the root of the subtree, then in addition perform Step 2.
98

6.3 Extension to other tree networks
2. Suppose that at current branch node a collection of path groups Q = {P i :
1 ≤ i ≤ k}, for k ≥ 0, with same orientation and from partitions of distinct
tree chains ending at the current node, have less than W paths. We merge
path groups of Q to obtain a new collection Q ′ = {P j ′ : 1 ≤ j ≤ h} of h
path groups, such that 0 ≤ h ≤ k, |P j| ′ ≤ W , and for any P s, ′ P q ′ ∈ Q ′ ,
W < |P s| ′ + |P q|, ′ for 1 ≤ s < q ≤ h. (6.1)
3. The type 0 paths are then coloured by reduction from PCM to ECBM as
in Section 6.2.2.
Lemma 2 For the set Q ′ of path groups defined in Step 2, at most one path group
has less than ⌈ ⌉
W
2 paths.
Proof : This follows from the condition of the Inequality 6.1, in Step 2.
Lemma 3 Let κ e = ∑ k
i=1 |P i| be the total number of paths for path groups of Q
in Step 2 traversing the edge e. Then h ≤ 2 · ⌈ κ e
W
⌉
.
Proof :
This follows as a consequence of Lemma 2 and application of the pigeonhole
principle.
6.3.2 Bounding performance of APR2 algorithm
The algorithm APR2 colours type 1 paths using the AP R1 algorithm for spider
trees. Thus for type 1 paths the analysis is similar to that of Section 6.2.3.
Algorithm AP R2 define path groups of at most W paths for type 0 paths. In
particular, Lemma 3 shows that at any link a partition has at most 2 · ⌈ κ e
w
⌉
path
groups with less than W paths, where κ e is the total number of paths for path
groups with less than W paths and same orientation traversing the link e. Let L e
be the total number of paths using link e. Furthermore, let P 0e and P 1e be a set of
type 0 and type 1 paths using link e respectively. The optimal algorithm OPT for
the PCM problem for the tree T with s(T ) = 3 uses at least ⌈ L e
⌉ ⌈ ⌉
W = |P0 e |+|P 1e |
W
optic fibers at link e.
In addition, we denote by P ′ 1 e
and P ′′
1 e
the path partitions at e for paths in P 1e ,
and P ′ 0 e
and P ′′
0 e
the path partitions at e for paths in P 0e . Let E 0 ⊆ E and E 1 ⊆ E
99

6.3 Extension to other tree networks
be the subset of edges of T , such that E 1 is the set of edges on the subtrees of
T rooted at the branch nodes labeled 1, and E 0 = E \ E 1 . By algorithm APR1
and Step 1 of algorithm ⌈ ⌉ APR2, ⌈ ⌉the ⌈ number ⌉ of ⌈ optic ⌉ fibers required at any link
|P ′
e ∈ E 1 is at most 1e | |P ′′ |P ′
+ + 0e | |P ′′
+ . Furthermore, by Step 2 of
W
W
1e |
W
W
algorithm APR2 and ⌈ Lemma ⌉ 3, the number of optic fibers required at any link
|P
e ′ ′ ⌈
0e ′
∈ E 0 is at most
|−κ′ e ′
|κ
⌉
⌉ ⌈
′
|κ
⌉
′′
+ 2 · + + 2 · , where κ ′ e and ′
e ′ |
W
0e |
W
⌈
|P ′′
0 e ′ |−κ′′ e ′
W
κ ′′
e are defined as in Lemma 3 for an edge ′ e′ ∈ E 0 .
Thus the total number of optic fibers C apr2 required by algorithm APR2 is,
≤ ∑ (⌈ |P
′
⌉ ⌈
1e
| |P
′′
⌉ ⌈
1
+ e
| |P
′
⌉ ⌈
+
0e
| |P
′′
⌉)
0
+ e
|
W W W W
e∈E 1
+ ∑ (⌈ |P
′ | − ⌉
0e ′ κ′ e ′
+ 2 ·
W
e ′ ∈E 0
≤ ∑ (⌈ |P
′
1e
| + |P 1 ′′
e
| + |P 0 ′ e
| + |P ′′
W
e∈E 1
0 e
|
⌈ ⌉ ⌈ κ
′
e ′ |P
′′
+
W
e ′ |
W
0 | − ⌉ ⌈ ⌉)
e ′ κ′′ e ′ κ
′′
e
+ 2 ·
′
W
W
⌉ )
+ 3 + ∑ ( ⌈ |P
′
2 ·
| ⌉ ⌈
0e ′ |P
′′
0
+ 2 ·
| ⌉)
e ′
W
W
e ′ ∈E 0
≤ ∑ (⌈ ⌉ )
|P1e | + |P 0e |
+ 3 + 2 · ∑ (⌈ ⌉ )
|P0e ′|
+ 1 .
W
W
e∈E 1 e ′ ∈E 0
Note that |P 1e | + |P 0e | and |P 0e ′| is the total number of requests traversing
and edge e ∈ E 1 and e ′ ∈ E 0 respectively. Hence,
C apr2 ≤ ∑ (⌈ ⌉ )
Le
+ 3 + 2 · ∑ (⌈ ⌉
Le ′
W
W
e∈E 1 e ′ ∈E 0
= ∑ (⌈ ⌉)
Le
+ 3 · |E 1 | + 2 · ∑ (⌈ ⌉)
Le ′
W
W
e∈E 1 e ′ ∈E 0
Theorem 4 APR2 is a 4-approximation algorithm.
)
+ 1
+ 2 · |E 0 |. (6.2)
Proof :
OPT is
The total number of optic fibers C opt require by the optimal algorithm
≥ ∑ ⌈ ⌉
Le
+ ∑ ⌈ ⌉
Le ′
.
W W
e∈E 1 e ′ ∈E 0
100

6.3 Extension to other tree networks
Let C opt
1 = ∑ ⌈
Le
⌉
W , and C
opt
0 = ∑ ⌈ ⌉
Le ′
. Note that |E
W
1 | ≤ C opt
1 and |E 0 | ≤
e∈E 1 e ′ ∈E 0
C opt
0 , since each active link requires at least one optic fiber. Substituting in the
Inequality 6.2 we obtain
C apr2 ≤ 4 · (C opt
1 + C opt
0 ) = 4 · C opt .
We can adapt the algorithm APR2 to a tree network T with s(T ) = 2 and
s(T ) > 3 respectively. We obtain the following results.
Corollary 1 For a tree T with s(T ) = 2, APR2 is a 3-approximation algorithm.
Proof :
Let T be a tree with s(T ) = 2 rooted at a degree 2 node on a path
between the two branch nodes (if none exists, create one by a subdivision of the
edge adjacent to the branch nodes).
The root node is labeled 0 and the two
branch nodes labeled 1. Apply the algorithm APR2 in Section 6.3.1 to T but
with type 0 paths all oriented in the same direction (this is possible since the root
node is of degree 2). The result follows by the analysis of Section 6.3.2 because
now all type 0 paths are in one partition.
Corollary ⌋ 2 For a tree T with s(T ) ≥⌋
3 odd and s(T ) ≥ 4 even, the APR2 is a
4· -approximation and a 4· −2-approximation algorithm respectively.
⌊
s(T )
2
Proof :
⌊
s(T )
2
We show by induction on s(T ). We describe for the case of s(T ) odd,
and for s(T ) even similar arguments apply in addition to arguments described in
Corollary 1 for root node and type 0 paths orientation. The result in Theorem 4
for a tree with s(T ) = 3 suffice for the basis case. Assume the result holds for an
arbitrary tree T with s(T ) ≤ h odd. We prove for the tree T with s(T ) = h + 2.
Let v be the root node of such tree T . The node v is labeled 0, where other
branch nodes labeled with respect to v as in Section 6.3, that is labeled from
the set {1, 2, . . . , h+1}.
Thus the paths of P is the union ⋃ q
2 k=0
{type k}, where
q = h+1.
2
Algorithm APR2 can be adapted to type 0 paths. Let P 0 ′ e
and P 0 ′′
e
be the
path partitions at e for type 0 paths. Let E 0 ⊆ E and E h+1 ⊆ E be the subset of
2
edges of T , such that E h+1 is the set of edges on the subtrees of T rooted at the
2
branch nodes labeled h+1,
and E 2 0 = E \ E h+1 . By Step 1 of algorithm APR2,
2
101

6.3 Extension to other tree networks
the number of optic fibers required for type 0 paths at any link e ∈ E h+1 is at
⌈ ⌉ ⌈ ⌉
2
|P ′
most 0 e | |P ′′
+ . Also by Step 2 of algorithm APR2 and Lemma 3, the
W
W
0e |
W
number ⌈ of ⌉ optic fibers required at any link e ′ ∈ E 0 for type 0 paths is at most
|P ′ ⌈
0e ′ |−κ′ e ′
|κ
⌉
⌉ ⌈
′
|κ
⌉
′′
+ 2 · +
+ 2 · , where κ ′ e ′ and κ′′ e are defined as
′
e ′ |
W
in Lemma 3 for edge e ′ ∈ E 0 .
⌈
|P ′′
0 e ′ |−κ′′ e ′
W
Let C apr3 be the total number of optic fibers required by algorithm APR2
for type 0 paths. Using the analysis in Section 6.3.2, C apr3 is bounded by the
inequality
≤ 2 · ∑
e ′ ∈E
(⌈
Le ′
e ′ |
W
⌉)
+ 2 · |E| ≤ 4 · OP T. (6.3)
W
On the other hand, for a subtree T ′ of T rooted at a branch node labeled
1 we have that s(T ′ ) ≤ h. By induction the algorithm APR2 require C apr4 ≤
4 · ⌊ ⌋ ⋃
h
2 · OP T optic fibers for the set of paths s h−1
j=0
{type j}, for s = , on such a
2
subtree. Thus the overall number of optic fiber required by the algorithm APR2
is C apr4 + C apr3 ≤ 4 · ⌊ ⌋ ⌊
h
2 · OP T + 4 · OP T = 4 · ( h
⌋ ⌊
2 + 1) · OP T = 4 · h+2
⌋
2 · OP T .
⌋
For complete k-ary tree T with n nodes,
⌊
s(T )
2
≤ log n. Thus APR2 is an logarithmic
factor approximation algorithm for tree networks that are homeomorphic
to the complete k-ary tree. For general trees, APR2 is an O(n)-approximation
algorithm, for example, on the caterpillar tree, where a third of the tree nodes
are of degree three and the remaining are leaf-nodes.
102

Chapter 7
Undirected Multi-fiber Rings
with Limited Wavelength
Conversion
In this chapter we consider undirected multi-fiber ring networks with limited
wavelength conversion at few nodes that optimally use the available wavelengths.
We also review the results in [Ramaswami and Sasaki (December 1998)] for singlefiber
ring networks with limited wavelength conversion and show that at least two
nodes with degree two conversion capabilities are required to optimally use the
available wavelengths. We propose a dual-fiber ring network with degree two
conversion capabilities at one node, and fixed conversion capabilities at two nodes
respectively that optimally use the available wavelengths. We also extend the
latter results to k-fiber rings with arbitrary k ≥ 2. As for single-fiber rings the
RWA problem for multi-fiber rings is NP-hard [Li and Simha (2000); Margara
and Simon (2000)]. For k-fiber rings where each link is realized using k optic
fibers with fixed number of wavelengths, the authors independently showed that
a 75% bandwidth utilization is best possible for any k ≥ 2. In dual-fiber rings
each link is realized using 2 optical fibers, see Figures 7.1 and 7.2.
We mentioned in Section 2.2.2 that wavelength convertible and multi-fiber
networks give alternative approach for improving bandwidth utilization for optical
networks. A Wavelength converter is modeled by a bipartite graph and the
maximum degree of the graph correspond to the complexity of the converter.
103

The cost of a wavelength converter increases with the complexity of conversion.
A limited wavelength converter can shift the input wavelength channel to a fixed
few other output wavelength channels, thus provides limited form of wavelength
conversion, whereas a full wavelength converter can shift the input wavelength
channel to any available output wavelength channels, thus provides an unlimited
form of wavelength conversion. Nevertheless, limited converters are relatively
cheaper compared to full converters.
0
1
2
3
w0
w1
w2
w3
w4
w5
Figure 7.1: A dual-fiber ring with four nodes and total of six wavelengths.
Each fiber has three wavelengths, where the first fiber has wavelengths labeled
w 0 , w 2 and w 4 , and the second fiber has wavelengths labeled w 1 , w 3 and w 5 . The
nodes 0 and 2 have fixed wavelength conversion capabilities. Dotted lines are the
wavelengths. At each node wavelength shifts may occur between pairs of same
wavelengths from distinct fibers (dotted lines) or pairs of distinct wavelengths
attached by optical converters (solid lines).
A fixed converter (degree 1 converter) is a variant of limited conversion in
which an input channel is shifted to one fixed output channel. Other variants
of limited conversion include a degree 2 converter, in which an input channel
can be shifted to two fixed output channels. Figure 7.1 shows a dual-fiber ring
network with fixed wavelength conversion capabilities at node 0 and node 2, and
Figure 7.2 shows a dual-fiber ring network with degree two wavelength conversion
capabilities at node 1. In the figures the dashed lines depict wavelength channels
and the solid lines the channel attachments through wavelength conversion.
Since the cost of an optical converter increases with the complexity of conversion,
a network using fixed conversion is relatively cheaper that one using degree 2
conversion, which is in turn less costly than one using full wavelength conversion.
104

7.1 Single-fiber rings with wavelength conversion
0
1
2
3
w0
w1
w2
w3
w4
w5
Figure 7.2: A dual-fiber ring with four nodes and total of six wavelengths.
Each fiber has three wavelengths, where the first fiber has wavelengths labeled
w 0 , w 2 and w 4 , and the second fiber has wavelengths labeled w 1 , w 3 and w 5 . The
node 1 has degree two wavelength conversion capabilities. Dotted lines are the
wavelengths. At each node wavelength shifts may occur between pairs of same
wavelengths from distinct fibers (dotted lines) or pairs of distinct wavelengths
attached by optical converters (solid lines).
Albeit, the latter provides a straightforward way to use the wavelength channels
optimally. We adapt the results of [Ramaswami and Sasaki (December 1998)]
for single-fiber rings with degree two wavelength conversion capabilities at two
nodes called paired wavelength (PW) ring network to dual-fiber rings with low
conversion capabilities at few nodes that optimally use the available wavelengths,
reducing the conversion cost.
7.1 Single-fiber rings with wavelength conversion
Single-fiber rings with wavelength conversion are considered in [Ramaswami and
Sasaki (December 1998)], where they show that if a single-fiber ring has full
wavelength conversion capabilities at one node then any instance can be satisfied
using optimal number of wavelengths. Furthermore, they show that such optimal
allocation of wavelengths can also be realized for a single-fiber ring with degree
two wavelength conversion capabilities at two nodes. However, one cannot guarantee
optimal wavelength allocation for single-fiber rings with fixed wavelength
105

7.1 Single-fiber rings with wavelength conversion
conversion at one or more nodes.
Given a ring network N and a routing R for an instance I on N, let L be
the network load. Without loss of generality, every link of N is traversed by L
paths, otherwise, at any link e traversed by L e < L paths, add L − L e dummy
paths of length one. The paths in R can be transformed into a set of paths with a
special structure relative to the ring as follows. A cut operation is applied to R by
splitting a fixed node of N into two new nodes, called split nodes, to obtain a new
set of paths R that corresponds to paths of a path (chain) network derived from
N with the split nodes as its end-nodes. We refer to a path of R as a cut route
if it traverses both of the two links incident to the split node of N. A cut route
corresponds to two requests in the chain network called residual routes. There is
an optimal algorithm for wavelength allocation for requests of a chain network in
[Golumbic (1980)]. Thus, if the split node of N is equipped with full wavelength
conversion capabilities, then optimal wavelength allocation for requests of the
chain network can be translated into optimal wavelength allocation for requests
of the ring network. This implies the following result,
Theorem 5 (Ramaswami and Sasaki (December 1998)) If a ring network
N has full wavelength conversion capabilities at one node then any instance has
optimal wavelength allocation.
Single-fiber rings with low degree conversion are also considered in [Ramaswami
and Sasaki (December 1998)]. They first define a path structure called multi-cycle
of routes (MCR) over the ring network. An MCR is a sequence of paths going
around the ring in clockwise orientation one or more times and has the same
start and end node. The number of times an MCR goes around the ring is its
multiplicity. A collection of MCRs for a set of paths such that each path in the
set belongs to exactly one MCR is called an MCR-partition. An MCR-partition
for a routing R of the ring network can be computed efficiently [Ramaswami and
Sasaki (December 1998)].
Analogous to MCR is a structure of wavelengths over the ring network called
multi cycles of channels (MCC). An MCC is a sequence of distinct attached
wavelengths that start at a node goes through the ring in clockwise direction one
or more times and ends at the start node. The multiplicity of an MCC is the
106

7.1 Single-fiber rings with wavelength conversion
number of times it goes around the ring.
Thus an MCC with multiplicity m
suffices for optimal wavelength allocation of an MCR of multiplicity m, for some
integer m, if both the MCC and MCR start and end at the same node.
A ring network is single-MCC ring if its wavelength channels can only be arranged
into an MCC of multiplicity L, and is multi-MCC ring, if for any integer
h ≥ 1, and a set of integers {m i } h−1
i=0 , such that ∑ h−1
i=0 m i = L, there is a collection
of h MCCs {H i } h−1
i=0 which are channel disjoint and that each MCC H i has
multiplicity m i . A single-fiber ring with fixed wavelength conversion capabilities
at one or more nodes is a single-MCC ring and a single-fiber ring with degree two
limited wavelength conversion at two nodes is a multi-MCC ring [Ramaswami
and Sasaki (December 1998)]. They show that,
Theorem 6 (Ramaswami and Sasaki (December 1998)) For a single-MCC
ring network with L wavelengths, any instance with load at most L − 1 can be
satisfied.
Theorem 7 (Ramaswami and Sasaki (December 1998)) For a multi-MCC
ring network with L wavelengths, any instance with load at most L can be satisfied.
A paired wavelength (PW) ring network is a single-fiber ring that uses degree
2 conversion at two nodes [Ramaswami and Sasaki (December 1998)]. Let
{w i : 0 ≤ i ≤ W − 1} be the set wavelength channels on a ring network, where W
is the total number of available wavelengths on a fiber. The wavelength channels
of a PW ring network are arranged in the following way. Channels at the same
wavelength are attached at every node and in addition at the first converting
node called, primary node, the wavelength pairs (w i , w i+1 ) and (w i+1 , w i ), for
i ≡ 0(mod 2) are attached, and at the second converting node called, secondary
node, the wavelength pairs (w i , w i+1 ) and (w i+1 , w i ), for i ≡ 1(mod 2) are attached,
the addition on wavelength labels is done modulo W . Such wavelength
attachment (w i , w i+1 ) and (w i+1 , w i ) is also referred to as the outgoing and incoming
attachment, respectively. In [Ramaswami and Sasaki (December 1998)]
they showed the following result for PW rings.
Proposition 1 (Ramaswami and Sasaki (December 1998)) For any two integers
j and h, such that, 0 ≤ j ≤ j + h ≤ W − 1, the PW ring network has
107

7.1 Single-fiber rings with wavelength conversion
an MCC with multiplicity h + 1, that uses the wavelength channels {w i : j ≤ i ≤
j + h}.
From Proposition 1, it follows that a PW ring is a multi-MCC ring network,
that is, for any arbitrary set of positive integers {m i } m−1
i=0 of cardinality m, such
that ∑ m−1
i=0 m i = W , there exists a set of MCCs, {H i } m−1
i=0 , and MCC H i has
multiplicity m i .
0
w0
w1
w2
w3
1 2 ... n−1
Figure 7.3: An n-node single-fiber ring network with four wavelengths labeled
w 0 , w 1 , . . . , w 3 , and degree two limited conversion capabilities at node 0. At each
node wavelength channels at the same wavelength are attached, and in addition at
the conversion node 0 the wavelength pairs (w i , w i+1 ) and (w i+1 , w i ), for 0 ≤ i ≤ 3,
and i ≡ 0(mod 2), are attached.
0
w0
w1
w2
w3
1 2 ... n−1
Figure 7.4: An n-node single-fiber ring network with four wavelengths labeled
w 0 , w 1 , . . . , w 3 , and degree two limited conversion capabilities at node 0. At each
node wavelength channels at the same wavelength are attached, and in addition
at the conversion node 0 the wavelength pairs (w i , w i+1 ), 0 ≤ i ≤ 3, are attached.
108

7.1 Single-fiber rings with wavelength conversion
By Theorem 6, a single-fiber ring cannot achieve optimal wavelength allocation
using fixed wavelength conversion. In the following we show that a singlefiber
ring also cannot achieve optimal wavelength allocation using degree 2 wavelength
conversion at only one node. Figure 7.3, shows a single-fiber ring with
degree two conversion at node 0 such that at each node channels at the same
wavelength are attached, in addition at the conversion node 0, the wavelength
pairs (w i , w i+1 ) and (w i+1 , w i ), for 0 ≤ i ≤ 3 and i ≡ 0(mod 2), are attached,
similarly, in Figure 7.4, at all other nodes wavelength channels at the same wavelength
are attached, and at the conversion node 0 the wavelength pairs (w i , w i+1 ),
for 0 ≤ i ≤ 3, are attached.
Proposition 2 For a single-fiber ring with L wavelengths and degree two conversion
capabilities at only one node there is an instance of load L that cannot be
satisfied.
Proof :
node.
We consider two possible wavelength attachments at the conversion
1. There exists at least one pair of consecutive wavelengths (w i , w i+1 ) that is
not attached at the conversion node (Figure 7.3),
2. The wavelength pairs (w i , w i+1 ), for 0 ≤ i ≤ L − 1, are attached at the
conversion node (Figure 7.4).
For the Case 1, no single-MCC exists since the wavelength pair (w i , w i+1 ) is
not attached at every network node. Thus an MCR-partition with single-MCR
cannot be satisfied. Figure 7.5 shows a set of paths on a ring network which
form a single-MCR. Now suppose that a single-MCC exists as in Case 2. That
is, starting from any node of the ring there is a sequence of wavelengths that
goes around the ring in clockwise direction L times and ends at the start node.
Without loss of generality, at the conversion node 0 the wavelength pair (w i , w i+1 ),
for 0 ≤ i ≤ L−1, are attached to realize a single-MCC. We consider the following
two possible cases for the other wavelength attachments at this node.
1. For 0 ≤ i ≤ L − 1, the wavelength pair (w i , w i ) is attached.
Consider an MCR-partition with an MCR of multiplicity L − 2 that starts
109

7.1 Single-fiber rings with wavelength conversion
and ends at the conversion node, and an MCR of multiplicity 2 that starts
and ends at a node other than the conversion node. The paths sequence
{1, 4}, {4, 2}, {2, 4} and {4, 1} in Figure 7.6 that starts and ends at node 1
is an example of such MCR. There is no MCC of multiplicity 2 that starts
and ends at node 1, because if an MCC starts say at wavelength w j from
node 1 it goes around the ring in clockwise direction, then enters wavelength
w j+1 through a (w j , w j+1 ) wavelength attachment. In the next round the
MCC should enter the wavelength w j through a (w j+1 , w j ) wavelength attachment,
to the start node 1. But at the conversion node the wavelength
pairs (w j , w j+1 ) and (w j+1 , w j ), cannot both be attached.
2. One or more wavelength pairs (w i , w i ) are not attached.
Consider an MCR-partition with L MCRs of multiplicity 1 that starts and
ends at distinct nodes other than the conversion node 0. The request paths
{{i, L + i}, {L + i, i} : 1 ≤ i ≤ L}, on a ring network with 2L + 1 nodes
form such MCR-partition. Clearly, if a channel (w i , w i ) is not attached,
then such MCR-partition cannot be satisfied using L wavelengths.
0 1 2 3 4 5 6
Figure 7.5: A 7-node ring network with seven request paths, {{0, 3}, {3, 6},
{6, 2}, {2, 5}, {5, 1}, {1, 4}, {4, 0}}, which form a single-MCR.
0
1
2
3
4
Figure 7.6: A 5-node ring network with four request paths, {{2, 4},
{1, 4}, {4, 1}, {4, 2}}.
Theorem 7 and Proposition 2 imply that for single-fiber rings two or more
nodes with degree two wavelength conversion capabilities are required for optimal
110

7.2 Dual-fiber rings with limited wavelength conversion
wavelength allocation. In the following sections we consider k-fiber rings that use
limited wavelength conversion for optimal wavelength allocation. We propose
dual-fiber rings with fixed conversion at two nodes and degree two conversion at
one node respectively, and extend to k-fiber rings with arbitrary k.
7.2 Dual-fiber rings with limited wavelength conversion
A dual-fiber ring employs two fibers per link. In this section we propose dual-fiber
rings with fixed conversion and degree two conversion for optimal wavelength allocation.
Figure 7.1 shows a dual-fiber ring with fixed conversion capabilities at
nodes 0 and 2, and Figure 7.2 shows a dual-fiber ring with degree two conversion
capabilities at node 1. In both figures all other nodes have no wavelength
conversion capabilities.
A fixed conversion dual-fiber ring (FDR) is a ring network on n nodes employing
two fibers at each link and fixed wavelength conversion at two nodes. Let w be
a fixed number of wavelengths supported by an optic fiber. The wavelengths on a
link of the FDR network are labeled w 0 , . . . , w W −1 , for W = 2·w, where (w i , w i+1 ),
is a pair of wavelength channels from distinct fibers at the same wavelength, for
0 ≤ i ≤ W − 1 and i ≡ 0(mod 2). The network nodes and links are labeled
0, . . . , n − 1, for integer n ≥ 4. Let s be a fixed integer, such that 0 ≤ s ≤ n − 1.
The nodes s and (s + 2) mod n, are equipped with fixed conversion capabilities,
such that for 0 ≤ i ≤ W − 1, the wavelength pair (w i+1 , w i ) and (w i , w i+1 )
is attached at the node s and (s + 2) mod n respectively, through wavelength
conversion. All other nodes have no wavelength conversion capabilities. At such
nodes the wavelength pairs (w i , w i ), (w i+1 , w i+1 ), (w i , w i+1 ) and (w i+1 , w i ), for
0 ≤ i ≤ W − 1 and i ≡ 0( mod 2) are attached as same wavelengths from distinct
fibers. We refer to the node s and (s+2) mod n, as co-primary and co-secondary
node, respectively, and the node (s+1) mod n and (s+3) mod n, as the primary
and secondary node, respectively. We relabel a wavelength channel at w i to w i+1 ,
for 0 ≤ i ≤ W − 1, for links between the co-primary and co-secondary nodes
in clockwise direction. With the relabeling we have that at the co-primary and
111

7.2 Dual-fiber rings with limited wavelength conversion
co-secondary nodes the wavelength pair (w i , w i ) is attached, for 0 ≤ i ≤ W − 1,
at the primary node the wavelength pairs (w i , w i ), (w i+1 , w i+1 ), (w i , w i+1 ) and
(w i+1 , w i ), for 0 ≤ i ≤ W − 1 and i ≡ 1(mod 2) are attached, and at the secondary
node the wavelength pairs (w i , w i ), (w i+1 , w i+1 ), (w i , w i+1 ) and (w i+1 , w i ),
for 0 ≤ i ≤ W − 1 and i ≡ 0(mod 2) are attached. Figure 7.1 shows a 4-node
FDR network with w = 3, where s = 0 (i.e., nodes 0 and 2 have fixed conversion
capabilities).
A limited conversion dual-fiber ring (LDR) is a ring network on n ≥ 2 nodes
employing two fibers at each link and degree two wavelength conversion at only
one node. Wavelengths, nodes and links of the LDR network are labeled as
for the FDR network, but there is no relabeling of any wavelength. At a node
other than the conversion node of the LDR network the wavelength pairs (w i , w i ),
(w i+1 , w i+1 ), (w i , w i+1 ) and (w i+1 , w i ), are attached, for 0 ≤ i ≤ W − 1 and i ≡
0(mod 2). Note that the wavelength channels at w i and w i+1 , for 0 ≤ i ≤ W − 1
and i ≡ 0(mod 2), are from distinct fibers and have the same wavelength. At
the conversion node the wavelength pairs (w i , w i ), (w i+1 , w i+1 ), (w i , w i+1 ) and
(w i+1 , w i ), are attached through wavelength conversion, for 0 ≤ i ≤ W − 1 and
i ≡ 1(mod 2). We refer to the conversion node as the primary node and its
clockwise neighbour as the secondary node. Figure 7.2 shows a 4-node LDR
network with w = 3, where node 1 is the primary node.
In the following we adapt the results for the PW ring described in Section
7.1 to the FDR and LDR networks and show that they are multi-MCC ring networks.
That is, starting from any node of the FDR and LDR networks, successive
wavelength channels can be arranged into an MCC of arbitrary multiplicity. We
describe the results for the FDR network since by construction the primary and
secondary nodes of both the FDR and LDR network have same role. Thus the
results for the LDR network follows from the results of the FDR network.
Lemma 4 Let j and h be arbitrary integers satisfying, 0 ≤ j ≤ j + h ≤ W − 1.
The FDR network has an MCC with multiplicity h + 1 that uses the wavelength
channels {w i : j ≤ i ≤ j + h}.
Proof : At each node of the FDR network the wavelength pair (w i , w i ), 0 ≤ i ≤
W −1 is attached. In addition, at the primary node the wavelength pairs (w i , w i+1 )
112

7.2 Dual-fiber rings with limited wavelength conversion
and (w i+1 , w i ), for i ≡ 1( mod 2), and at the secondary node the wavelength pairs
(w i , w i+1 ) and (w i+1 , w i ), for i ≡ 0(mod 2), are attached, for 0 ≤ i ≤ W − 1. We
prove by induction on the number of multiplicity h. The lemma holds for h = 0,
since the sequence of wavelength channels at w i form an MCC of multiplicity 1
as required.
Assume that the lemma holds for h > 0, that is, there is an MCC, H, of
multiplicity h + 1 that only uses channels at wavelengths {w i } j+h
i=j . We show that
the lemma also holds for h + 1. Without loss of generality, the MCC H starts
and ends at the counter-clockwise neighbour v of the primary node and that the
first channel of the MCC H is at wavelength w j . Thus starting at wavelength w j ,
the MCC H goes around the ring in clockwise direction, and enters a channel at
wavelength w j+1 (at the primary node if j ≡ 1(mod 2), at the secondary node,
otherwise), where the wavelength pair (w j , w j+1 ) is attached, and so on. Since
the MCC H has channels with wavelengths from the set {w i : j ≤ i ≤ j + h}, the
last outgoing wavelength attachment used by the MCC H is the wavelength pair
(w j+h−1 , w j+h ). We consider two cases for j + h − 1.
1. j + h − 1 ≡ 1(mod 2).
The wavelength pair (w j+h−1 , w j+h ) is attached at the primary node and
at secondary node the wavelength pair (w j+h , w j+h+1 ) is attached. Let H ′
be a subset of wavelength channels in the MCC H that starts at node v
with wavelength w j , goes around the ring one or more times until the the
wavelength at w j+h is encountered at the secondary node, and let H ′′ =
H \ H ′ . At the secondary node the wavelength pairs (w j+h , w j+h+1 ) and
(w j+h+1 , w j+h ) are attached, thus an MCC that starts with the subset of
channels H ′ , then at the secondary node goes around the ring once using the
wavelength at w j+h+1 (since the wavelength pair (w j+h , w j+h+1 ) is attached
at the secondary node), then continues with the channel subsequence H ′′
after the incoming wavelength attachment (w j+h+1 , w j+h ) at the secondary
node, is the MCC of multiplicity h + 2 that uses the wavelength channels
{w i : j ≤ i ≤ j + h + 1}, as required.
113

7.3 k-fiber rings with limited wavelength conversion for k > 2
2. j + h − 1 ≡ 0(mod 2).
The proof is similar to Case 1 above, but the role of the primary and
secondary node is interchanged.
Now we show that the FDR and LDR networks are multi-MCC rings.
Lemma 5 The FDR and LDR are multi-MCC ring networks.
Proof : Let {m 0 , . . . , m h−1 } be a set of positive integers such that ∑ h−1
i=0 m i = W ,
for integer h > 0. We show that there is a collection of MCCs, {H 0 , . . . , H h−1 },
such that an MCC, H i , has multiplicity m i , 0 ≤ i ≤ h − 1. Note that if the
wavelength channels of the network are partitioned into subsets {W 0 , . . . , W h−1 },
such that, W 0 = {w 0 , . . . , w m0 −1}, W 1 = {w m0 , . . . , w m0 +m 1 −1},. . ., W h−1 =
{w mh−2 , . . . , w mh−2 +m h−1 −1}, then the number of wavelengths in W i is m i , for
0 ≤ i ≤ h − 1. By Lemma 4, there is an MCC, H i , of multiplicity m i that only
uses the wavelengths in W i , 0 ≤ i ≤ h−1, which give the desired MCCs collection
{H 0 , . . . , H h−1 }.
By Lemma 5 we obtain the following results.
Theorem 8 There is a dual-fiber ring network with four or more nodes, L wavelengths
and fixed wavelength conversion at two nodes such that any instance of
load at most L has optimal wavelength allocation.
Theorem 9 There is a dual-fiber ring network with L wavelengths and degree
two wavelength conversion at only one node such that any instance of load at
most L has optimal wavelength allocation.
7.3 k-fiber rings with limited wavelength conversion
for k > 2
In this section we extend the results for the LDR and FDR networks described in
Section 7.2 to k-fiber rings for arbitrary k > 2. As in Section 7.2, w is the number
of wavelengths supported on a fiber. For k ≡ 0( mod 2), the wavelengths on links
114

7.3 k-fiber rings with limited wavelength conversion for k > 2
0
1
2
3 4
w0
w5
w5
w4
w3
w2
w4
w3
w2
w1
w1
w0
w0
w1
w2
w3
w4
w5
(a)
0
w0
w9
w8
w7
w6
w5
w4
w3
w2
w1
1
2
w9
w8
w7
w6
w5
w4
w3
w2
w1
w0
3 4
w0
w1
w2
w3
w4
w5
w6
w7
w8
w9
(b)
Figure 7.7: An FDR adaptation to k-fiber ring for k ≡ 1(mod 2), with nodes
0, 1, 2, 3 and 4 as the co-primary, primary, co-secondary, secondary and tertiary
node respectively. (a) A 3-fiber ring, and (b) a 5-fiber ring, with w = 2.
115

7.3 k-fiber rings with limited wavelength conversion for k > 2
0 1 2
w0
w1
w2
w3
w4
w5
(a)
0 1 2
w0
w1
w2
w3
w4
w5
w6
w7
w8
w9
(b)
Figure 7.8: An LDR adaptation to k-fiber ring for k ≡ 1(mod 2), with nodes
0, 1 and 2 as the primary, tertiary and secondary node respectively. (a) A 3-fiber
ring, and (b) a 5-fiber ring, with w = 2.
116

7.3 k-fiber rings with limited wavelength conversion for k > 2
of the k-fiber ring are labeled w 0 , w 1 , . . . , w W −1 , where W = k · w, such that the
wavelength pair (w i , w i+1 ), for 0 ≤ i ≤ W − 1 and i ≡ 0(mod 2), consists of
the wavelength channels from distinct fibers with the same wavelength. Thus for
k-fiber rings with k ≡ 0(mod 2) the results for the FDR and LDR networks in
Lemmas 4 and 5 can be adapted in an obvious way.
For k ≡ 1(mod 2), wavelengths are labeled as follows. For the first k − 3
fibers, the wavelengths on the links of a k-fiber ring are labeled as in the case of
k ≡ 0( mod 2), that is, with labels w 0 , w 1 , . . . , w W ′ −1, where W ′ = (k −3)·w, and
the wavelength pair (w i , w i+1 ), for 0 ≤ i ≤ W ′ − 1 and i ≡ 0(mod 2), consists
of channels from distinct fibers with the same wavelength. For the remaining
three fibers, the wavelengths are labeled w W ′, w W ′ +1, . . . , W − 1, where W =
k · w, such that the wavelength triple (w j , w j+1 , w j+2 ), for W ′ ≤ j ≤ W − 3 and
j − W ′ ≡ 0(mod 3), are wavelength channels from distinct fibers with the same
wavelength.
A k-fiber ring adaptation of the FDR network is described as follows. Let
n ≥ 5 be the number of nodes on a k-fiber ring. For a fixed s, 0 ≤ s ≤ n − 1, the
nodes s and (s + 2) mod n, on a k-fiber ring have fixed conversion capabilities.
At the node s and (s + 2) mod n the wavelength pair (w i+1 , w i ) and (w i , w i+1 ),
respectively, for 0 ≤ i ≤ W − 1, is attached through wavelength conversion.
Other nodes have no wavelength conversion capabilities, thus only channels at
same wavelength are attached. The node s and (s + 2) mod n, is the co-primary
and co-secondary node respectively. The node (s+1) mod n and (s+3) mod n, is
the primary and secondary node respectively. In addition the node (s+4) mod n
is distinguished as the tertiary node.
For 0 ≤ i ≤ W ′ − 2, the wavelength pairs (w i , w i+1 ) and (w i+1 , w i ), are
attached at the primary and secondary nodes. In addition, the wavelength pairs
(w i+1 , w i+2 ) and (w i+2 , w i+1 ), for W ′ ≤ i ≤ W − 3 and i − W ′ ≡ 0(mod 3) are
attached. At the tertiary node the wavelength pairs (w i , w i+1 ) and (w i+1 , w i ), for
W ′ ≤ i ≤ W −3 and i−W ′ ≡ 0( mod 3) are attached. We relabel the wavelength
at w i to w i+1 for links between the co-primary and co-secondary nodes in clockwise
direction, for 0 ≤ i ≤ W − 1. With the relabeling we have that the wavelength
pair (w i , w i ), for 0 ≤ i ≤ W − 1 is attached at each node, and in addition the
following cases for attachments of consecutive wavelengths w q and w q+1 .
117

7.3 k-fiber rings with limited wavelength conversion for k > 2
1. For q ≡ 1(mod 2) and 1 ≤ q ≤ W ′ − 1 the wavelengths pairs (w q , w q+1 )
and (w q+1 , w q ) are attached at the primary node.
2. For q ≡ 0(mod 2) and 0 ≤ q ≤ W ′ − 1 the wavelengths pairs (w q , w q+1 )
and (w q+1 , w q ) are attached at the secondary node.
3. For q − W ′ ≡ 0(mod 3) and W ′ ≤ q ≤ W − 3 the wavelengths pairs
(w q , w q+1 ) and (w q+1 , w q ) are attached at the tertiary node.
4. For q − W ′ ≡ 1(mod 3) and W ′ ≤ q ≤ W − 2 the wavelengths pairs
(w q , w q+1 ) and (w q+1 , w q ) are attached at the secondary node.
5. For q − W ′ ≡ 2(mod 3) and W ′ ≤ q ≤ W − 1 the wavelengths pairs
(w q , w q+1 ) and (w q+1 , w q ) are attached at the primary node.
Figure 7.7 (a)-(b) shows an FDR adaption for 3-fiber and 5-fiber rings, with
w = 2.
A k-fiber ring adaptation of the LDR network is described as follows. Let n ≥
3 be the number of nodes on the k-fiber ring. One node has degree two wavelength
conversion capabilities, referred to as the primary node. At the primary node the
wavelength pairs (w j , w j+1 ) and (w j+1 , w j ), for 0 ≤ j ≤ W ′ −1 and j ≡ 1( mod 2),
are attached through conversion. In addition, the wavelength pairs (w j+2 , w j+3 )
and (w j+3 , w j+2 ), for W ′ ≤ j ≤ W −3 and j −W ′ ≡ 0( mod 3), also are attached.
Let the node s be the primary node. The nodes (s + 1) mod n and (s +
2) mod n, for a fixed s, 0 ≤ s ≤ n − 1, are referred to as the secondary and
tertiary node respectively. At the secondary node the wavelength pairs (w j , w j+1 )
and (w j+1 , w j ), for 0 ≤ j ≤ W ′ − 1 and j ≡ 0(mod 2), are attached. In addition
the wavelength pairs (w j , w j+1 ) and (w j+1 , w j ), for W ′ ≤ j ≤ W −3 and j −W ′ ≡
0(mod 3) are attached. At the tertiary node the wavelength pairs (w j+1 , w j+2 )
and (w j+2 , w j+1 ), for W ′ ≤ j ≤ W − 3 and j − W ′ ≡ 0(mod 3), are attached. At
each node the wavelength pair (w i , w i ), for 0 ≤ i ≤ W − 1, is attached, and in
addition we have the following cases for attachments of consecutive wavelengths
w q and w q+1 .
1. For q ≡ 1(mod 2) and 1 ≤ q ≤ W ′ − 1 the wavelengths pairs (w q , w q+1 )
and (w q+1 , w q ) are attached at the primary node.
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7.3 k-fiber rings with limited wavelength conversion for k > 2
2. For q ≡ 0(mod 2) and 0 ≤ q ≤ W ′ − 1 the wavelengths pairs (w q , w q+1 )
and (w q+1 , w q ) are attached at the secondary node.
3. For q − W ′ ≡ 2(mod 3) and W ′ ≤ q ≤ W − 4 the wavelengths pairs
(w q , w q+1 ) and (w q+1 , w q ) are attached at the primary node.
4. For q − W ′ ≡ 0(mod 3) and W ′ ≤ q ≤ W − 3 the wavelengths pairs
(w q , w q+1 ) and (w q+1 , w q ) are attached at the secondary node.
5. For q − W ′ ≡ 1(mod 3) and W ′ ≤ q ≤ W − 2 the wavelengths pairs
(w q , w q+1 ) and (w q+1 , w q ) are attached at the tertiary node.
Figure 7.8 (a)-(b) shows the LDR adaption for 3-fiber and 5-fiber rings, for w = 2.
The results for the FDR and LDR networks in Lemmas 4 and 5, can be
adapted to k-fiber rings where k ≡ 1(mod 2) using the outgoing and incoming
attachments at the primary, secondary and tertiary nodes as specified by consecutive
wavelength attachments given for FDR (Cases 1-5) and LDR (Cases 1-5)
adaptations.
119

Chapter 8
Traffic Grooming on Path
Networks - General Instances
In this chapter we study traffic grooming problem for unidirectional path networks
with grooming ratio 2. We consider general instances with unit size requests
(traffic) and propose several heuristic algorithms for grooming such instances, extending
the work in [Bermond et al. (June 2005)] for all-to-all instances. Previous
chapters assumed that a request would generally require the full bandwidth of a
wavelength channel, hence, the main objective became to optimize wavelength allocation
to requests. It became apparent, however, that current state-of-art optic
fibers support several hundreds wavelengths each in turn supporting an enormous
bandwidth (in the order of tens of Gigabyte), while most network traffic
require just a fraction of a wavelength’s bandwidth. Combining several requests
to share a wavelength channel is an elegant way to efficiently use the wavelength’s
bandwidth and also improve network connectivity.
Traffic grooming combine several requests to share a wavelength channel (lightpath),
see Section 2.2.2.3. When requests are combined into a lightpath an Add-
Drop Multiplexor (ADM) is needed for opto-electronic conversion at each node
where a request is added or dropped from the lightpath. A request is added
into a lightpath at a node for transmission and dropped from the lightpath at
its destination node or some intermediate node for retransmission (i.e., because
a lightpath to its destination node is unavailable). An ADM device uses Time
Division Multiplexing technique to add or drop several requests onto a lightpath
120

8.1 General lower bounds
at a node. If traffic is added or dropped on the same lightpath at the node then
an ADM may be shared. The traffic grooming problem seek to satisfy an instance
using a minimum number of ADMs and wavelengths. This problem involves the
routing and wavelength allocation as subproblems, so is at least as hard as the
RWA problem. In particular, the traffic grooming problem is NP-hard even for
path networks [Dutta et al. (June, 2003)], where the RWA problem is optimally
solved.
In [Bermond et al. (June 2005)] exact methods are presented for the traffic
grooming problem on unidirectional path networks with all-to-all instances (in
which there is a connection request of unit size for every node pair (i, j), for
i < j), and grooming ratio 1 and 2. We consider general instances with unit
size requests on unidirectional path networks and propose two main grooming
algorithms for grooming ratio 2. The first algorithm is based on interval graph
partitioning where requests are viewed as intervals, and the second algorithm is
based on graph edge decomposition of request graph. The two algorithms are
experimentally compared for performance and extended to path networks with
grooming ratios higher than 2. We use randomly generated instances described
in Section 2.4.1 for experiments. We show that the graph edge decomposition
based algorithms achieve near optimal performance for most instances.
8.1 General lower bounds
Given an instance R for path network P n on n nodes and grooming ratio g, let
Γ o (k) and Γ i (k) be the number of requests in R that begin and end at node k
when traversing the nodes of ⌈P n from left to⌉
right, 0 ≤ k ≤ n − 1. Obviously,
each node k requires at least max{Γo(k),Γi (k)}
ADMs, since an ADM at node k
g
can be shared by at most g requests that begin at k, and at most g requests that
end at k. Thus ∑ ⌈ ⌉
n−1 max{Γo (k),Γ i (k)}
k=0
is the general lower bound on the number
g
of ADMs required by instance R. The load of a network, L, is the maximum
number of requests using any network link. Clearly, at least ⌈ L ⌉ wavelengths are
g
needed for any routing instance inducing load L on the network, since at most g
requests may share a wavelength at any network link.
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8.2 Algorithms for path networks with g = 2
8.2 Algorithms for path networks with g = 2
In this section we describe different grooming heuristics for path networks with
grooming ratio 2 and general instances of unit size requests. We describe the interval
based algorithm in Section 8.2.1 that uses the interval graph algorithm in
[Golumbic (1980)] and weighted graph matching algorithms in [Edmonds (1965a);
Edmonds (1965b)]. We describe the edge decomposition based algorithm in Section
8.2.3 that adapts the techniques for finding a minimum circuit on the graph
in [Itai and Rodeh (1977)]. The graph edge decomposition problem is defined
as follows. Given a fixed graph G, a G-decomposition of a graph H(V, E) is a
partition of E into subgraphs isomorphic to G. The G-decomposition problem is
to determine whether an input graph H admits a G-decomposition. This problem
is NP-complete for general graphs [Dor and Tarsi (1997)].
8.2.1 Interval partition algorithm
Given an instance R for P n , let L be the network load induced by the instance
R on P n . The interval partition algorithm (IPA) grooms the requests in the
following way.
1. Partition the requests in R into L sets S i , for 1 ≤ i ≤ L, using the interval
graph algorithm in [Golumbic (1980)], such that no two requests in S i share
any link of P n .
2. Let V Si be end-nodes of requests in S i , for 1 ≤ i ≤ L. Define a complete
weighted graph, H = (V H , E H ), where V H = {1, 2, . . . , L}, a node i corresponds
to the set S i , and an edge e = {i, j} has weight w(e) = |V Si ∩ V Sj |,
for 1 ≤ i < j ≤ L. Note that the edge weight w{i, j} is equal to the number
of common end-nodes between the partition sets S i and S j obtained in Step
1.
3. Find a maximum weight matching (MWM), M, for the graph H. Denote
by Q be the set of nodes in H not covered by M (note that |Q| = 1, if L is
odd and |Q| = 0, otherwise).
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8.2 Algorithms for path networks with g = 2
4. For each edge e = {i, j} ∈ M, the end-nodes i and j correspond to the sets
S i and S j obtained in Step 1. The requests in S i and S j are combined into
a wavelength of P n . For any node k ∈ Q, one wavelength of P n is used for
requests in the set S k .
In Step 4, the algorithm IPA uses ⌈ L ⌉ optimal number of wavelengths. For
2
each edge e = {i, j} ∈ M, the wavelength allocated to requests in S i ∪S j uses |V Si ∪
V Sj | ADMs, in addition for any node k ∈ Q, the wavelength allocated to requests
in S k uses |V Sk | ADMs. Thus the algorithm IPA uses ∑ e={i,j}∈M |V S i
∪ V Sj | + ɛ
ADMs, where ɛ = 0, if L ≡ 0(mod 2), and ɛ = |V Sk |, otherwise, where k ∈ Q.
8.2.2 IPA variant algorithms
In this section we describe the variants of the IPA algorithm that have different
run-time requirements. The run-time requirement for the IPA algorithm framework
in Section 8.2.1 is dominated by the computation of the weighted matching
M. For example, the blossom shrinking algorithm of [Edmonds (1965a); Edmonds
(1965b)], that is used in our implementations computes the weighted matching
M in O(L 3 ) time since the graph H defined in Step 2 has L nodes. We propose
two alternative greedy approaches for matching the sets S i obtained in Step 1
to improve the worst case run-time performance for algorithm IPA called, large
weight first matching (LWM) and simple greedy matching (SGM).
1. LWM matching:
From the graph H defined in Step 2, find the weighted matching M by
repeatedly selecting an edge of largest weight in H incident to unmatched
nodes. The LWM matching sorts the edges of H by weight. Since the
number of edges in H is bounded by O(L 2 ), its worst case run-time is
O(L 2 log L).
2. SGM matching:
The sets S i obtained in Step 1 are merged a pair at a time by taking
unmatched sets S j and S k with least indices. This takes O(L) time since
there are L such sets.
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8.2 Algorithms for path networks with g = 2
We consider three variants of the IPA algorithm; using the maximum weight
matching algorithm of [Edmonds (1965a); Edmonds (1965b)] (IPA-MWM), the
LWM matching algorithm (IPA-LWM) and the SGM matching algorithm (IPA-
SGM). All three variants compute optimal number of wavelengths.
8.2.3 Edge decomposition algorithm
The input is an instance R on P n with unidirectional requests, but each request
represented as directed pair (i, j), where i < j. From the set R define the request
digraph D(V, A) where the node set V are the nodes of P n and the set A =
{(i, j) : (i, j) ∈ R} are the requests.
obtained by replacing the arcs of D with edges.
The underlying graph G(V, E) for D is
Note that D is acyclic since
every arc (i, j) ∈ D, satisfies i < j. In addition, since the numbering of nodes in
D is the numbering of nodes of P n , they induce a topological ordering for D. The
terms request, arc, and edge are used interchangeably.
Let W be total number of wavelengths to be determined by this algorithm.
The edge decomposition algorithm (EDA) seek to partition the arcs of the request
digraph D into W subgraphs D j = (V j , A j ), 0 ≤ j ≤ W − 1, such that
• for each j = 0, . . . , W − 1 and each pair i, i + 1 of consecutive nodes
on P n , no more than 2 arcs in A j cross the link (i, i + 1) on P n , that is,
|{(h, k) ∈ A j : h ≤ i, k ≥ i + 1}| ≤ 2 (i.e., grooming ratio constraint of 2 on
P n );
• A i ∩ A j = ∅, ⋃ W −1
i=0
A i = A;
and minimize ∑ W −1
i=0
|V i |. The parameter W (not part of the input, but to be
determined as part of the output) is the total number of used wavelengths. The
requests in D i are allocated to wavelength i and need |V i | ADMs, one at each node
which is an end node of a request in D i . Thus the total number of ∑ W −1
i=0
|V i |
ADMs are required. Intuitively, since we aim at minimizing ∑ W −1
i=1
|V i | - the
number of ADMs required to support the allocation of requests to wavelengths,
we should look for partitionings of D with small W and small ratios |V i|
. |A i |
The subgraph D i is also referred to as valid subgraph. A subgraph of D is
called a circuit subgraph if its underlying graph is of degree 2. A circuit subgraph
124

8.2 Algorithms for path networks with g = 2
is a closed circuit if each vertex on the underlying graph has degree 2, and open
circuit, otherwise. An open circuit is called linear circuit if no more than one arc
cross any link of P n . A circuit subgraph is said to be valid circuit if it satisfies
the grooming ratio constraint g = 2 on P n .
Let v be a node in D. The node v is said to be source node if arcs are incident
(outgoing) from v. Analogously, the node v is said to be sink node if arcs are
incident (incoming) to v. The algorithm EDA partitions the arcs of D into arc
disjoint valid subgraphs in the following way.
1. While more arcs remain in D, choose the least label source node v (with
empty incoming arcs but non-empty outgoing arcs) as the root node for D.
Use the DFS traversal on D starting from the node v to find a valid circuit
C i (described in Section 8.2.3.1), and remove the arcs of C i from D.
2. Combine (merge) sets of compatible valid circuit subgraphs and resulting
compatible valid subgraphs into maximal valid subgraphs (described in Section
8.2.3.2).
The following Lemma 6 gives a characterization for valid circuit subgraphs
of D, showing that only closed circuit subgraphs with unique source and sink
nodes are valid circuit subgraphs of D. This property also applies to open circuit
subgraphs that can be transformed into such closed circuit subgraphs by an additional
arc of the form (x, y) where x and y are nodes of degree 1 on the underlying
graph of the open circuit subgraph, with x < y.
Lemma 6 A closed circuit subgraph of D is valid circuit subgraph iff has unique
source and sink node.
Proof : Let C(V C , A C ) be a circuit subgraph of D. Without loss of generality,
C is a closed circuit, otherwise, we replace C with a closed circuit C ∪ (x, y) by
adding a new arc (x, y), where x and y are nodes of degree 1 on the underlying
graph of C with x < y. Clearly, the subgraph C has at least one source and one
sink node since D is acyclic and so its subgraph C. Let a and z be source and
sink node of C with least and highest label respectively among all source and
sink nodes of C.
125

8.2 Algorithms for path networks with g = 2
Now suppose that there is a source node f ∈ C other than a, choose such f
with the smallest label greater than a. Consider the link f = (f, f + 1) in P n .
Since f is a source node of C, there are two arcs of C say, (f, q) and (f, s), incident
from the node f where f < q and f < s (since every arc (i, j) of D satisfies i < j).
We have that the arcs (f, q) and (f, s) both cross the link (f, f + 1) on P n , and
that q and s are non-adjacent nodes of C, since C is a circuit subgraph containing
a. Now consider the edge disjoint paths p{a, q} and p{a, s} from node a to q and s
respectively, on the underlying graph of C. Such paths exist since the underlying
graph of C is a cycle. Since a < f < f + 1 ≤ q and a < f < f + 1 ≤ s, each of
the paths p{a, q} and p{a, s} cross the link (f, f + 1) of P n . Consequently, the
link (f, f + 1) on P n is crossed four times by the arcs of C, implying that C is an
invalid circuit subgraph.
Analogously, if there is a sink node h ∈ C other than z, choose such h with
the largest label less than z. Consider the link h − 1 = (h − 1, h) in P n . Since h
is a sink node of C, there are two arcs of C say, (q, h) and (s, h), incident to the
node h, where q < h and s < h (since every arc (i, j) of D satisfies i < j). We
have that the arcs (q, h) and (s, h) both cross the link (h − 1, h) on P n , and that
q and s are non-adjacent nodes of C, since C is a circuit subgraph containing z.
Now consider the edge disjoint paths p{q, z} and {s, z} from node q and s to z
respectively, on the underlying graph of C. Such paths exist since the underlying
graph of C is a cycle. Since q ≤ h − 1 < h < z and s ≤ h − 1 < h < z, each of
the paths p{q, z} and p{s, z} cross the link (h − 1, h) of P n . Consequently, the
link (h − 1, h) on P n is crossed four times by the arcs of C implying that C is an
invalid circuit subgraph.
Now let a and z be the unique source and sink nodes of the circuit subgraph
C. Since every arc (i, j) ∈ D satisfies that i < j, the source a and sink z are the
nodes of C with least and highest label. Thus the two p(a, z) arc disjoint paths
in C each cross the links (i, i + 1) of P n exactly once, where a ≤ i ≤ z − 1, and
none of the other links of P n are crossed.
We have shown that the existence of more than one sink and source nodes on
a closed circuit subgraph of D violates the grooming ratio constraint of 2 on P n ,
where a link of P n is crossed at least 4 times. If the circuit subgraph considered
is an open circuit subgraph the requirement to add the arc (x, y) (which leads to
126

8.2 Algorithms for path networks with g = 2
a closed acyclic subgraph) does not invalidate the proof of this lemma because
by removing the arcs (x, y) still leaves a link of P n crossed at least 3 times, that
also violates the grooming ratio constraint of 2 on P n .
Figure 8.1, shows a closed circuit graph which is not a valid circuit since four
arcs cross the links (i, i+1) of P n , where f ≤ i ≤ h−1. Note that in this example
there are two source nodes {a, f} and sink nodes {h, z}.
a
f
h
z
Figure 8.1: An example of invalid closed circuit with nodes a < f < h < z.
8.2.3.1 Finding valid circuit subgraphs
In this section we describe how Step 1 of the EDA algorithm finds a valid circuit
subgraph of D. In Step 1 of the EDA algorithm, a DFS traversal on the nodes of
D starts at the least label source node i, and explore arcs out of the node most
recently reached by the search. When arcs out of the node have been exhausted
the DFS backtracks to the node from which the current node was reached. The
DFS traversal on D induces two numbering of the nodes, one in the order in
which the nodes are reached by the DFS, DFSnum, and the other in the order in
which the calls to DFS are completed, COMPnum. Moreover, the DFS traversal
partitions the arcs of D into four classes; Tree, Forward, Backward, and Cross
arcs. Let i be a node in D, and DFSnum(i) and COMPnum(i) the respective
DFSnum and COMPnum of i for this DFS traversal on D. By the DFS traversal
on D an arc (i, j) ∈ D belongs to one of the following three classes.
1. Tree or Forward arcs:
iff DFSnum(i) < DFSnum(j) and COMPnum(i) > COMPnum(j).
2. Backward arcs:
iff DFSnum(i) ≥ DFSnum(j) and COMPnum(i) ≤ COMPnum(j).
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8.2 Algorithms for path networks with g = 2
3. Cross arcs:
iff DFSnum(i) > DFSnum(j) and COMPnum(i) > COMPnum(j).
For the request digraph D we always have empty set of Backward arcs since
the graph D is acyclic. That is, COMPnum(i) > COMPnum(j) for every arc
(i, j) ∈ D, as each arc has a node of higher and lower completion number as its
source and sink node respectively. Hence the COMPnum is also a topological
numbering for D. Note that the DFS traversal considers each arc of D exactly
once, therefore, runs in linear time in the number of arcs and nodes in D.
We use the arc partitions returned by the DFS traversal on D to find a circuit
subgraph of D by considering one of the following three possible cases in this
order.
1. There is an arc (i, j) ∈ Forward arcs.
The arc (i, j) and the set of arcs obtained by backtracking from j to i on
the Tree arcs is a closed circuit with unique source node i and sink node j.
2. There is an arc (i, j) ∈ Cross arcs.
By backtracking from both node i and j, if a common node k is reached,
then the arc (i, j) and the set of arcs obtained from backtracking is a closed
circuit with unique source node k and sink node j. Otherwise, the arc (i, j)
and the set of arcs obtained by backtracking is an open circuit with two
source nodes (i.e., the nodes where backtracking from i and j ended), and
sink node j.
3. Both Forward arcs and Cross arcs are empty.
If a single leaf node exists in the tree, backtrack from this leaf node to the
root of the tree and obtain a linear circuit with the root node as the source
and the leaf node as the sink. Otherwise, choose two leaf nodes i and j from
a connected tree component. Backtrack from each to the first node k that
is on the same backtrack path for node i and j. The set of arcs obtained
from backtracking is an open circuit with two sink nodes i and j, and a
unique source node k.
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8.2 Algorithms for path networks with g = 2
By Lemma 6, the circuit subgraphs obtained by Cases 1, 2 and 3 above are
valid circuit subgraphs of D.
At each DFS traversal on D a valid circuit is found and its arcs removed from
D. A closed circuit subgraph is most preferred than an open circuit subgraph (to
minimize the ratios |V i|
|A i
), therefore the Cases 1, 2 and 3 above are considered in
|
that order. That is, Forward arcs first and Cross arcs next unless both are empty,
in which case the last Case 3 is used. Since a circuit subgraph found by the three
cases has one or more arcs, at most |A(D)| calls to DFS search are made by Step
1 of the EDA algorithm to partition the arcs of D into valid circuits of D.
The Forward arcs and Cross arcs classes may contain zero or more arcs following
the DFS traversal on D, but only one such arc may be used to find a
circuit subgraph of D in the Cases 1 and 2 above. The selection of such arc is
based on the following heuristics. For arcs in the Forward arcs class, we use two
different approaches based on the arc length (given by the difference between the
label of the arc source and sink nodes). The first approach always selects the
shortest arc and the second approach always selects the longest arc among the
arcs in the Forward arcs class. For the arcs in the Cross arcs class, on the other
hand, we always select the arc whose source node has the least COMPnum, as
the Cross arc with source node j leads into a closed circuit if COMPnum(i) >
COMPnum(j) (i.e., node j is on same component as the DFS root), where i is
the node the DFS traversal started.
We consider two variants of the EDA algorithm based on the selection approaches
of the arcs in the Forward arcs class.
The EDA-SF algorithm that
always selects the shortest Forward arc and the EDA-LF algorithm that always
selects the longest Forward arc to find a circuit subgraph of D. We remark that
the EDA-SF algorithm is likely to partition the arcs of D into short circuit subgraphs
(in terms of number of arcs in the circuit subgraph) hence find more closed
circuit subgraphs, whereas the EDA-LF algorithm is likely to partition the arcs
of D into long circuit subgraphs (in terms of number of arcs in the subgraph)
hence find fewer closed circuit subgraphs but more open circuit subgraphs. Consequently,
the EDA-SF algorithm is likely to achieve better grooming results than
the EDA-LF algorithm, since closed circuit subgraphs give minimum ratios |V i|
|A i |
than open circuit subgraphs.
129

8.2 Algorithms for path networks with g = 2
8.2.3.2 Merging compatible valid subgraphs
Step 2 of the EDA algorithm merges sets of compatible valid circuit subgraphs
and compatible valid subgraphs into maximal valid subgraphs. A pair of valid
circuit subgraphs is said to be compatible if they cross no link of P n in common.
Let C i and C j be two valid circuit subgraphs of D such that as closed circuits each
has the node u i and u j as source and v i and v j as sink respectively. The circuit
subgraphs C i and C j are compatible if v i ≤ u j or v j ≤ u i . A set of mutually
compatible valid circuit subgraphs can be merged into a valid subgraph. Note
that a valid subgraph is possibly disconnected. Let D i and D j be a pair of valid
subgraphs of D such that each has the node u i and u j as the least label node and
v i and v j as the highest label node respectively. The subgraphs D i and D j are
compatible if v i ≤ u j or v j ≤ u i . A set of mutually compatible valid subgraphs
can be merged into a valid subgraph. A maximal valid subgraph is a valid circuit
or subgraph that is not compatible with any other valid circuit or subgraph. Note
that merging compatible circuits and subgraphs lead to minimization of ADMs
(if a node is shared by a pair of compatible components) and wavelengths.
We find sets of compatible valid subgraphs (valid circuits) by using the valid
subgraphs digraph D vs (V, A) defined as follows. The set V are the valid subgraphs.
There is an arc a = (i, j) ∈ A (i.e., without loss of generality v i ≤ u j ) for every
pair of compatible valid subgraphs D i and D j . The graph D vs is acyclic and may
have several components including trivial ones. The set of nodes on the path of
D vs correspond to a set of compatible valid subgraphs. The longest path in D vc
gives a set of compatible subgraphs of maximum cardinality. We give priority to
compatible components in which adjacent subgraphs in D vs share a node of D to
minimize ADMs requirement.
We describe a heuristic called compatible subgraphs partition (CSP) that takes
as input the graph D vc and iteratively partitions the nodes of D vc by finding a
longest path in D vc and removing all nodes on this path from D vc to form a
partition. The CSP algorithm partitions the nodes of D vc in the following way.
1. While there are more nodes in D vc ,
(a) find a longest path of D vc ;
130

8.2 Algorithms for path networks with g = 2
(b) put the nodes on this path into a new partition and remove the longest
path found from D vc .
Each node partition obtained from the CSP algorithm form a set of compatible
valid subgraphs that is merged into a new valid subgraph.
We say a valid circuit (from Section 8.2.3.1) is type-1 if it is an open circuit
and crosses at least one link of P n twice, and type-2 if it is a linear circuit. We
merge compatible sets of valid circuits and valid subgraphs to obtain a set of
maximal valid subgraphs using the following sequence.
1. Merge compatible valid circuits where adjacent circuits in D vs share a node
of D.
The graph D vs (V, A) has valid circuit subgraphs as nodes and an arc (C i , C j ) ∈
A, for every pair of compatible circuits C i and C j sharing a node of D (i.e.,
without loss of generality v i = u j ). Use the CSP algorithm to obtain valid
subgraphs.
2. Merge any remaining type-1 and type-2 circuits if are compatible.
We say a type-2 circuit C j is compatible to a type-1 circuit C i , if the nodes
{a, b} ∈ C i , and {f, h} ∈ C j , such that a < b and f < h are the degree
one nodes in the underlying graph for C i and C j , where f ≤ a, and b ≤ h.
We define a weighted bipartite graph H = (A ˙∪B, E), where the nodes
for A and B are the type-1 and type-2 circuits respectively, and an edge
e = {i, j} ∈ E, if the type-2 circuit C j is compatible with the type-1 circuit
C i . The edge weight w({C i , C j }) = |V (C i ) ∩ V (C j )|. Use the weighted
matching algorithm in [Edmonds (1965a); Edmonds (1965b)] to find the
weighted matching M for H.
circuit subgraphs C i and C j into valid subgraph.
For each edge {C i , C j } ∈ M, merge the
3. Merge any remaining pairs of type-2 circuits (since they are compatible by
virtue of grooming ratio g = 2).
We define a complete weighted graph H = (V, E) where the nodes are the
type-2 circuits. The edge weight w({C i , C j }) = |V (C i ) ∩ V (C j )|, where C i
and C j is a pair of type-2 circuits. Use the weighted matching algorithm in
[Edmonds (1965a); Edmonds (1965b)] to find the weighted matching M for
131

8.3 Numerical results and performance evaluation
H. For each edge {C i , C j } ∈ M, merge the circuit subgraphs C i and C j to
a valid subgraph.
4. Merge valid subgraphs obtained in Cases 1, 2 and 3 to maximal subgraphs.
The graph D vs (V, A) has the valid subgraphs as nodes and an arc (D i , D j ) ∈
A, for every pair of compatible subgraphs D i and D j (i.e., without loss
of generality v i ≤ u j ). Use the CSP algorithm to obtain maximal valid
subgraphs.
The CSP algorithm iteratively uses the shortest path algorithm for acyclic
digraphs to find longest path on the graph D vs (with negative arc weights), which
is a linear time algorithm in the number of nodes and arcs of the input graph.
In addition, the merge Cases 2 and 3, use the weighted matching algorithms in
[Edmonds (1965a); Edmonds (1965b)], which are cubic in the number of nodes
of the input graph, but are expected to generate small instances.
Therefore
it is likely that the run-time for the EDA algorithm is dominated by merging
valid subgraphs.
The maximal valid subgraphs obtained partition the arcs of
the request digraph D. Let {D i = (V i , A i ) : 0 ≤ i ≤ W − 1}, be the set of
such maximal valid subgraphs. The algorithm EDA outputs W wavelengths and
∑ W −1
i=0
|V i | ADMs as the number wavelengths and ADMs required to satisfy the
requests in D.
8.3 Numerical results and performance evaluation
In this section we report numerical results for ADMs and wavelengths requirements
for the different variants of the algorithms IPA and EDA. The types of
instances which we use in our experiments are described in Section 2.4.1. Note
that for request instances on path networks no routing decisions are needed since
each request is routed on the unique path of the network between the request
end-nodes. We consider UL-Uniform, BL-Uniform, and Gaussian distribution of
requests. We also consider all-to-all instances and instances with FL-Uniform
distribution of requests.
132

8.3 Numerical results and performance evaluation
Distribution EDA IPA
n |R| EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
512 337 397 391 399 490
16 1024 660 794 756 774 969
2048 1289 1567 1444 1479 1903
512 335 385 426 432 480
32 1024 634 755 803 815 937
2048 1215 1487 1535 1556 1832
512 410 433 518 523 561
64 1024 711 803 961 972 1066
2048 1290 1542 1810 1830 2059
512 469 482 566 572 606
96 1024 822 939 1044 1054 1133
2048 1419 1612 1969 1989 2191
1024 881 923 1110 1121 1194
128 2048 1547 1689 2084 2102 2280
Table 8.1: Average number of ADMs for instances with BL-Uniform distribution
of requests.
133

8.3 Numerical results and performance evaluation
Distribution EDA IPA
n |R| EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
512 426 494 456 471 606
16 1024 812 984 869 895 1201
2048 1600 1972 1691 1733 2412
512 487 521 515 530 648
32 1024 885 997 969 1003 1262
2048 1665 1982 1856 1915 2495
512 561 569 586 597 676
64 1024 1031 1069 1095 1122 1315
2048 1895 2059 2055 2125 2577
512 604 607 630 640 706
96 1024 1108 1135 1173 1192 1350
2048 2077 2158 2205 2258 2631
1024 1172 1186 1228 1243 1376
128 2048 2181 2229 2309 2348 2668
Table 8.2: Average number of ADMs for instances with UL-Uniform distribution
of requests.
134

8.3 Numerical results and performance evaluation
Tables 8.1, 8.2 and 8.3 report the numerical results for the BL-Uniform, UL-
Uniform, and Gaussian instances on path networks with 16, 32, 64, 96 and 128
nodes. The network and instance size and the average number of ADMs used
by each algorithm is presented.
The average number of wavelengths used by
our algorithms are reported in Table 8.4, where the average lower bound (W-
LB)– obtained by the general lower bound relation ⌈ L ⌉ in Section 8.1, also is
2
shown. The average is taken using ten random instances. All variants of the
algorithm IPA use optimal number of wavelengths.
Also, all variants of the
EDA algorithm use the same number of wavelengths in most instances, and for
instances where they use different number of wavelengths they differ by only 1 or 2
wavelengths. Thus in Table 8.4 we show only the average number of wavelengths
used by the algorithms EDA-SF and IPA-MWM. For instances with FL-Uniform
distribution of requests where all requests have the same length, Table 8.5 shows
the average number of ADMs used by the algorithms EDA-SF and IPA-MWM.
For this instance type the algorithm EDA-SF always computes optimal number
of ADMs that is equal to the lower bound ADM-LB.
We also consider all-to-all instances for path network P n , for 20 ≤ n ≤ 160,
and algorithms IPA-MWM and EDA-SF. Table 8.6 reports the number of ADMs
used by the algorithms EDA-SF and IPA-MWM, and the optimal bound (LB)
computed by the exact method of [Bermond et al. (June 2005)].
The different variants of algorithm IPA use optimal number of wavelengths,
but differ in ADMs performance significantly. Tables 8.1 - 8.3, show that among
the variants of algorithm IPA the algorithm IPA-MWM has the best ADMs performance
with up to 43% better than the worst performance algorithm IPA-SGM.
The ADMs performance for the algorithm IPA-LWM is very close to that of algorithm
IPA-MWM, differing by just up to 3%. The variants of algorithm EDA
have very close wavelength requirements where occasionally the algorithm EDA-
SF uses 1 or 2 more wavelengths than the algorithm EDA-LF. However, the
algorithm EDA-SF has the best ADMs performance with up to 23% better than
the algorithm EDA-LF.
We now compare the performance of the best performance algorithms amongst
the variants of EDA and IPA, the algorithms EDA-SF and IPA-MWM. Table 8.1
shows that algorithm EDA-SF has the best ADMs performance for instances with
135

8.3 Numerical results and performance evaluation
Distribution EDA IPA
n |R| EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
512 418 469 428 436 576
16 1024 816 898 834 848 1151
2048 1620 1799 1643 1660 2296
512 418 457 427 435 578
32 1024 821 900 834 846 1143
2048 1615 1825 1641 1660 2271
512 416 453 427 435 571
64 1024 818 910 834 846 1136
2048 1624 1790 1647 1664 2246
512 417 463 428 436 572
96 1024 821 939 835 847 1149
2048 1619 1797 1644 1661 2301
1024 818 923 834 845 1153
128 2048 1619 1783 1643 1661 2284
Table 8.3: Average number of ADMs for instances with Gaussian distribution of
requests.
Distribution |R| W-LB EDA-SF IPA-MWM
512 27 29 27
BL-Uniform 1024 52 54 52
2048 99 101 99
512 136 139 136
UL-Uniform 1024 265 270 265
2048 528 532 528
512 146 148 146
Gaussian 1024 288 290 288
2048 575 577 575
Table 8.4: Average number of wavelengths on 64-node path network.
136

8.3 Numerical results and performance evaluation
n ADM-LB EDA-SF IPA-MWM
16 1171 1171 1272
32 1037 1037 1149
64 967 967 1063
96 959 959 1094
Table 8.5: Average number of ADMs for instances with FL-Uniform distribution
of requests, where each request traverses 4 links. The instances have 2048
requests.
n 20 30 40 64 80 96 112 128 144 160
LB 180 408 727 1867 2920 4208 5731 7488 9480 11707
IPA-MWM 195 443 790 2032 3180 4584 6244 8160 10332 12760
EDA-SF 195 445 797 2070 3250 4730 6398 8417 10606 13140
Table 8.6: ADMs requirements for all-to-all instances.
Distribution EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
UL-Uniform 18.81 18.41 88.09 2.67 0.03
Gaussian 58.20 55.41 128.93 6.31 0.03
BL-Uniform 19.08 18.45 67.52 2.08 0.03
Table 8.7: Average time used (in seconds) for path network with 64 nodes and
instance of 2048 requests (using 1.0GHZ AMD Duron Processor with 112.0MB
RAM).
137

8.3 Numerical results and performance evaluation
BL-Uniform distribution of requests with up to 40% better than the algorithm
IPA-MWM. The algorithm EDA-SF maintains good ADMs performance even
on instances with UL-Uniform (see Table 8.2), and Gaussian (see Table 8.3)
distribution of requests. For UL-Uniform instances the ADMs performance for
the algorithm EDA-SF is up to 12% better than that of the algorithm IPA-
MWM. For Gaussian instances, on the other hand, the ADMs performance for the
algorithm EDA-SF is up to 3% better than that of the algorithm IPA-MWM. For
FL-Uniform instances the algorithm EDA-SF always computed optimal number
of ADMs that is equal to the lower bound ADM-LB for all instances we considered
(see Table 8.5). The algorithm IPA-MWM used up to 14% more ADMs over the
optimal number of ADMs for instances with FL-Uniform distribution of requests.
However, the algorithm IPA-MWM is relatively slow than other algorithms as
Table 8.7 shows for path network with 64 nodes and instance of 2048 requests.
For all-to-all instances the algorithm IPA-MWM turned out to outperform
the algorithm EDA-SF slightly, as Table 8.6 shows. Overall both algorithms have
ADMs performance close to the optimal bound (LB), where the algorithm IPA-
MWM and EDA-SF uses at most 8% and 10% more ADMs than the optimal
number of ADMs respectively. We should also mention that the algorithm IPA-
MWM always uses optimal number of wavelengths. But as Table 8.4 shows, the
algorithm EDA-SF uses up to 5 more wavelengths than the optimal number of
wavelengths. Nevertheless, minimizing the number of ADMs is considered as
the primary objective than minimizing the number of wavelengths as ADMs are
considered to be more costly devices.
We report additional numerical results for different grooming algorithms for
path networks in Appendix A.4. We considered networks with 16, 32, 64 and 96
nodes. The various variants of the algorithm IPA all always computed optimal
wavelength allocations, using the number of wavelengths equal to W-LB for all
instances we considered. The variants of algorithm EDA all used at most 5 more
wavelengths than the optimal number of wavelengths W-LB on BL-Uniform and
UL-Uniform instances, but at most 1 more wavelength than the optimal number
of wavelengths W-LB for Gaussian instances. The differences between the ADMs
performance of different algorithms grow when the size of the network increases
for all instance types we considered except for Gaussian instances where the
138

8.4 Extension to path networks with arbitrary grooming ratios
ADMs performance of the algorithms EDA-LF and IPA-MWM converge (see
Tables A.11, A.12, A.13 and A.14).
We summarize the performance results for the algorithms EDA-SF, EDA-
LF, IPA-MWM, IPA-LWM and IPA-SGM in the plots in Figures 8.2, 8.3 and
8.4, for path networks with 64 nodes and the BL-Uniform, UL-Uniform and
Gaussian distribution of requests (note that in Figure 8.4 we do not show the
wavelengths performance for Gaussian instances because all variants of algorithm
IPA use optimal number of wavelengths and all variants of algorithm EDA use
number of wavelengths that differ by at most 1 wavelength from the optimal
number of wavelengths). The plots show that algorithm EDA-SF gives overall
the best performance, followed by the algorithms EDA-LF and IPA-MWM. The
good performance of the edge decomposition based algorithms, and especially the
EDA-SF algorithm, could be attributed to the fact that when finding a circuit
subgraph EDA-SF gives priority to shortest arcs of the Forward arcs class that
lead to more closed circuit subgraphs being found with minimum ratios |V i|
|A i | .
8.4 Extension to path networks with arbitrary
grooming ratios
In this section we extend the algorithms EDA and IPA to path networks with
arbitrary grooming ratios. We describe the extension for the algorithm EDA,
but similar description apply for the algorithm IPA. We use the term grooming
solution on grooming ratio q(or grooming solution for q) to refer to a set of
subgraphs that each satisfies the grooming constraint of q on the path network
P n . Let S be a grooming solution on grooming ratio q. We say H ⊆ S is a partial
grooming solution on grooming ratio q with respect to f denoted by Hq f , when
we wish to transform the set H into grooming solution H ′ on grooming ratio f.
Such H ′ is said to be the partial solution for f.
Given a path network with arbitrary grooming ratio g. We compute the
grooming solution for grooming ratio g by starting with the grooming solution
S on grooming ratio 2 (i.e., determined using the EDA algorithm). We find
the grooming solution on grooming ratio g by recursively using the weighted
139

8.4 Extension to path networks with arbitrary grooming ratios
# ADMs/ADM-LB
1.75
1.7
1.65
1.6
1.55
1.5
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
EDA-SF
EDA-LF
IPA-MWM
IPA-LWM
IPA-SGM
151 301 451 601 751 901 1051 1201 1351 1501
ADM-LB
(a)
1.04
# WAVELENGTHS/W-LB
1.03
1.02
1.01
1
EDA-SF & EDA-LF
IPA-MWM, IPA-LWM & IPA-SGM
0.99
101 201 301 401 501
W-LB
(b)
Figure 8.2: ADMs and Wavelength performance for instances with BL-Uniform
requests on 64-node path network (a) ADMs (b) Wavelengths.
140

8.4 Extension to path networks with arbitrary grooming ratios
# ADMs/ADM-LB
1.7
1.65
1.6
1.55
1.5
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
EDA-SF
EDA-LF
IPA-MWM
IPA-LWM
IPA-SGM
151 301 451 601 751 901 1051 1201 1351 1501 1651
ADM-LB
(a)
1.05
# WAVELENGTHS/W-LB
1.04
1.03
1.02
1.01
1
EDA-SF & EDA-LF
IPA-MWM, IPA-LWM & IPA-SGM
0.99
101 201 301 401 501 601
W-LB
(b)
Figure 8.3: ADMs and Wavelength performance for instances with UL-Uniform
distribution of requests on 64-node path network (a) ADMs (b) Wavelengths.
141

8.4 Extension to path networks with arbitrary grooming ratios
1.5
1.425
# ADMs/ADM-LB
1.35
1.275
1.2
1.125
EDA-SF
EDA-LF
IPA-MWM
IPA-LWM
IPA-SGM
1.05
151 301 451 601 751 901 1051 1201 1351 1501 1651
ADM-LB
Figure 8.4: ADMs performance for instances with Gaussian distribution of requests
on 64-node path network.
142

8.4 Extension to path networks with arbitrary grooming ratios
matching algorithm. That is let g = x + y, for some integer x and y. We obtain
the grooming solution for g by first determining the partial grooming solutions
for x and y from the partial solution on grooming ratio 2 with respect to x and
y respectively. We always choose the number x = 2 t as the largest integer less
or equal to g, for some integer t ≥ 0. From the set S we compute the partial
solution on grooming ratio 2, with respect to x and y, by partitioning the set S
(can be done arbitrarily) into the sets X2
x and Y y
2 with cardinality ⌈ x · |S|⌉ and
g
|S| − ⌈ x · |S|⌉ respectively. Now let X and Y be the partial solutions for x and y.
g
The following algorithm uses the weighted matching algorithm for bipartite
graphs to compute the grooming solution Z for g from the partial solutions X
and Y .
1. Define a complete weighted bipartite graph, G = (A ˙∪B, E), where the nodes
in A are the subgraphs in X and the nodes in B are the subgraphs in Y . The
edge weight w({D i , D j }) = |V (D i ) ∩ V (D j )|, for every pairs of subgraphs
D i ∈ A and D j ∈ B.
2. Find a maximum weight matching M, for the graph G. Denote by Q be
the set of nodes in H not covered by M.
3. For each edge {D i , D j } ∈ M merge the subgraphs D i and D j into a new
subgraphs D ij . Add D ij into Z. For every node k ∈ Q, add the subgraph
D k into Z.
Now consider x = 2 t , where t ≥ 0. For t ≥ 2, compute the set X from X x 2
by using k − 1 iterations of the weighted matching algorithm for general graph
in [Edmonds (1965a); Edmonds (1965b)] with X2
x as the initial input. In each
iteration the solution obtained at the i th iteration is the input to the next (i+1) th
iteration, for 1 ≤ i ≤ k − 1. Let S 0 = X x 2 . The set X is obtained from the output
of the following operations.
1. For 1 ≤ h ≤ k − 1,
(a) S h ← S h−1 , Z ← ∅.
143

8.4 Extension to path networks with arbitrary grooming ratios
(b) Define a complete weighted graph, G = (V, E), where the nodes in V
are the subgraphs in S h . The edge weight w({D i , D j }) = |V (D i ) ∩
V (D j )|, for every distinct pairs of subgraphs D i , D j ∈ V .
(c) Find a maximum weight matching M, for the graph G. Denote by Q
be the set of nodes in H not covered by M.
(d) For each edge {D i , D j } ∈ M merge the subgraphs D i and D j into a
2. Output S h .
new subgraphs D ij . Add D ij into Z. For every node k ∈ Q, add the
subgraph D k into Z. S h ← Z.
For the case t = 1, the set X is the set X x 2 . It remain to consider the case t = 0,
that is x = 1. We define a split operation that transforms the solution X 1 2
solution X. The split operation partitions the arcs of each valid subgraph in X 1 2
into two sets such that each satisfies the grooming ratio constraint of 1.
to
8.4.1 Splitting valid subgraphs
Recall that a valid subgraph consists of a set of valid circuit subgraphs. Let D i
be a subgraph in X 1 2. We adapt the merge sequence in Section 8.2.3.2 to split
D i into its valid circuit subgraph components. By Lemma 6, any valid circuit
subgraph as a closed circuit has unique source and sink node. Thus the pair of
arc disjoint paths between the unique source and sink nodes are split into two
sets as required.
Finally, the set Y is computed from Y y
2 recursively in the same way the solution
for g is obtained from S.
144

Chapter 9
Traffic Grooming on
Unidirectional Path Networks -
All-to-all Instances
In this chapter we extend the work in [Bermond et al. (June 2005)] where they
study the traffic grooming problem for unidirectional path networks with all-to-all
instances and grooming ratio 1 and 2. We establish lower bounds on the number
of ADMs needed for path networks with all-to-all instances and grooming ratio
3 and 4, and give an upper bound for the case of grooming ratio 3.
9.1 Problem definition
We consider unidirectional path network P n with n nodes, numbered 0, . . . , n−1.
Let g be the grooming ratio for wavelengths on the links of P n , that is, each
wavelength in P n can support up to g requests. An all-to-all instance has one
request of unit size (i.e., requires one unit of wavelength bandwidth), for every
node pair (i, j) in P n , 0 ≤ i < j ≤ n − 1. Note that we consider unidirectional
requests, but represent them as directed pairs (i, j), where i < j. We view the
all-to-all instance for P n as a complete graph, G = (V, E), where the node set V
consists of the nodes of P n and the edge set E = {(i, j) : 0 ≤ i < j ≤ n − 1}
represents the requests. We will denote a complete graph on n nodes by K n . Let
145

9.2 Lower bounds
A(P n , g) denote the number of ADMs needed to satisfy the all-to-all instance on
the network P n with grooming ratio g.
The grooming problem for P n and the all-to-all requests can be stated in the
following way. Partition the edges of the complete graph G into W subgraphs
G i
= (V i , E i ), 0 ≤ i ≤ W − 1, where W (not part of the input, but to be
determined as part of the output) is the total number of used wavelengths, such
that
• for each i = 0, . . . , W − 1 and each pair h, h + 1 of consecutive nodes on
P n , no more than g edges in E i cross the link (h, h + 1) on P n , that is,
|{(j, k) ∈ E i : j ≤ h, k ≥ h + 1}| ≤ g (i.e., grooming ratio constraint of g
on P n );
• E i ∩ E j = ∅, ⋃ W −1
i=0
E i = E;
and minimize ∑ W −1
i=0
|V i |. The requests in G i are allocated to wavelength i and
need |V i | ADMs, one at each node which is an end node of a request in G i . Thus
the total number of ∑ W −1
i=0
|V i | ADMs are required. The minimum ∑ W −1
i=0
|V i |
is equal to A(P n , g) since any feasible solution to our grooming problem can be
viewed as a partitioning of graph G satisfying the above two conditions.
For network P n with all-to-all instance on unit rate requests the exact values
of A(P n , 1) and A(P n , 2), are shown in [Bermond et al. (June 2005)]. They show
that A(P n , 1) = 3n2 −2n−ɛ, where ɛ = 1 when n ≡ 1(mod 2) and 0 otherwise,
4
and that A(P n , 2) = ⌈ 11n2 −8n−3⌉ for n ≡ 1(mod 2), and A(P
24 n , 2) = 11n2 −4n
+ ɛ
24 n
for n ≡ 0(mod 2), where ɛ n = 1 when n ≡ 2 or 6(mod 12), ɛ 2 n = 1 when 3
n ≡ 4(mod 12), ɛ n = 5 when n ≡ 10(mod 12) and 0 when n ≡ 0 or 8(mod
6
6). We establish the lower bound for A(P n , 3) and A(P n , 4), and give upper
bound for A(P n , 3) using graph decomposition methods presented in [Bermond
and Schönheim (1977)].
9.2 Lower bounds
We derive the lower bounds for A(P n , 3) and A(P n , 4) following the approach presented
in [Bermond et al. (June 2005)]. Let W denote the number of subgraphs
146

9.2 Lower bounds
G i (V i , E i ) in a partitioning of the request graph G(V, E), a p the number of subgraphs
with exactly p nodes, and A = ∑ W −1
i=0
|V i | the total number of ADMs
required. We have the following relations.
A =
n∑
pa p (9.1)
p=2
W∑
−1
i=0
n∑
a p = W (9.2)
p=2
|E i | = |E| =
n(n − 1)
2
(9.3)
Let L denote the network load induced by the all-to-all instance on P n with
grooming ratio g. The lower bound for the required number of wavelengths, and
the required number of subgraphs partitioning G(V, E) is L . The network load
g
for P n with all-to-all instance is ⌈ n2 −ɛ⌉. Hence we have that
4
⌈ ⌉ n 2 − ɛ
W ≥
(9.4)
4g
where ɛ is 1 for n odd, and 0 otherwise.
Intuitively, since we aim at minimizing ∑ W −1
i=1
|V i | - the number of ADMs
required to support the allocation of requests to wavelengths, we should look for
partitionings of G with small W and small ratios |V i|
|E i
. We first establish the
|
upper bound for the number of edges |E i | in a subgraph G i with |V i | nodes. Let
γ(g, p) be the maximum number of edges possible on a p-node subgraph of the
request graph G(V, E) for a network P n with grooming ratio g. Equation (9.2)
and Inequality (9.4) imply
n∑
⌈ ⌉ n 2 − ɛ
a p ≥ . (9.5)
4g
p=2
Equation (9.3) implies
n∑
a p γ(g, p) ≥
p=2
n(n − 1)
. (9.6)
2
Lemma 7 For p ≥ 2, γ(3, p) = 2p − 3.
147

9.2 Lower bounds
Proof :
This lemma holds for p = 2 since the request graph K 2 has a maximum
of 1 edge. For p = 3 the maximum number of edges is 3, and the graph K 3
achieves the equality. Analogously, for p = 4 we have a maximum of 5 edges,
and the graph K 4 − e (the graph obtained by removing one edge from the graph
K 4 ) achieves the equality. Now assume that the lemma holds for some p ≥ 4,
that is, γ(3, p) = 2p − 3. We show that it also holds for p + 3. Let the vertices
of a p + 3 node subgraph satisfying the grooming ratio constraint be u 0 < . . . <
u p−1 < x 0 < x 1 < x 2 . In this subgraph we can have at most 2p − 3 edges of the
type {u i , u j }, where 0 ≤ i < j ≤ p − 1 (the inductive hypothesis); at most three
edges of the type {u i , x j }, where 0 ≤ i ≤ p − 1 and 0 ≤ j ≤ 2; and additional
three edges of the type {x j , x k }, where 0 ≤ j ≠ k ≤ 2. Thus this subgraph has
at most 2p − 3 + 6 edges. Hence γ(3, p + 3) ≤ 2(p + 3) − 3 , so, by induction , for
each p ≥ 2, γ(3, p) ≤ 2p − 3.
We now show a subgraph of G which has p ≥ 3 nodes, 2p − 3 edges, and
satisfies the grooming ratio constraint. The nodes of this subgraph are u 0 < u 1 <
... < u p−1 . The 2p − 3 edges are (u i , u i+1 ), for i = 0, 1, ..., p − 2, and (u i , u i+2 ), for
i = 0, 1, ..., p−3. To see that this subgraph satisfies the grooming ratio constraint,
take an arbitrary link (j, j + 1) on P n such that u 0 ≤ j < u p−1 and the index
i such that u i ≤ j < u i+1 . This link is crossed by at most three request from
this subgraph: (u i , u i+1 ), (u i , u i+2 ), if i ≤ p − 3, and (u i−1 , u i+1 ), if i ≥ 1. Thus
γ(3, p) ≥ 2p − 3.
Theorem 10 A(P n , 3) ≥
otherwise.
Proof :
Therefore
⌈ ⌈ ⌉⌉
n(n−1)
+ 3 n 2 −ɛ
, where ɛ is 1 for n odd, and 0
4 2 12
By Lemma 7, Inequality (9.6) can be written as
n∑
n(n − 1)
(2p − 3)a p ≥ . (9.7)
2
p=2
n∑
pa p ≥
p=2
n(n − 1)
4
+ 3 2
n∑
a p . (9.8)
Thus by Equation (9.1) and Inequality (9.5) we have
⌈ n(n − 1)
A ≥ + 3 ⌈ ⌉⌉ n 2 − ɛ
. (9.9)
4 2 12
p=2
148

9.2 Lower bounds
Lemma 8 For p ≥ 2, γ(4, p) = ⌊ ⌋
7p−10
3 .
Proof :
For p = 2, 3 and 4 complete graphs K 2 , K 3 and K 4 show that the
equality holds. Now we assume that the lemma holds for some p ≥ 4, that is,
γ(4, p) = ⌊ ⌋
7p−10
3 . We show that it also holds for p + 3. Let the vertices of a p + 3
node subgraph satisfying the grooming ratio constraint be u 0 < . . . < u p−1 <
x 0 < x 1 < x 2 . In this subgraph we can have at most ⌊ ⌋
7p−10
3 edges of the type
{u i , u j }, where 0 ≤ i ≤ p − 1 (the inductive hypothesis); at most four edges of
the type {u i , x j }, where 0 ≤ i ≤ p − 1 and 0 ≤ j ≤ 2; and additional three
edges of the type {x j , x k }, where 0 ≤ i ≠ j ≤ 2. Thus this subgraph has at
most ⌊ ⌋ ⌊ ⌋
7p−10
3 + 7 edges. Hence γ(4, p + 3) ≤ 7(p+3)−10
, so by induction, for each
3
p ≥ 2, γ(4, p) ≤ ⌊ ⌋
7p−10
3 .
We now show a subgraph of G which has p ≥ 4 nodes, ⌊ ⌋
7p−10
3 edges, and
satisfies the grooming ratio constraint. The nodes of this subgraph are u 0 <
u 1 < ... < u p−1 . The edges are (u i , u i+1 ), for i = 0, 1, ..., p − 2, (u i , u i+2 ), for
i = 0, 1, ..., p − 3, and (u 3i , u 3(i+1) ), for i = 0, 1, ..., ⌊(p − 4)/3⌋; see Figure 9.1.
The number of edges in this subgraph is 2p − 3 + ⌊(p − 4)/3⌋ + 1 = ⌊ ⌋
7p−10
3 . To
see that this subgraph satisfies the grooming ratio constraint, take an arbitrary
link (j, j + 1) on P n such that u 0 ≤ j < u p−1 and the index i such that u i ≤
j < u i+1 .
This link is crossed by at most four request from this subgraph:
(u i , u i+1 ), (u i , u i+2 ), if i ≤ p − 3, (u i−1 , u i+1 ), if i ≥ 1, and (u 3⌊i/3⌋ , u 3(⌊i/3⌋+1) ), if
3⌊i/3⌋ ≤ p − 4. Thus γ(4, p) ≥ ⌊ ⌋
7p−10
3 .
u 0
u 1
... ...
u 2
u 3
u 4
u 5
u
p−k
u
p−k+1
u p−k+2
u p−3
u p−2
u p−1
Figure 9.1: A construction for the graph on p nodes that satisfies the grooming
ratio constraint of 4 and achieves the upper bound γ(4, p) on number of edges.
Theorem 11 A(P n , 4) ≥
otherwise.
⌈
3n(n−1)
14
+ 10 7
⌈
⌉⌉
n 2 −ɛ
, where ɛ is 1 for n odd, and 0
16
149

9.3 Upper bound for A(P n , 3)
Proof :
Therefore
By Lemma 8, Inequality (9.6) can be written as
n∑
⌊ ⌋ 7p − 10 n(n − 1)
a p ≥ . (9.10)
3
2
p=2
n∑
pa p ≥
p=2
3n(n − 1)
14
+ 10 7
n∑
a p . (9.11)
Thus by Equation (9.1) and Inequality (9.5) we have
⌈ 3n(n − 1)
A ≥
+ 10 ⌈ ⌉⌉ n 2 − ɛ
. (9.12)
14 7 16
9.3 Upper bound for A(P n , 3)
The complete graph K n is said to have a G-decomposition if it is the union of
edge disjoint subgraphs each isomorphic to G. The G-decomposition problem is
to determine the set of naturals N(G), such that K n has a G-decomposition if
and only if n ∈ N(G). The solution of this problem for small graphs G with
up to four nodes is given in [Bermond and Schönheim (1977)]. In this section
we use some of the decomposition results for K n into subgraphs isomorphic to
K 4 − e in [Bermond and Schönheim (1977)] to find subgraph partitions for the
request graph on P n . We show that the obtained subgraph partitions satisfy the
grooming ratio constraint of g = 3 on P n .
The nodes of K n are labeled from the set {0, 1, . . . , n−1} denoted by Z n as the
numbering of nodes in P n . A sequence of nodes (a, b, c, d) where {a, b, c, d} ⊆ Z n ,
represents a subgraph of K n with four nodes a, b, c, and d, three edges between
the three consecutive pairs of nodes, and the edge between the first node and
the last one. We denote the set of graphs (a + j, b + j, c + j, d + j), j ∈ Z n , by
(a, b, c, d)(mod n), but additions of node labels are done mod n. An over-brace
between non-consecutive nodes a and c denotes the edge {a, c}. That is, ( { }} {
a, b, c, d)
represents the graph isomorphic to K 4 − e with edges {a, b}, {b, c}, {c, d}, {d, a}
and {a, c}.
The following Lemmas 9-10 describe some direct constructions for the decomposition
of K n into subgraphs isomorphic to K 4 − e, given in [Bermond and
Schönheim (1977)].
p=2
150

9.3 Upper bound for A(P n , 3)
Lemma 9 (Bermond and Schönheim (1977)) If n = 10k + 1 then K n has
a K 4 − e decomposition. For k ≥ 4 the following k(10k + 1) subgraphs form a
K 4 − e decomposition of K n :
(1, { 1 + 2i, 2k + }} 2, 4k + i + 1), {
i = 3, 4, . . . , ⌊ k⌋
2
(1, { 2k + 3 − 2i, 2k }} + 2, 5k + 2 − {
i), i = 1, 2, . . . , ⌈ k⌉
2
( { 1, 3, }} 4k + 1, {
6k)
( { 1, 5, }} 2k + 2, {
6k + 1)
⎫
⎪⎬
(mod 10k + 1)
Lemma 10 (Bermond and Schönheim (1977)) If n = 10k+6, the following
k(10k + 6) + 5k + 3 subgraphs form a K 4 − e decomposition of K n :
(1, { 2i, 2 + 2k, }} 4k + 3 + {
⎫
i), i = 1, 2, . . . , ⌈ k⌉
⎪⎬
2
{ }} {
(1, 2i + 2⌈ k (mod 10k + 6)
2 ⌉, 2 + 2k, 4k + 3 + ⌈k 2 ⌉ + i), i = 1, 2, . . . , ⌊ k⌋ ⎪ ⎭
2
( { 1 + i, 2 + 2k }} + i, 5k + 4 + {
i, 7k + 5 + i), i = 0, 1, . . . , 5k + 2.
We are interested in subgraphs which satisfy the grooming ratio constraint g = 3
on P n . Lemma 11 says that the graphs in the set (a, b, c, d)(mod n) satisfy the
grooming ratio constraint g = 2 on P n , if a < b < c < d.
Lemma 11 Let (a, b, c, d)(mod n) be the set of graphs and i, i + 1 a pair of
consecutive nodes in P n .
⎪⎭
If a < b < c < d then the edges of any graph in
(a, b, c, d)(mod n) cross the link (i, i + 1) on P n at most twice.
Proof :
If a < b < c < d, then a link (i, i + 1) on P n is crossed twice by the
edges of subgraph (a, b, c, d), if a ≤ i < d, and is not crossed by any of the
edges of (a, b, c, d), if i < a or i ≥ d. For a fixed integer j ∈ Z n , let the graph
(a + j, b + j, c + j, d + j) be denoted by (ā, ¯b, ¯c, ¯d), that is, ā = (a + j) mod n,
etc. Since a < b < c < d, the nodes of the graph (ā, ¯b, ¯c, ¯d) satisfies one of the
following four possible cases ā < ¯b < ¯c < ¯d, ¯d < ā < ¯b < ¯c, ¯c < ¯d < ā < ¯b,
and ¯b < ¯c < ¯d < ā. In each case, graph (ā, ¯b, ¯c, ¯d) is graph (a ′ , b ′ , c ′ , d ′ ) for some
a ′ < b ′ < c ′ < d ′ , so it crosses a link (i, i + 1) on P n twice or does not cross it at
all.
If a < b < c < d, Lemma 11 implies that the edges of each graph in
( { }} {
a, b, c, d)(mod n) cross any link on P n at most thrice. The following holds in
151

9.3 Upper bound for A(P n , 3)
Lemma 9, for k ≥ 4: 1 < 1 + 2i < 2k + 2 < 4k + i + 1, for 3 ≤ i ≤ ⌊ k 2 ⌋,
1 < 2k + 3 − 2i < 2k + 2 < 5k + 2 − i, for 1 ≤ i ≤ ⌈ k ⌉, 1 < 3 < 4k + 1 < 6k
2
and 1 < 5 < 2k + 2 < 6k + 1. Analogously, the following holds in Lemma 10, for
k ≥ 1: 1 < 2i < 2 + 2k < 4k + 3 + i, for 1 ≤ i ≤ ⌈ k 2 ⌉, 1 < 2i + 2⌈ k 2 ⌉ < 2 + 2k <
4k + 3 + ⌈ k⌉ + i, for 1 ≤ i ≤ ⌊ k ⌋, and 1 + i < 2 + 2k + i < 5k + 4 + i < 7k + 5 + i,
2 2
for 0 ≤ i ≤ 5k + 2. By Lemma 11, it follows that the subgraph partitions in
Lemmas 9 and 10 admit the grooming ratio constraint g = 3 on P n .
We now show the upper bound for A(P n , 3). We denote by K n1 ,n 2
the complete
bipartite graph with n 1 and n 2 nodes for the two node classes of the graph,
respectively. The edges of K n are partitioned into a set of small subgraphs, including
subgraphs isomorphic to K 2 (denoted by (a, b)), subgraphs isomorphic to
K 3 (denoted by (a, b, c)), subgraphs isomorphic to the circle graph on four nodes
(denoted by (a, b, c, d)), subgraphs isomorphic to K 4 − e (denoted by ( { }} {
a, b, c, d)),
and subgraphs isomorphic to K 2,3 (denoted by node partition {{a, b}, {c, d, e}}).
We show that each subgraph of K n satisfies the grooming ratio constraint g = 3 on
P n . This is obvious for the subgraphs (a, b) and (a, b, c). If a < b < c < d, Lemma
11 implies that the subgraphs (a, b, c, d) and ( { }} {
a, b, c, d) satisfy the grooming ratio
constraint g = 3 on P n . Figures 9.2-9.3 show constructions for subgraphs isomorphic
to K 4 − e and K 2,3 respectively, that satisfy the grooming ratio constraint
g = 3 on P n . Figure 9.2 (a)-(c), shows that subgraphs ( { }} {
a, b, c, d) for a < b < d < c,
b < a < c < d, and b < c < d < a satisfy the grooming ratio constraint g = 3
on P n . Figure 9.3 (a)-(c), shows that if a < b < c < d < e, then subgraphs
{{b, d}{a, c, e}}, {{a, e}{b, c, d}}, and {{a, d}{b, c, e}} isomorphic to K 2,3 satisfy
the grooming ratio constraint g = 3 on P n . These facts will be used in the proofs
of Lemmas 12 and 13.
Lemmas 12 and 13 describe direct constructions for the subgraph partitions
of K n that give the upper bound for A(P n , 3). The subgraph partitions described
satisfy the grooming ratio constraint g = 3 on P n , which follows either by Lemma
11 or the constructions given in Figures 9.2 and 9.3. A wavelength is reserved
for each subgraph of K n . The number of ADMs required on each wavelength is
equal to the number of nodes on the subgraph allocated to this wavelength. The
total number of ADMs is the sum of ADMs used on each wavelength.
152

9.3 Upper bound for A(P n , 3)
a b d c
b
a
c
d
b
c
d
a
(a)
(b)
(c)
Figure 9.2: Graphs isomorphic to K 4 − e where (a) a < b < d < c, (b) b < a <
c < d, and (c) b < c < d < a.
a
b
c
d
e
a
b c d e
a
b c d e
(a)
(b)
(c)
Figure 9.3: Graphs isomorphic to K 2,3 where (a) {{b, d}{a, c, e}}, (b)
{{a, e}{b, c, d}}, and (c) {{a, d}{b, c, e}}, for a < b < c < d < e.
153

9.3 Upper bound for A(P n , 3)
Lemma 12 For 4 ≤ n ≤ 39, we have that
Proof :
A(P 4 , 3) ≤ 6, A(P 5 , 3) ≤ 10, A(P 6 , 3) ≤ 12, A(P 7 , 3) ≤ 19,
A(P 8 , 3) ≤ 24, A(P 9 , 3) ≤ 30, A(P 10 , 3) ≤ 36, A(P 11 , 3) ≤ 47,
A(P 12 , 3) ≤ 54, A(P 13 , 3) ≤ 64, A(P 14 , 3) ≤ 76, A(P 15 , 3) ≤ 87,
A(P 16 , 3) ≤ 96, A(P 17 , 3) ≤ 114, A(P 18 , 3) ≤ 124, A(P 19 , 3) ≤ 139,
A(P 20 , 3) ≤ 156, A(P 21 , 3) ≤ 168, A(P 22 , 3) ≤ 188, A(P 23 , 3) ≤ 205,
A(P 24 , 3) ≤ 224, A(P 25 , 3) ≤ 244, A(P 26 , 3) ≤ 260, A(P 27 , 3) ≤ 285,
A(P 28 , 3) ≤ 304, A(P 29 , 3) ≤ 328, A(P 30 , 3) ≤ 352, A(P 31 , 3) ≤ 377,
A(P 32 , 3) ≤ 402, A(P 33 , 3) ≤ 430, A(P 34 , 3) ≤ 456, A(P 35 , 3) ≤ 485,
A(P 36 , 3) ≤ 504, A(P 37 , 3) ≤ 544, A(P 38 , 3) ≤ 566, A(P 39 , 3) ≤ 597.
K n , for 4 ≤ n ≤ 39.
We exhibit case by case the subgraph partitions for the request graph
1. A(P 4 , 3) ≤ 6: K 4 is the union of the subgraphs ( { }} {
0, 1, 2, 3) and (1, 3). Thus
2 wavelengths and 6 ADMs are sufficient to support the request graph K 4 .
That is, 1 wavelength uses 4 ADMs for the subgraph isomorphic to K 4 − e
and 1 wavelength uses 2 ADMs for the subgraph isomorphic to K 2 .
2. A(P 5 , 3) ≤ 10: K 5 is the union of the subgraphs (0, 2, 4), (1, 3, 4), and
(0, 1, 2, 3). Thus 3 wavelengths and 10 ADMs are sufficient to support the
request graph K 5 . That is, 2 wavelengths uses 3 ADMs each for the subgraphs
isomorphic to K 3 , and 1 wavelength uses 4 ADMs for the subgraph
isomorphic to C 4 (i.e., the circle graph on four nodes).
3. A(P 6 , 3) ≤ 12: K 6 is the union of the subgraphs ( { 0, }} 1, 5, {
3), ( 2, { }} 0, 4, {
5) and
( 1, { }} 2, 3, {
4). Thus 3 wavelengths and 12 ADMs are sufficient to support the
request graph K 6 , such that each of the wavelengths uses 4 ADMs for the
subgraphs isomorphic to K 4 − e.
4. A(P 7 , 3) ≤ 19: K 7 is the union of the subgraphs K 5 (on the set of nodes
{0, 2, 3, 4, 6}), ( { }} {
1, 2, 5, 4), and K 2,3 (with node partition {{1, 5}, {0, 3, 6}}).
Thus 5 wavelengths and 19 ADMs are sufficient to support the request
graph K 7 . That is, 3 wavelengths use 10 ADMs for the subgraphs of K 5
(from Case 2), 1 wavelength uses 4 ADMs for the subgraph isomorphic to
K 4 − e, and 1 wavelength uses 5 ADMs for the subgraph K 2,3 .
154

9.3 Upper bound for A(P n , 3)
5. A(P 8 , 3) ≤ 24: K 8 is the union of the subgraphs ( 0, { }} 2, 4, {
6), ( 1, { }} 3, 6, {
7),
( 6, { }} 1, 2, {
5), ( { 7, }} 0, 3, {
4), (0, 1, 4, 5) and (2, 3, 6, 7). Thus 6 wavelengths and 24
ADMs are sufficient to support the request graph K 8 , such that 4 wavelengths
uses 4 ADMs each for the subgraphs isomorphic to K 4 − e, and 2
wavelengths uses 4 ADMs each for the subgraphs isomorphic C 4 .
6. A(P 9 , 3) ≤ 30: K 9 is the union of the subgraphs K 6 (on nodes {1, 2, 3, 5, 6, 7}),
(0, 4, 8) and K 6,3 (with node partition {{1, 2, 3, 5, 6, 7}, {0, 4, 8}}). The subgraph
K 6,3 is the union of 3 subgraphs isomorphic to K 2,3 , with node partitions
{{k, k + 4}, {0, 4, 8}}, for 1 ≤ k ≤ 3. Thus 7 wavelengths and 30
ADMs are sufficient to support the request graph K 9 .
That is, 3 wavelengths
use 12 ADMs for the subgraphs of K 6 (from Case 3), 1 wavelength
uses 3 ADMs for the subgraph isomorphic to K 3 , and 3 wavelengths uses 5
ADMs each for the subgraphs of K 6,3 .
7. A(P 10 , 3) ≤ 36: K 10 is the union of the subgraphs K 6 (on nodes {0, 2, 4, 6, 8,
9}), K 4 (on nodes {1, 3, 5, 7}) and K 6,4 (with node partition {{0, 2, 4, 6, 8, 9},
{1, 3, 5, 7}}). The subgraphs K 4 and K 6,4 are the union of the subgraphs
( 1, { }} 0, 3, {
6), ( { 1, }} 2, 5, {
8), ( { 1, }} 4, 7, {
9), ( 3, { }} 4, 5, {
9), ( 3, { }} 2, 7, {
8), and ( 5, { }} 0, 7, {
6). Thus
9 wavelengths and 36 ADMs are sufficient to support the request graph K 10 .
That is, 3 wavelengths use 12 ADMs for the subgraphs of K 6 (from Case
3), and 6 wavelengths uses 4 ADMs each for the subgraphs of K 4 and K 6,4 .
8. A(P 11 , 3) ≤ 47: K 11 is the union of the subgraphs K 6 (on nodes {1, 2, 3, 6, 7,
8}), K 5 (on nodes {0, 4, 5, 9, 10}), and K 6,5 (with node partition {{1, 2, 3, 6,
7, 8}, {0, 4, 5, 9, 10}}). The subgraph K 6,5 is the union of 5 subgraphs isomorphic
to K 2,3 , with node partitions {{0, 10}, {5k + 1, 5k + 2, 5k + 3}},
for k ∈ {0, 1}, and {{k, k + 5}, {4, 5, 9}}, for 1 ≤ k ≤ 3. Thus 11 wavelengths
and 47 ADMs are sufficient to support the request graph K 11 . That
is, 3 wavelengths use 12 ADMs for the subgraphs of K 6 (from Case 3), 3
wavelengths use 10 ADMs for the subgraphs of K 5 (from Case 2), and 5
wavelengths uses 5 ADMs each for the subgraphs of K 6,4 .
9. A(P 12 , 3) ≤ 54: K 12 is the union of the subgraphs K 10 (on nodes {1, 2, 3, 4,
5, 6, 7, 8, 9, 11}), (0, 10, 11), and K 9,2 (with node partition {{1, . . . , 9, 11},
155

9.3 Upper bound for A(P n , 3)
{0, 10}}). The subgraph K 9,2 is the union of 3 subgraphs isomorphic to
K 2,3 , with node partitions {{0, 10}, {3k + 1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ 2.
Thus 13 wavelengths and 54 ADMs are sufficient to support the request
graph K 12 . That is, 9 wavelengths use 36 ADMs for the subgraphs of K 10
(from Case 7), 1 wavelength uses 3 ADMs for the subgraph isomorphic to
K 3 , and 3 wavelengths uses 5 ADMs each for the subgraphs of K 9,2 .
10. A(P 13 , 3) ≤ 64: K 13 is the union of the subgraphs K 10 (on nodes {1, . . . , 5,
7, . . . , 11}), (0, 6, 12), and K 10,3 (with node partition {{1, . . . , 5, 7, . . . , 11},
{0, 6, 12}}). The subgraph K 10,3 is the union of 5 subgraphs isomorphic to
K 2,3 , with node partitions {{k, k + 6}, {0, 6, 12}}, for 1 ≤ k ≤ 5. Thus 15
wavelengths and 64 ADMs are sufficient to support the request graph K 13 .
That is, 9 wavelengths use 36 ADMs for the subgraphs of K 10 (from Case
7), 1 wavelengths uses 3 ADMs for the subgraph isomorphic to K 3 , and 5
wavelengths uses 5 ADMs each for the subgraphs of K 10,3 .
11. A(P 14 , 3) ≤ 76: K 14 is the union of the subgraphs K 10 (on nodes {2, . . . , 10,
13}), K 5 (on nodes {0, 1, 11, 12, 13}), and K 9,4 (with node partition {{2, . . . ,
10}, {0, 1, 11, 12}}). The subgraph K 9,4 is the union of 6 subgraphs isomorphic
to K 2,3 , with node partitions {{0, 11}, {3k + 2, 3k + 3, 3k + 4}}, and
{{1, 12}, {3k + 2, 3k + 3, 3k + 4}}, for 0 ≤ k ≤ 2. Thus 18 wavelengths and
76 ADMs are sufficient to support the request graph K 14 . That is, 9 wavelengths
use 36 ADMs for the subgraphs of K 10 (from Case 7), 3 wavelengths
use 10 ADMs for the subgraphs of K 5 (from Case 2), and 6 wavelengths
uses 5 ADMs each for the subgraphs of K 9,4 .
12. A(P 15 , 3) ≤ 87: K 15 is the union of the subgraphs K 13 (on nodes {1, 2,
. . . , 12, 14}, (0, 13, 14), and K 12,2 (with node partition {{1, . . . , 12, }, {0,
13}}). The subgraph K 12,2 is the union of 4 subgraphs isomorphic to K 2,3 ,
with node partitions {{0, 13}, {3k + 1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ 3. Thus
20 wavelengths and 87 ADMs are sufficient to support the request graph
K 15 . That is, 15 wavelengths use 64 ADMs for the subgraphs of K 13 (from
Case 10), 1 wavelength uses 3 ADMs for the subgraph isomorphic to K 3 ,
and 4 wavelengths uses 5 ADMs each for the subgraphs of K 12,2 .
156

9.3 Upper bound for A(P n , 3)
13. A(P 16 , 3) ≤ 96: K 16 is the union of 24 subgraphs isomorphic to K 4 −e, given
by Lemma 10, for k = 1. Thus 24 wavelengths and 96 ADMs are sufficient
to support the request graph K 16 , such that each of the wavelengths uses 4
ADMs for the subgraphs isomorphic to K 4 − e.
14. A(P 17 , 3) ≤ 114: K 17 is the union of the subgraphs K 14 (on nodes {0, . . . , 6,
10, . . . , 16}), (7, 8, 9), and K 14,3 (with node partition {{0, . . . , 6, 10, . . . , 16},
{7, 8, 9}}). The subgraph K 14,3 is the union of 7 subgraphs isomorphic to
K 2,3 , with node partitions {{k, k + 10}, {7, 8, 9}}, for 0 ≤ k ≤ 6. Thus 26
wavelengths and 114 ADMs are sufficient to support the request graph K 17 .
That is, 18 wavelengths use 76 ADMs for the subgraphs of K 14 (from Case
11), 1 wavelength uses 3 ADMs for the subgraph isomorphic to K 3 , and 7
wavelengths uses 5 ADMs each for the subgraphs of K 14,3 .
15. A(P 18 , 3) ≤ 124: K 18 is the union of the subgraphs K 16 (on nodes {1, 2, . . . ,
15, 17}), (0, 16, 17), and K 15,2 (with node partition {{1, 2, . . . , 15}, {0, 16}}).
The subgraph K 15,2 is the union of 5 subgraphs isomorphic to K 2,3 , with
node partitions {{0, 16}, {3k + 1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ 4. Thus 30
wavelengths and 124 ADMs are sufficient to support the request graph K 18 .
That is, 24 wavelengths use 96 ADMs for the subgraphs of K 16 (from Case
13), 1 wavelength uses 3 ADMs for the subgraph isomorphic to K 3 , and 5
wavelengths uses 5 ADMs each for the subgraphs of K 15,2 .
16. A(P 19 , 3) ≤ 139: K 19 is the union of the subgraphs K 16 (on nodes {0, . . . , 7,
11, . . . , 18}), (8, 9, 10), and K 16,3 (with node partition {{0, 1, . . . , 7, 11, . . . ,
18}, {8, 9, 10}}). The subgraph K 16,3 is the union of 8 subgraphs isomorphic
to K 2,3 , with node partitions {{k, k + 11}, {8, 9, 10}}, for 0 ≤ k ≤ 7. Thus
33 wavelengths and 139 ADMs are sufficient to support the request graph
K 19 . That is, 24 wavelengths use 96 ADMs for the subgraphs of K 16 (from
Case 13), 1 wavelength uses 3 ADMs for the subgraph isomorphic to K 3 ,
and 8 wavelengths uses 5 ADMs each for the subgraphs of K 16,3 .
17. A(P 20 , 3) ≤ 156: K 20 is the union of the subgraphs K 16 (on nodes {2, . . . ,
16, 19}), K 5 (on nodes {0, 1, 17, 18, 19}), and K 15,4 (with node partition
157

9.3 Upper bound for A(P n , 3)
{{2, . . . , 16}, {0, 1, 17, 18}}). The subgraph K 15,4 is the union of 10 subgraphs
isomorphic to K 2,3 , with node partitions {{0, 17}, {3k+2, 3k+3, 3k+
4}}, and {{1, 18}, {3k + 2, 3k + 3, 3k + 4}} for 0 ≤ k ≤ 4. Thus 37 wavelengths
and 156 ADMs are sufficient to support the request graph K 20 . That
is, 24 wavelengths use 96 ADMs for the subgraphs of K 16 (from Case 13),
3 wavelengths use 10 ADMs for the subgraphs of K 5 , and 10 wavelengths
uses 5 ADMs each for the subgraphs of K 15,4 .
18. A(P 21 , 3) ≤ 168: K 21 is the union of the subgraphs ( { 1, 3, }} 11, {
15)(mod 21)
and ( { 1, }} 2, 7, {
10)(mod 21), [Bermond and Schönheim (1977)]. By Lemma 11,
each of the subgraphs admits the grooming ratio constraint g = 3 on P n .
Thus 42 wavelengths and 168 ADMs are sufficient to support the request
graph K 21 , such that each of the wavelengths uses 4 ADMs for the subgraph
isomorphic to K 4 − e.
19. A(P 22 , 3) ≤ 188: K 22 is the union of the subgraphs K 16 (on nodes {0, . . . , 7,
14, . . . , 21}), K 6 (on nodes {8, . . . , 13}), and K 16,6 (with node partition
{{0, . . . , 7, 14, . . . , 21}, {8, . . . , 13}}). The subgraph K 16,6 is the union of 16
subgraphs isomorphic to K 2,3 , with node partitions {{k, k +14}, {8, 9, 10}},
and {{k, k + 14}, {11, 12, 13}} for 0 ≤ k ≤ 7. Thus 43 wavelengths and
188 ADMs are sufficient to support the request graph K 22 . That is, 24
wavelengths use 96 ADMs for the subgraphs of K 16 (from Case 13), 3 wavelengths
use 12 ADMs for the subgraphs of K 6 , and 16 wavelengths uses 5
ADMs each for the subgraphs of K 16,6 .
20. A(P 23 , 3) ≤ 205: K 23 is the union of the subgraphs K 21 (on nodes {1, . . . , 21}),
(0, 22), and K 21,2 (with node partition {{1, . . . , 21}, {0, 22}}). The subgraph
K 21,2 is the union of 7 subgraphs isomorphic to K 2,3 , with node
partitions {{0, 22}, {3k + 1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ 6. Thus 50
wavelengths and 205 ADMs are sufficient to support the request graph K 23 .
That is, 42 wavelengths use 168 ADMs for the subgraphs of K 21 (from Case
18),1 wavelength uses 2 ADMs for the subgraph isomorphic to K 2 , and 7
wavelengths uses 5 ADMs each for the subgraphs of K 21,2 .
158

9.3 Upper bound for A(P n , 3)
21. A(P 24 , 3) ≤ 224: K 24 is the union of the subgraphs K 21 (on nodes {0, . . . , 9,
13, . . . , 23}), K 4 (on nodes {10, 11, 12, 23}), and K 20,3 (with node partition
{{0, . . . , 9, 13, . . . , 22}, {10, 11, 12}}). The subgraph K 20,3 is the union of 10
subgraphs isomorphic to K 2,3 , with node partitions {{k, k+13}, {10, 11, 12}},
for 0 ≤ k ≤ 9. Thus 54 wavelengths and 224 ADMs are sufficient to support
the request graph K 24 . That is, 42 wavelengths use 168 ADMs for
the subgraphs of K 21 (from Case 18), 2 wavelengths use 6 ADMs for the
subgraphs of K 4 (from Case 1), and 10 wavelengths uses 5 ADMs each for
the subgraphs of K 20,3 .
22. A(P 25 , 3) ≤ 244: K 25 is the union of the subgraphs K 21 (on nodes {2, . . . , 22}),
K 4 (on nodes {0, 1, 23, 24}), and K 21,4 (with node partition {{2, . . . , 22},
{0, 1, 23, 24}}). The subgraph K 21,4 is the union of 14 subgraphs isomorphic
to K 2,3 , with node partitions {{0, 23}, {3k + 1, 3k + 2, 3k + 3}} and
{{1, 24}, {3k + 2, 3k + 3, 3k + 4}}, for 0 ≤ k ≤ 6. Thus 58 wavelengths
and 244 ADMs are sufficient to support the request graph K 25 . That is,
42 wavelengths use 168 ADMs for the subgraphs of K 21 (from Case 18), 2
wavelengths use 6 ADMs for the subgraphs of K 4 (from Case 1), and 14
wavelengths uses 5 ADMs each for the subgraphs of K 21,4 .
23. A(P 26 , 3) ≤ 260: K 26 is the union of 65 subgraphs isomorphic to K 4 − e,
given by Lemma 10, for k = 2. Thus 65 wavelengths and 260 ADMs are sufficient
to support the request graph K 26 , such that each of the wavelengths
uses 4 ADMs for the subgraphs isomorphic to K 4 − e.
24. A(P 27 , 3) ≤ 285: K 27 is the union of the subgraphs K 21 (on nodes {3, . . . , 23}),
K 6 (on nodes {0, 1, 2, 24, 25, 26}), and K 21,6 (with node partition {{3, . . . , 23},
{0, 1, 2, 24, 25, 26}}). The subgraph K 21,6 is the union of 21 subgraphs isomorphic
to K 2,3 , with node partitions {{0, 24}, {3k + 3, 3k + 4, 3k + 5}},
{{1, 25}, {3k + 3, 3k + 4, 3k + 5}}, and {{2, 26}, {3k + 3, 3k + 4, 3k + 5}},
for 0 ≤ k ≤ 6. Thus 66 wavelengths and 285 ADMs are sufficient to support
the request graph K 27 . That is, 42 wavelengths use 168 ADMs for
the subgraphs of K 21 (from Case 18), 3 wavelengths use 12 ADMs for the
159

9.3 Upper bound for A(P n , 3)
subgraphs of K 6 (from Case 3), and 21 wavelengths uses 5 ADMs each for
the subgraphs of K 21,6 .
25. A(P 28 , 3) ≤ 304: K 28 is the union of the subgraphs K 26 (on nodes {1, 2, . . . , 25,
27}), ( 0, { 1, }} 26, {
27), and K 24,2 (with node partition {{2, . . . , 25}, {0, 26}}).
The subgraph K 24,2 is the union of 8 subgraphs isomorphic to K 2,3 , with
node partitions {{0, 26}, {3k + 2, 3k + 3, 3k + 4}}, for 0 ≤ k ≤ 7. Thus 74
wavelengths and 304 ADMs are sufficient to support the request graph K 28 .
That is, 65 wavelengths use 260 ADMs for the subgraphs of K 26 (from Case
23), 1 wavelength uses 4 ADMs for the subgraph isomorphic to K 4 − e, and
8 wavelengths uses 5 ADMs each for the subgraphs of K 24,2 .
26. A(P 29 , 3) ≤ 328: K 29 is the union of the subgraphs K 26 (on nodes {0, . . . , 12,
16, . . . , 28}), (13, 14, 15), and K 26,3 (with node partition {{0, 1, . . . , 12, 16,
. . . , 28}, {13, 14, 15}}). The subgraph K 26,3 is the union of 13 subgraphs
isomorphic to K 2,3 , with node partitions {{k, k + 16}, {13, 14, 15}}, for
0 ≤ k ≤ 12. Thus 79 wavelengths and 328 ADMs are sufficient to support
the request graph K 29 . That is, 65 wavelengths use 260 ADMs for
the subgraphs of K 26 (from Case 23), 1 wavelength uses 3 ADMs for the
subgraph isomorphic to K 3 , and 13 wavelengths uses 5 ADMs each for the
subgraphs of K 26,3 .
27. A(P 30 , 3) ≤ 352: K 30 is the union of the subgraphs K 28 (on nodes {1, . . . , 27,
29}, (0, 28, 29), and K 27,2 (with node partition {{1, . . . , 27}, {0, 28}}). The
subgraph K 27,2 is the union of 9 subgraphs isomorphic to K 2,3 , with node
partitions {{0, 28}, {3k + 1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ 8. Thus 84
wavelengths and 352 ADMs are sufficient to support the request graph K 30 .
That is, 74 wavelengths use 304 ADMs for the subgraphs of K 28 (from Case
25), 1 wavelength uses 3 ADMs for the subgraph isomorphic to K 3 , and 9
wavelengths uses 5 ADMs each for the subgraphs of K 27,2 .
28. A(P 31 , 3) ≤ 377: K 31 is the union of the subgraphs K 29 (on nodes {1, . . . , 28,
30}), ( { }} {
0, 1, 29, 30), and K 27,2 (with node partition {{2, . . . , 28}, {0, 29}}).
The subgraph K 27,2 is the union of 9 subgraphs isomorphic to K 2,3 , with
node partitions {{0, 29}, {3k + 2, 3k + 3, 3k + 4}}, for 0 ≤ k ≤ 8. Thus 89
160

9.3 Upper bound for A(P n , 3)
wavelengths and 377 ADMs are sufficient to support the request graph K 31 .
That is, 79 wavelengths use 328 ADMs for the subgraphs of K 29 (from Case
26), 1 wavelength uses 4 ADMs for the subgraph isomorphic to K 4 − e, and
9 wavelengths uses 5 ADMs each for the subgraphs of K 27,2 .
29. A(P 32 , 3) ≤ 402: K 32 is the union of the subgraphs K 26 (on nodes {0, 1, . . . , 12,
19, . . . , 31}), K 6 (on nodes {13, . . . , 18}), and K 26,6 (with node partition
{{0, . . . , 12, 19, . . . , 31}, {13, . . . , 18}}). The subgraph K 26,6 is the union of
26 subgraphs isomorphic to K 2,3 , with node partitions {{k, k+19}, {13, 14, 15}},
and {{k, k + 19}, {16, 17, 18}}, for 0 ≤ k ≤ 12. Thus 94 wavelengths and
402 ADMs are sufficient to support the request graph K 32 . That is, 65
wavelengths use 260 ADMs for the subgraphs of K 26 (from Case 23), 3
wavelengths use 12 ADMs for the subgraphs of K 6 (from Case 3), and 26
wavelengths uses 5 ADMs each for the subgraphs of K 26,6 .
30. A(P 33 , 3) ≤ 430: K 33 is the union of the subgraphs K 30 (on nodes {0, 1, . . . , 14,
18, . . . , 32}), (15, 16, 17), and K 30,3 (with node partition {{0, 1, . . . , 14, 18,
. . . , 32}, {15, . . . , 17}}). The subgraph K 30,3 is the union of 15 subgraphs
isomorphic to K 2,3 , with node partitions {{k, k + 18}, {15, 16, 17}}, for
0 ≤ k ≤ 14. Thus 100 wavelengths and 430 ADMs are sufficient to support
the request graph K 33 . That is, 84 wavelengths use 352 ADMs for the subgraphs
of K 30 (from Case 27), 1 wavelength uses 3 ADMs for the subgraph
isomorphic to K 3 , and 15 wavelengths uses 5 ADMs each for the subgraphs
of K 30,3 .
31. A(P 34 , 3) ≤ 456: K 34 is the union of the subgraphs K 28 (on nodes {0, . . . , 13,
20, . . . , 33}), K 6 (on nodes {14, . . . , 19}), and K 28,6 (with node partition
{{0, . . . , 13, 20, . . . , 33}, {14, . . . , 19}}). The subgraph K 28,6 is the union of
28 subgraphs isomorphic to K 2,3 , with node partitions {{k, k+20}, {14, 15, 16}}
and {{k, k + 20}, {17, 18, 19}}, for 0 ≤ k ≤ 13. Thus 105 wavelengths and
456 ADMs are sufficient to support the request graph K 34 . That is, 74
wavelengths use 304 ADMs for the subgraphs of K 28 (from Case 25), 3
wavelengths use 12 ADMs for the subgraphs of K 6 , and 28 wavelengths
uses 5 ADMs each for the subgraphs of K 28,6 .
161

9.3 Upper bound for A(P n , 3)
32. A(P 35 , 3) ≤ 485: K 35 is the union of the subgraphs K 32 (on nodes {0, . . . , 15,
19, . . . , 34}), (16, 17, 18), and K 32,3 (with node partition {{0, 1, . . . , 15, 19,
. . . , 34}, {16, 17, 18}}). The subgraph K 32,3 is the union of 16 subgraphs
isomorphic to K 2,3 , with node partitions {{k, k + 19}, {16, 17, 18}}, for
0 ≤ k ≤ 15. Thus 111 wavelengths and 485 ADMs are sufficient to support
the request graph K 35 . That is, 94 wavelengths use 402 ADMs for the subgraphs
of K 32 (from Case 29), 1 wavelength uses 3 ADMs for the subgraph
isomorphic to K 3 , and 16 wavelengths uses 5 ADMs each for the subgraphs
of K 32,3 .
33. A(P 36 , 3) ≤ 504: K 36 is the union of 126 subgraphs isomorphic to K 4 − e,
given by Lemma 10, for k = 3. Thus 126 wavelengths and 504 ADMs are
sufficient to support the request graph K 36 , such that each of the wavelengths
uses 4 ADMs for the subgraphs isomorphic to K 4 − e.
34. A(P 37 , 3) ≤ 544: K 37 is the union of the subgraphs K 34 (on nodes {0, . . . , 16,
20, . . . , 36}), (17, 18, 19), and K 34,3 (with node partition {{0, 1, . . . , 16, 20,
. . . , 36}, {17, 18, 19}}). The subgraph K 34,3 is the union of 17 subgraphs
isomorphic to K 2,3 , with node partitions {{k, k + 20}, {17, 18, 19}}, for
0 ≤ k ≤ 16. Thus 123 wavelengths and 544 ADMs are sufficient to support
the request graph K 37 . That is, 105 wavelengths use 456 ADMs for
the subgraphs of K 34 (from Case 31), 1 wavelength uses 3 ADMs for the
subgraph isomorphic to K 3 , and 17 wavelengths uses 5 ADMs each for the
subgraphs of K 34,3 .
35. A(P 38 , 3) ≤ 566: K 38 is the union of the subgraphs K 36 (on nodes {1, . . . , 36}),
(0, 37), and K 36,2 (with node partition {{1, . . . , 36}, {0, 37}}). The subgraph
K 36,2 is the union of 12 subgraphs isomorphic to K 2,3 , with node
partitions {{0, 37}, {3k + 1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ 11. Thus 139
wavelengths and 566 ADMs are sufficient to support the request graph K 38 .
That is, 126 wavelengths use 504 ADMs for the subgraphs of K 36 (from
Case 33), 1 wavelength uses 2 ADMs for the subgraph isomorphic to K 2 ,
and 12 wavelengths uses 5 ADMs each for the subgraphs of K 36,2 .
162

9.3 Upper bound for A(P n , 3)
36. A(P 39 , 3) ≤ 597: K 39 is the union of the subgraphs K 36 (on nodes {0, . . . , 17,
21, . . . , 38}), (18, 19, 20), and K 36,3 (with node partition {{0, . . . , 17, 21, . . . ,
38}, {18, 19, 20}}). The subgraph K 36,3 is the union of 18 subgraphs isomorphic
to K 2,3 , with node partitions {{k, k+21}, {18, 19, 20}}, for 0 ≤ k ≤ 17.
Thus 145 wavelengths and 597 ADMs are sufficient to support the request
graph K 39 . That is, 126 wavelengths use 504 ADMs for the subgraphs of
K 36 (from Case 33), 1 wavelength uses 3 ADMs for the subgraph isomorphic
to K 3 , and 18 wavelengths uses 5 ADMs each for the subgraphs of K 36,3 .
Lemma 13 For n ≥ 40 we have that
n A(P n , 3) ≤ ɛ
n ≡ 0(mod 5)
n ≡ 1(mod 5)
n ≡ 2(mod 5)
n ≡ 3(mod 5)
n ≡ 4(mod 5)
10, if n ≡ 1(mod 3)
6n 2 −4n+ɛ
15
20, if n ≡ 2(mod 3)
0, if n ≡ 0(mod 3)
2n 2 −2n
5
2n 2 −n−6
5
16, if n ≡ 2(mod 3)
6n 2 −5n+ɛ
15
6, if n ≡ 0(mod 3)
−4, if n ≡ 1(mod 3)
3, if n ≡ 1(mod 2)
4n 2 −3n+ɛ
10
8, if n ≡ 0(mod 2)
Proof : We consider separately the five cases defined by the value of n mod 5.
1. For n ≡ 1(mod 5), A(P n , 3) ≤ 2n2 −2n
5
.
Note that for n ≡ 1(mod 5) we have that n ≡ 1(mod 10) or n ≡ 6(mod 10).
By Lemmas 9 (for n ≡ 1(mod 10)) and 10 (for n ≡ 6(mod 10)), the graph
K n is the union of n2 −n
subgraphs isomorphic to K
10 4 − e. Therefore n2 −n
10
wavelengths and 2n2 −2n
ADMs are sufficient to support the request graph
5
K n .
2. For n ≡ 0(mod 5), A(P n , 3) ≤ 6n2 −4n+ɛ
15
, where ɛ = 10, 20, and 0, for
n ≡ 1(mod 3), n ≡ 2(mod 3), and n ≡ 0(mod 3), respectively.
We consider the cases n ≡ 1(mod 3), n ≡ 2(mod 3), and n ≡ 0(mod 3) as
follows.
163

9.3 Upper bound for A(P n , 3)
(a) n ≡ 1(mod 3): K n is the union of the subgraphs K n−4 (on nodes
{2, . . . , n−3}), K 4 (on nodes {0, 1, n−2, n−1}), and K n−4,4 (with node
partition {{2, . . . , n − 3}, {0, 1, n − 2, n − 1}}). The subgraph K n−4,4 is
the union of 2× n−4 subgraphs isomorphic to K
3 2,3 , with node partitions
{{j, j +n−2}, {3k +2, 3k +3, 3k +4}}, for 0 ≤ j ≤ 1, 0 ≤ k ≤ n−4 −1. 3
Since n − 4 ≡ 1(mod 5), by Case 1 of this lemma the subgraph K n−4
is the union of n2 −9n+20
subgraphs isomorphic to K
10 4 − e. By Case 1
of Lemma 12 the subgraph K 4 is the union of two subgraphs. Thus
3n 2 −7n+40
wavelengths and 6n2 −4n+10
ADMs are sufficient to support
30 15
the request graph K n . That is, n2 −9n+20
wavelengths use 2n2 −18n+40
10 5
ADMs for the subgraphs of K n−4 , 2 wavelengths use 6 ADMs for the
subgraphs of K 4 , and 2n−8
3
wavelengths uses 5 ADMs each for the
subgraphs of K n−4,4 .
(b) n ≡ 2(mod 3): K n is the union of the subgraphs K n−4 (on nodes
{2, . . . , n−4, n−3}), K 5 (on nodes {0, 1, n−3, n−2, n−1}), and K n−5,4
(with node partition {{2, . . . , n−4}, {0, 1, n−2, n−1}}). The subgraph
K n−5,4 is the union of 2× n−5
3
subgraphs isomorphic to K 2,3 , with node
partitions {{j, j+n−2}, {3k+2, 3k+3, 3k+4}}, for 0 ≤ j ≤ 1, 0 ≤ k ≤
n−5
−1. Since n−4 ≡ 1(mod 5), by Case 1 of this lemma the subgraph
3
K n−4 is the union of n2 −9n+20
subgraphs isomorphic to K
10 4 −e. By Case
2 of Lemma 12 the subgraph K 5 is the union of three subgraphs. Thus
3n 2 −7n+50
wavelengths and 6n2 −4n+20
ADMs are sufficient to support
30 15
the request graph K n . That is, n2 −9n+20
wavelengths use 2n2 −18n+40
10 5
ADMs for the subgraphs of K n−4 , 3 wavelengths use 10 ADMs for the
subgraphs of K 5 , and 2n−10
3
wavelengths uses 5 ADMs each for the
subgraphs of K n−5,4 .
(c) n ≡ 0(mod 3): K n is the union of the subgraphs K n−4 (on nodes
{2, . . . , n − 5, n − 4, n − 3}), K 6 − e (on nodes {0, 1, n − 4, n − 3, n −
2, n − 1}, where e = (n − 4, n − 3)), and K n−6,4 (with node partition
{{2, . . . , n − 5}, {0, 1, n − 2, n − 1}}). The subgraph K n−6,4 is the
union of 2 × n−6
3
subgraphs isomorphic to K 2,3 , with node partitions
{{j, j +n−2}, {3k +2, 3k +3, 3k +4}}, for 0 ≤ j ≤ 1, 0 ≤ k ≤ n−6
3 −1.
164

9.3 Upper bound for A(P n , 3)
Since n − 4 ≡ 1(mod 5), by Case 1 of this lemma the subgraph K n−4
is the union of n2 −9n+20
subgraphs isomorphic to K
10 4 − e. By Case
3 of Lemma 12 the subgraph K 6 − e is the union of three subgraphs.
Thus 3n2 −7n+30
wavelengths and 6n2 −4n
ADMs are sufficient to support
30 15
the request graph K n . That is, n2 −9n+20
wavelengths use 2n2 −18n+40
10 5
ADMs for the subgraphs of K n−4 , 3 wavelengths use 12 ADMs for the
subgraphs of K 6 − e, and 2n−12
3
wavelengths uses 5 ADMs each for the
subgraphs of K n−6,4 .
3. For n ≡ 2(mod 5), A(P n , 3) ≤ 2n2 −n−6
5
.
We consider the cases n ≡ 2(mod 10) and n ≡ 7(mod 10).
(a) n ≡ 2(mod 10): We have that n ≡ 0(mod 2).
K n is the union of
the subgraphs K n−6 (on nodes {0, . . . , n−6 − 1, n−6 + 6, . . . , n − 1}),
2 2
K 6 (on nodes { n−6,
. . . , n−6 + 5}) and K
2 2 n−6,6 (with node partition
{{0, . . . , n−6 n−6
n−6
−1, +6, . . . , n−1}, { , . . . , n−6 +5}}). The subgraph
2 2 2 2
K n−6,6 is the union of n − 6 subgraphs isomorphic to K 2,3 , with node
partitions {{k, k+ n−6 n−6
+6}, {
2
, n−6
2 2
n−6
n−6
+1, +2}}, and {{k, k+ +
2 2
6}, { n−6
2
+ 3, n−6
2
+ 4, n−6
2
+ 5}}, for 0 ≤ k ≤ n−6
2
− 1.
(b) n ≡ 7(mod 10): We further consider the cases n ≡ 0(mod 3), n ≡
1(mod 3) and n ≡ 2(mod 3).
i. n ≡ 0(mod 3): K n is the union of the subgraphs K n−6 (on nodes
{3, . . . , n − 4}), K 6 (on nodes {0, . . . , 2, n − 3, . . . , n − 1}), and
K n−6,6 (with node partition {{3, . . . , n−4}, {0, . . . , 2, n−3, . . . , n−
1}}). The subgraph K n−6,6 is the union of n − 6 subgraphs isomorphic
to K 2,3 , with node partitions {{j, j + n − 3}, {3k + 3, 3k +
4, 3k + 5}}, for 0 ≤ j ≤ 2, 0 ≤ k ≤ n−6
3
− 1.
ii. n ≡ 1(mod 3): K n is the union of the subgraphs K n−6 (on nodes
{0, 1, 5, . . . , n−6, n−2, n−1}), K 6 (on nodes {2, . . . , 4, n−5 . . . , n−
3}), and K n−6,6 (with node partition {{0, 1, 5, . . . , n − 6, n − 2, n −
1}, {2, . . . , 4, n − 5 . . . , n − 3}}). The subgraph K n−6,6 is the union
of n−6 subgraphs isomorphic to K 2,3 , with node partitions {{j, j+
165

9.3 Upper bound for A(P n , 3)
n − 7}, {3k + 5, 3k + 6, 3k + 7}}, for 2 ≤ j ≤ 4, 0 ≤ k ≤ n−10
3
− 1,
and {{k, k + n − 2}, {j, j + 1, j + 2}}, for j ∈ {2, n − 5}, 0 ≤ k ≤ 1.
iii. n ≡ 2(mod 3): K n is the union of the subgraphs K n−6 (on nodes
{0, 4, . . . , n − 5, n − 1}), K 6 (on nodes {1, . . . , 3, n − 4 . . . , n − 2}),
and K n−6,6 (with node partition {{0, 4, . . . , n−5, n−1}, {1, . . . , 3, n−
4 . . . , n−2}}). The subgraph K n−6,6 is the union of n−6 subgraphs
isomorphic to K 2,3 , with node partitions {{j, j + n − 5}, {3k +
4, 3k + 5, 3k + 6}}, for 1 ≤ j ≤ 3, 0 ≤ k ≤ n−8
3
− 1, and
{{0, n − 1}, {j, j + 1, j + 2}}, for j ∈ {1, n − 4}.
Since n ≡ 2(mod 5), we have that n − 6 ≡ 1(mod 5). By Case 1 of this
lemma the subgraph K n−6 is the union of n2 −13n+42
10
subgraphs isomorphic
to K 4 − e. By Case 3 of Lemma 12 the subgraph K 6 is the union of three
subgraphs. Thus in each case n2 −3n+12
wavelengths and 2n2 −n−6
ADMs
10 5
are required for the request graph K n . That is, n2 −13n+42
wavelengths use
10
2n 2 −26n+84
ADMs for the subgraphs of K
5 n−6 , 3 wavelengths use 12 ADMs
for the subgraphs of K 6 , and n − 6 wavelengths uses 5 ADMs each for the
subgraphs of K n−6,6 .
4. For n ≡ 3(mod 5), A(P n , 3) = 6n2 −5n+ɛ
15
, where ɛ = 16, 6, and −4, for
n ≡ 2(mod 3), n ≡ 0(mod 3), and n ≡ 1(mod 3), respectively.
We consider the cases n ≡ 2(mod 3), n ≡ 0(mod 3), and n ≡ 1(mod 3) as
follows.
(a) n ≡ 2(mod 3): K n is the union of the subgraphs K n−2 (on nodes
{1, . . . , n − 2}), (0, n − 1), and K n−2,2 (with node partition ({{1,. . . ,n-
2}, {0,n-1} }). The subgraph K n−2,2 is the union of n−2
3
subgraphs
isomorphic to K 2,3 , with node partitions {{0, n − 1}, {3k + 1, 3k +
2, 3k + 3}}, for 0 ≤ k ≤ n−2 − 1. Since n ≡ 3(mod 5), by Case 1 of
3
this lemma the subgraph K n−2 is the union of n2 −5n+6
subgraphs isomorphic
to K 4 − e. Thus 3n2 −5n+28
10
wavelengths and 6n2 −5n+16
ADMs
30 15
are sufficient to support the request graph K n . That is, n2 −5n+6
wavelengths
use 2n2 −10n+12
10
ADMs for the subgraphs of K
5 n−2 , 1 wavelength
166

9.3 Upper bound for A(P n , 3)
uses 2 ADMs for the subgraph (0, n − 1), and n−2
3
wavelengths uses 5
ADMs each for the subgraphs of K n−2,2 .
(b) n ≡ 0(mod 3): K n is the union of the subgraphs K n−2 (on nodes
{1, . . . , n − 3, n − 2}), (0, n − 2, n − 1), and K n−3,2 (with node partition
{{1, . . . , n − 3}, {0, n − 1}}. The subgraph K n−3,2 is the union of n−3
3
subgraphs isomorphic to K 2,3 , with node partitions {{0, n − 1}, {3k +
1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ n−3 − 1. Since n ≡ 3(mod 5), by Case
3
1 of this lemma the subgraph K n−2 is the union of n2 −5n+6
subgraphs
10
isomorphic to K 4 −e. Thus 3n2 −5n+18
wavelengths and 6n2 −5n+6
ADMs
30 15
are sufficient to support the request graph K n . That is, n2 −5n+6
wavelengths
use 2n2 −10n+12
10
ADMs for the subgraphs of K
5 n−2 , 1 wavelength
uses 3 ADMs for the subgraph (0, n − 2, n − 1), and n−3 wavelengths
3
uses 5 ADMs each for the subgraphs of K n−3,2 .
(c) n ≡ 1(mod 3): K n is the union of the subgraphs K n−2 (on nodes
{1, . . . , n − 4, n − 3, n − 2}), ( { }} {
0, n − 3, n − 1, n − 2), and K n−4,2 (with
node partition {{1, . . . , n − 4}, {0, n − 1}}). The subgraph K n−4,2 is
the union of n−4
3
subgraphs isomorphic to K 2,3 , with node partitions
{{0, n − 1}, {3k + 1, 3k + 2, 3k + 3}}, for 0 ≤ k ≤ n−4
3
− 1. Since
n ≡ 3(mod 5), by Case 1 of this lemma the subgraph K n−2 is the union
of n2 −5n+6
subgraphs isomorphic to K
10 4 −e. Thus 3n2 −5n+8
wavelengths
30
and 6n2 −5n−4
ADMs are sufficient to support the request graph K
15 n .
That is, n2 −5n+6
wavelengths use 2n2 −10n+12
ADMs for the subgraphs of
10 5
K n−2 , 1 wavelength uses 4 ADMs for the subgraph ( 0, { n − }} 3, n − 1, {
n−
2), and n−4
3
wavelengths uses 5 ADMs each for the subgraphs of K n−4,2 .
5. For n ≡ 4(mod 5), A(P n , 3) ≤ 4n2 −3n+ɛ
10
, where ɛ = 3, and 8 for n ≡
1(mod 2), and n ≡ 0(mod 2) respectively.
We consider the cases n ≡ 1(mod 2), and n ≡ 0(mod 2) as follows.
(a) n ≡ 1(mod 2): K n is the union of the subgraphs K n−3 (on nodes
{0, . . . , n−3
2
−1,
n−3
2
n−3
+3, . . . , n−1}), ( , n−3 n−3
+1, +2), and K 2 2 2 n−3,3
(with node partition {{0, . . . , n−3
2
− 1, n−3
2
+ 3, . . . , n − 1}, { n−3
2 , n−3
2
+
167

9.3 Upper bound for A(P n , 3)
1, n−3
2
+ 2}}). The subgraph K n−3,3 is the union of n−3
2
subgraphs isomorphic
to K 2,3 , with node partitions {{k, k + n−3
2
+ 3}, { n−3
2 , n−3
2
+
1, n−3
2
+ 2}}, for 0 ≤ k ≤ n−3
2
− 1. Since n − 3 ≡ 1(mod 5), by Case 1
of this lemma the subgraph K n−3 is the union of n2 −7n+12
10
subgraphs
isomorphic to K 4 − e. Thus n2 −2n+7
10
wavelengths and 4n2 −3n+3
10
ADMs
are sufficient to support the request graph K n . That is, n2 −7n+12
10
wavelengths
use 2n2 −14n+24
5
ADMs for the subgraphs of K n−3 , 1 wavelength
uses 3 ADMs for the subgraph ( n−3
2 , n−3
2
+ 1, n−3
2
+ 2), and n−3
2
wavelengths
uses 5 ADMs each for the subgraphs of K n−3,3 .
(b) n ≡ 0(mod 2): K n is the union of the subgraphs K n−3 (on nodes
{0, . . . , n−4 n−4
−1, +3, . . . , n−2, n−1}), K 2 2 4 (on nodes { n−4,
. . . , n−4 +
2 2
2, n − 1}), and K n−4,3 (with node partition {{0, . . . , n−4 − 1, n−4 +
2 2
3, . . . , n−2}, { n−4,
. . . , n−4 +2}}). The subgraph K 2 2 n−4,3 is the union of
n−4
subgraphs isomorphic to K
2 2,3 , with node partitions {{k, k + n−4 +
2
3}, { n−4,
n−4 n−4
n−4
+1, +2}}, for 0 ≤ k ≤ −1. Since n−3 ≡ 1(mod 5)
2 2 2 2
by Case 1 of this lemma the subgraph K n−3 is the union of n2 −7n+12
10
subgraphs isomorphic to K 4 −e. By Case 1 of Lemma 12 the subgraph
K 4 is the union of two subgraphs. Thus n2 −2n+12
wavelengths and
10
4n 2 −3n+8
ADMs are sufficient to support the request graph K
10 n . That
n
is, 2 −7n+12
wavelengths use 2n2 −14n+24
ADMs for the subgraphs of
10 5
K n−3 , 2 wavelengths use 6 ADMs for the subgraphs of K 4 , and n−4
2
wavelengths uses 5 ADMs each for the subgraphs of K n−4,3 .
To the best of our knowledge there is no known exact method for grooming
all-to-all instances for networks P n with g > 2. The lower and upper bounds for
g = 3 presented in this chapter lead to an asymptotic ratio of 16
15 ≈ 1.07.
168

Chapter 10
Traffic Grooming on Ring
Networks - General Instances
In this chapter we study the traffic grooming problem for unidirectional ring network
N with grooming ratio g = 2. We consider instances with unit size requests
(traffic) and propose a heuristic algorithm for grooming arbitrary instances on
such networks. We experimentally compare our algorithm with related grooming
algorithms for ring networks in [Zhang and Qiao (2000)] and [Călinescu and
Wan (2002)] for performance. The current state-of-art optic fiber technology
support several hundreds of wavelengths per fiber, each in turn offer an enormous
bandwidth (in the order of tens of Gigabyte), while a great deal of network
traffic require just a fraction of the wavelength bandwidth to be satisfied. The
wavelength channels are more efficiently used when several low rate requests are
combined to share the wavelength bandwidth, which also improves the network
connectivity.
The traffic grooming problem was introduced in Chapter 8 where we also
mentioned that an ADM is needed for opto-electronic conversion at a node whenever
a request is added or dropped at the node from the shared wavelength. In
traffic grooming we are given a set of requests to be satisfied on the network N
with grooming ratio g, where at each link the wavelength of N can be shared by
no more than g requests (the grooming ratio constraint). Requests are satisfied
by combining a subset of them into a wavelength to honour the grooming ratio
169

10.1 General lower bounds
constraint. The traffic grooming problem seek to satisfy any set of requests using
minimum number of ADMs and wavelengths.
The traffic grooming problem is NP-hard for ring networks [Dutta et al. (June,
2003)] with general instances. For all-to-all instances with unit size requests the
grooming problem has been optimally solved for ring networks with grooming
ratio g ≤ 4 and g = 6 by different authors referred in Section 2.5. For arbitrary
instances a 3 -approximation algorithm is presented in [Călinescu and Wan (2002)]
2
for minimizing the ADM requirement on ring networks with grooming ratio g = 1.
A more generic grooming algorithm is presented in [Zhang and Qiao (2000)] for
ring networks with arbitrary instances and any grooming ratio, g. We propose a
grooming algorithm for unidirectional rings with g = 2 and arbitrary instances
with unit size requests. Our proposed algorithm adapts the EDA-SF algorithm
for unidirectional path networks with grooming ratio 2, described in Section 8.2.3.
We experimentally compare with the algorithms in [Călinescu and Wan (2002);
Zhang and Qiao (2000)] adapted for this network for performance, and show that
our algorithm achieves better performance for most grooming instances.
10.1 General lower bounds
Given an instance R for an n nodes ring network N with grooming ratio g, where
the nodes and links are labeled i, for 0 ≤ i ≤ n − 1. Let Γ o (k) and Γ i (k) be
the number of requests in R that begin and end at node k, when traversing the
nodes of the network N in ⌈clockwise direction, ⌉ for 0 ≤ k ≤ n − 1. Obviously,
the node k requires at least max{Γo (k),Γ i (k)}
number of ADMs, since an ADM at
g
node k can be shared by at most g requests that begin and end at k respectively.
Thus, ∑ ⌈ ⌉
n−1 max{Γo(k),Γi (k)}
k=0
, is the general lower bound for number of ADMs
g
required by the instance R in N. This was also noted in [Călinescu and Wan
(2002); Gerstel et al. (1998a)] for ring networks. The load, L, of a network is
the maximum number of requests using any network link. Clearly, at least ⌈ L g ⌉
wavelengths are needed for an instance R inducing load L on the network, since
at most g requests can share a wavelength at any link.
170

10.2 Grooming Algorithms
10.2 Grooming Algorithms
In this section we describe different algorithms for grooming arbitrary instances
on unidirectional rings with grooming ratio 2. In Section 10.3 we present our proposed
algorithm that adapts the EDA-SF algorithm for path networks in Section
8.2.3. We describe the algorithms of [Zhang and Qiao (2000)] and [Călinescu and
Wan (2002)] in Section 10.4, and show how they are adapted for ring networks
with g = 2. The algorithms are experimentally compared for their grooming
performance in Section 10.5.
10.3 Fixed edge with edge decomposition algorithm
We are given an arbitrary instance R with unit size requests on unidirectional
ring network N with grooming ratio 2. Note that the instances of a unidirectional
ring have all requests oriented in the same direction (clockwise direction) since
there is a unique path joining any pair of nodes. The request digraph D(V, A) is
defined as follows. The nodes in V are nodes of the network N, and the arcs in
A represent the requests in R. Let R i be the subset of requests in R traversing
the link i ∈ N, for 0 ≤ i ≤ n − 1. For a fixed i the subset of requests R \ R i
corresponds to the instance on a unidirectional path network P n = N\i, obtained
by removing the link i from the ring network N.
The fixed edge with edge decomposition (FE-EDA) algorithm takes as input the
request digraph D and partition the arcs of D into valid subgraphs. A subgraph
of D is a valid subgraph if no more than g arcs of the subgraph cross any link of N.
We consider valid subgraphs on unidirectional rings with g = 2. A subgraph of
D is a circuit subgraph if its underlying graph is of degree 2. A circuit subgraph
is a closed circuit if in the underlying graph all nodes have degree 2, but an
open circuit otherwise. The FE-EDA algorithm partition the arcs of D in two
phases. Let i be a fixed link of N. The first phase uses Breadth First Search
(BSF) and greedy approaches to partition arcs of D into set of closed and open
circuit subgraphs such that each circuit subgraph has one arc corresponding to
a request in R i . Each of such circuit subgraphs also consists of arcs that cross
171

10.3 Fixed edge with edge decomposition algorithm
any link of N no more than once. Thus pairs of circuit subgraphs obtained this
way are combined into valid subgraphs using the weighted matching algorithm
in [Edmonds (1965a); Edmonds (1965b)]. By the end of the first phase all arcs
of D that correspond to requests in R i have been considered. The second phase
considers the remaining D as the requests graph for unidirectional path network.
The EDA-SF algorithm for path networks is used to partition the remaining arcs
into valid subgraphs.
The FE-EDA algorithm distinguishes a link i ∈ N as the pivot link, and an
arc in D that corresponds to request traversing the pivot link is marked red, the
arc is marked blue, otherwise. In addition, each arc of D is associated with an
integer value (arc length) which is equal to the number of links of N traversed by
the request corresponding to this arc. The arcs of D are partitioned into valid
subgraphs of D using the following four steps.
1. For each red arc in D try to find a closed circuit containing this arcs as the
only red arc. If exists, then remove the arcs of this closed circuit from D
(described in Section 10.3.1).
2. For each red arc remaining in D find an open circuit containing this arc.
Remove the arcs of this open circuit from D (described in Section 10.3.2).
3. Merge pairs of circuit subgraphs obtained from Steps 1 and 2 into subgraphs
of D (described in Section 10.3.3).
4. Partition the remaining arcs of D (now D consists of only blue arcs) into
valid subgraphs using the EDA-SF algorithm in Section 8.2.3.
10.3.1 Finding closed circuits
Let (i, j) be a red arc of D. Starting from the node j perform a BFS traversal of
D exploring only the blue arcs to find a path p j,i from node j to node i using the
blue arcs of D. If such path p j,i exists, then the set of arcs in p j,i ∪ (i, j) form a
closed circuit of D containing the red arc (i, j), where all other arcs of this circuit
are blue arcs. Note that such closed circuit satisfies the requirements of Step 1
of the FE-EDA algorithm that closed circuits should contain exactly one red arc
172

10.3 Fixed edge with edge decomposition algorithm
of D. Moreover, since none of the blue arcs may cross the pivot link it follows
that the set of arcs on a closed circuit found this way cross the links of N exactly
once.
10.3.2 Finding open circuits
We find open circuits that form connected subgraph components of D, by operating
directly on requests of N corresponding to the remaining arcs of D, where the
red arcs in D correspond to requests on N traversing the pivot link and the blue
arcs in D correspond to requests on N avoiding the pivot link. In the following we
say red and blue request to refer to the corresponding red and blue arc. We find
a set of requests C r on N containing only one red requests where no two requests
share any link of N greedily from given red request r = (i, j), in two phases as
follows.
1. Phase 1
(a) Given the red request r = (i, j), C r ← {r}.
(b) Let (i 1 , j 1 ), (i 2 , j 2 ), . . . , (i k , j k ), be the requests in clockwise orientation
already added to C r . If there is a request (j k , x) between j k and i 1 ,
then select the shortest possible request (with x closest to j k ). C r ←
C r ∪ {(j k , x)}. Otherwise, go to Phase 2.
(c) Repeat Step 1b.
2. Phase 2
(a) Let (i 1 , j 1 ), (i 2 , j 2 ), . . . , (i k , j k ), be the requests in clockwise orientation
already added to C r . If there is a request (x, i 1 ) between j k and i 1 ,
then select the shortest possible request (with x closest to i 1 ). C r ←
C r ∪ {(x, i 1 )}. Otherwise, return C r .
(b) Repeat Step 2a.
The set of requests C r returned after Phase 2 above has the red request r as
the only request using the pivot link, where any other request in C r avoids the
pivot link (are blue requests). Moreover, any request added into C r should share a
173

10.3 Fixed edge with edge decomposition algorithm
node with a request already in C r and share no link of N with any request already
in C r . Therefore the set C r correspond to a set of arcs of D that form a connected
circuit subgraph that has exactly one red arc. The arcs on this circuit subgraph
cross the links of N no more than once. Note that this circuit subgraph is an
open circuit, since otherwise a closed circuit containing the red arc corresponding
to the red request r could have been found by Step 1 of the FE-EDA algorithm.
10.3.3 Merging circuit subgraphs into valid subgraphs
Step 3 merges pairs of circuits subgraphs obtained in Steps 1 and 2 of the FE-
EDA algorithm to form valid subgraphs of D using the weighted graph matching
algorithm in [Edmonds (1965a); Edmonds (1965b)]. Note that the set of arcs in
each circuit subgraph found by Steps 1 and 2 of the FE-EDA algorithm cross the
links of N no more than once, hence, the set of arcs on subgraphs obtained by
merging pairs of such circuit subgraphs cross the link of N no more than twice
as required. We define a complete weighted graph G = (V, E) where the nodes
in V are the circuit subgraphs. The edge weight w({C i , C j }) = |V (C i ) ∩ V (C j )|
for every pair of distinct circuits C i and C j . That is, the edge weight is equal to
the number of common nodes between the adjacent pair of circuits. Let M be
the maximum weight matching for G and Q a set of unmatched nodes of G (note
that |Q| ≤ 1). For every edge {C i , C j } ∈ M merge the pair of circuit subgraphs
C i and C j into a valid subgraph of D. Any unmatched node C k ∈ Q forms a
valid subgraphs of D.
10.3.4 Finding valid subgraphs from blue arcs
A request digraph D with only blue arcs corresponds to request digraph of path
network P n obtained by removing the pivot link from N. The EDA-SF algorithm
in Section 8.2.3, is used to partition blues arcs of D into valid subgraphs of D.
10.3.5 ADMs and wavelengths requirement
Let {D i : 0 ≤ i ≤ W −1} be the set of valid subgraph partitionings of D obtained
by the FE-EDA algorithm. The number of wavelengths computed by the FE-
174

10.4 Related algorithms
EDA algorithm is W since each valid subgraph is assigned a new wavelength.
The number of ADMs computed by the FE-EDA algorithm ∑ W −1
i=0
|V (D i )|.
10.4 Related algorithms
In this section we describe two related algorithms proposed in [Zhang and Qiao
(2000)] and [Călinescu and Wan (2002)] for ring networks with g ≥ 1 and g = 1
respectively and arbitrary instances with unit requests.
We adapt these algorithms
to ring networks with g = 2 and compare with the FE-EDA algorithm for
performance.
In [Zhang and Qiao (2000)] several grooming algorithms for ring networks with
arbitrary grooming ratios are presented.
The algorithms in [Zhang and Qiao
(2000)] grooms requests in two greedy phases: the phase that defines circuits
(circles) of requests from the input instance, referred to as algorithm IV, and
the phase that group the sets of circles found from previous phase into sets of
cardinality at most g, referred to as algorithm VI. The algorithms in [Zhang and
Qiao (2000)] consider ring networks with arbitrary grooming ratio g.
In [Călinescu and Wan (2002)] they present a 3 -approximation algorithm that
2
partitions the instance of a unidirectional ring network into chains, such that any
two requests in a chain do not overlap at any link of the network. They refer
to their algorithm as the preprocessed iterative matching (PIM) algorithm. The
algorithms in [Călinescu and Wan (2002)] consider ring networks with grooming
ratio g = 1.
In the following sections we describe the above algorithms and show how they
are adapted to rings with grooming ratio 2.
10.4.1 Algorithms IV and VI
1. Algorithm IV:
The algorithm IV partitions requests on a ring network into circles by considering
one network node a time. For requests incident from the node i,
0 ≤ i ≤ n − 1, a longest request is considered first and added either into an
175

10.4 Related algorithms
existing circle, a new circle or a gap maker list. An existing circle is a nonempty
set of requests such that each request in the set shares an end-node
with at least one other request already in the set, and no two requests share
any network link (overlap). A new circle is an empty set of requests. A
request is a gap maker if cannot be added into an existing circle because it
does not share a node with requests on existing circles. A request is added
into an existing circle if it does not overlap and create a gap to the existing
circle. It is added into a new circle if it overlaps on every existing circle,
and is added to the gap maker list if it creates a gap on every existing circle.
The requests on the gap maker list are further considered to achieve either
of the following goals. The goal to minimize the number of circles where
we try to add a request from the gap maker list into an existing circle. If it
overlaps on every existing circle, we add the request into a new circle. The
goal to minimize the number of end-nodes in circles where we try to add
the request from the gap maker list into an existing circle, if it overlaps or
creates a gap in every existing circle, we add the request into a new circle.
2. Algorithm VI:
The algorithm VI combine up to g circles to share a wavelength as follows.
A new wavelength w is first assigned to an unsigned circle with largest
number of end-nodes. While more circles remain unassigned and w has less
than g circles, find an unassigned circle resulting into a maximum overlap
of end-nodes with circles already assigned with wavelength w. Assign this
circle with w.
10.4.2 Preprocessed iterative matching algorithm
1. Preprocessing phase:
The preprocessing phase finds a set of closed chains as follows. The link
e is fixed about which requests in R are partitioned into two sets, R e and
R ′ = R \ R e , with requests that traverse the link e and those that do not.
An acyclic request digraph for R ′ , D AG (V, A), is defined as follows. The
set V are the nodes of the ring network and the arcs in A represents the
requests in R ′ .
176

10.4 Related algorithms
For every request (i, j) ∈ R e , use the BSF traversal on D AG to find a path,
p j,i , from node j to node i. If such path p j,i exists, then the request (i, j)
and requests in R ′ corresponding to arcs of p j,i is a closed chain. Requests
on this closed chain are removed from the sets R e and R ′ .
2. Iterative matching phase:
The iterative matching phase finds a set of open chains from remaining
requests as follows. Each request remaining in R e and R ′ forms an open
chain with one request. Let P be the set of open chains defined this way.
We define a graph F (P) called fit graph from the set P, such that there is
a node in F (P) for each open chain in P, and there is an edge between a
pair of open chains if their corresponding requests do not overlap at any
link. While the fit graph F (P) for P has non-empty set of edges, we find
a maximum matching M for F (P), and merge each matched pair of chains
into a new chain to obtain a new set P. If F (P) has no edges, then we
output the set P of open chains.
Note that the notions of circuit, circle and chain used by the algorithms considered
in this chapter are similar, since each refers to a subset of requests where
no two requests share any link of the network. Thus these terms can be used
interchangeably. Note also that Step 1 of the FE-EDA algorithm finds the set
of closed circuits in the same way the PIM algorithm does in the preprocessing
phase. The FE-EDA algorithm differs, however, from the PIM algorithm, when
finding open circuits in which the former uses a greedy approach whereas the
latter uses iterative matching. In addition, unlike the PIM algorithm which considers
all the remaining requests by iterative matching, the FE-EDA algorithm
finds open circuits with respect to requests traversing the pivot link, where any
remaining requests are considered using the EDA-SF algorithm.
The algorithm VI and the weighted matching algorithm in [Edmonds (1965a);
Edmonds (1965b)] can be used to merge up to g sets of circuits to obtain a set
of requests that satisfy the grooming ratio constraint of g. For our experiments
we adapt the algorithms in [Zhang and Qiao (2000)] and [Călinescu and Wan
(2002)] to ring networks with grooming ratio g = 2 as follows. We consider
the algorithm that combines algorithm IV and algorithm VI, denoted by A1-g2,
177

10.5 Numerical results and performance evaluation
and the algorithm that combines algorithm IV and the weighted graph matching
algorithm [Edmonds (1965a); Edmonds (1965b)], denoted by A2-g2. Also we
consider the algorithm that combines algorithm PIM and the weighted graph
matching algorithm [Edmonds (1965a); Edmonds (1965b)], denoted by PIM-g2.
The algorithms FE-EDA and PIM-g2 both require a fixed link (i.e., pivot link)
as part of the input. Distinct choices of pivot links may lead to different output
results. For our experiments each algorithm iterates over two pivot links, the link
traversed by the least number and the maximum number of requests and outputs
the best result.
10.5 Numerical results and performance evaluation
We use randomly generated instances with BL-Uniform, UL-Uniform and Gaussian
distributions of requests that are described in Section 2.4.1 for experiments.
Note that for request instances of unidirectional ring networks no routing decisions
are needed since each request is routed on the unique path of the ring
between the request end-nodes. Tables 10.1, 10.2 and 10.3 show the average
number of ADMs for BL-Uniform, UL-Uniform and Gaussian instances for ring
networks with 16, 32 and 64 nodes and instances with 512, 1024, 1536 and 2048
requests. Table 10.4 shows the average number of wavelengths for each instance
type on 64-node ring networks. The tables also show the average general lower
bounds ADM-LB and W-LB for number of ADMs and wavelengths determined
using the general lower bounds relations in Section 10.1. The average is taken
using ten random instances.
The algorithms A1-g2 and A2-g2 differ considerably on their ADMs performance
where the A2-g2 algorithm that uses the maximum weight matching algorithm
has up to 5%, 7%, and 5% better ADMs performance for BL-Uniform,
UL-Uniform and Gaussian instances, than the A1-g2 algorithm which uses algorithm
VI for merging circuits. The two algorithms, however, always computed
the same number of wavelengths.
178

10.5 Numerical results and performance evaluation
n |R| ADM-LB FE-EDA PIM-g2 A2-g2 A1-g2
512 321 344 348 357 369
16 1024 571 604 608 627 646
1536 1020 1038 1045 1069 1084
2048 1123 1159 1168 1199 1219
512 317 398 408 424 446
32 1024 607 715 730 747 779
1536 903 1035 1065 1070 1122
2048 1380 1481 1525 1511 1564
512 330 469 479 508 527
64 1024 618 845 871 921 960
1536 935 1211 1253 1308 1363
2048 1274 1599 1672 1701 1778
Table 10.1: Average number of ADMs for instances with BL-Uniform distribution
of requests.
n |R| ADM-LB FE-EDA PIM-g2 A2-g2 A1-g2
512 286 349 343 364 391
16 1024 553 630 622 654 695
1536 818 894 889 928 971
2048 1081 1160 1155 1206 1247
512 299 444 435 467 491
32 1024 573 786 770 825 877
1536 838 1072 1057 1131 1210
2048 1106 1359 1343 1434 1533
512 321 560 550 574 590
64 1024 598 1002 982 1043 1081
1536 872 1407 1378 1469 1527
2048 1140 1765 1730 1852 1935
Table 10.2: Average number of ADMs for instances with UL-Uniform distribution
of requests.
179

10.5 Numerical results and performance evaluation
Nevertheless, the ADMs performance for the A2-g2 algorithm is not as good
compared to that of the FE-EDA and PIM-g2 algorithms. For example, Table
10.1 shows that the ADMs performance for the FE-EDA and PIM-g2 algorithms
is up to 9% and 6% better than that of the A2-g2 algorithm for instances with
BL-Uniform distribution of requests. For instances with UL-Uniform distribution
of requests, on the other hand, Table 10.2 shows that the ADMs performance of
the FE-EDA and PIM-g2 algorithms is up to 5.5% and 7.3% better than that of
the A2-g2. The ADMs performance differences are small for all algorithms on
instances with Gaussian distribution of requests, where both the FE-EDA and
PIM-g2 algorithms have ADMs performance that is up to 3% better than that of
the A2-g2 algorithm as Table 10.3 shows.
The A2-g2 algorithm also have higher wavelengths requirement compared to
that of the FE-EDA and PIM-g2 algorithms as Table 10.4 shows. For example, for
some BL-Uniform instances the A2-g2 uses a significant number of wavelengths
that is up to 39% more than the number of wavelengths the FE-EDA algorithm
uses. It is likely that such high wavelength requirement for the A2-g2 algorithm
occurs when most requests do not mutually share their end-nodes, hence more
circles are found by algorithm IV of [Zhang and Qiao (2000)], especially when
considering requests on the gap maker list. This is because in our experiments we
adopted the goal to minimize the number of end-nodes in circles. The wavelength
performance for the A2-g2 may improve if the goal to minimize number of circles
is adopted, but this may lead to poor ADMs performance.
The FE-EDA and PIM-g2 algorithms gave an impressive ADMs performance,
where the FE-EDA algorithm has best ADMs performance for instances with BL-
Uniform distribution of requests, and on the other hand, the PIM-g2 algorithm
has the best ADMs performance for instances with UL-Uniform distribution of
requests. Table 10.1 shows that the ADMs performance of the FE-EDA algorithm
is up to 5% better than that of the PIM-g2 algorithm for BL-Uniform instances.
And Table 10.2 shows that the ADMs performance of the PIM-g2 algorithm
is up to 2% better than that of the FE-EDA for UL-Uniform instances. For
instances with Gaussian distribution of requests, however, Table 10.3 shows that
both FE-EDA and PIM-g2 algorithms have very close ADMs performance, where
180

10.5 Numerical results and performance evaluation
n |R| ADM-LB FE-EDA PIM-g2 A2-g2 A1-g2
512 276 303 301 312 328
16 1024 534 566 564 579 595
1536 794 830 828 848 867
2048 1067 1110 1107 1131 1150
512 273 297 295 304 320
32 1024 534 569 566 579 597
1536 794 831 829 847 867
2048 1052 1093 1091 1109 1128
512 273 298 297 303 316
64 1024 533 568 564 580 596
1536 810 853 850 874 892
2048 1050 1097 1093 1112 1131
Table 10.3: Average number of ADMs for instances with Gaussian distribution
of requests.
Distribution |R| W-LB FE-EDA PIM-g2 A1-g2 & A2-g2
512 74 78 75 99
BL-Uniform 1024 145 147 146 184
1536 213 220 219 284
2048 283 290 287 402
512 137 150 146 151
UL-Uniform 1024 268 291 284 291
1536 399 429 421 428
2048 529 567 558 565
512 132 134 133 134
Gaussian 1024 260 263 262 264
1536 393 398 397 400
2048 518 523 522 525
Table 10.4: Average number of wavelengths for different instance types on 64-
node ring network.
181

10.5 Numerical results and performance evaluation
Distribution FE-EDA PIM-g2 A1-g2 A2-g2
UL-Uniform 41.09 51.41 35.51 9.66
Gaussian 43.05 47.79 37.95 16.11
BL-Uniform 13.45 15.69 24.82 7.18
Table 10.5: Average time used (in seconds) for ring network with 64 nodes and
instance of 2048 requests (using 1.0GHZ ADM Duron Processor with 112.0MB
RAM).
the FE-EDA algorithm always used from 1 to 4 more ADMs than the number of
ADMs the PIM-g2 algorithm used.
Table 10.4 shows that the algorithms FE-EDA and PIM-g2 also have close
wavelength performance for instances with BL-Uniform and Gaussian distribution
of requests, where the FE-EDA algorithm always used from 1 to 3 more
wavelengths than the number of wavelengths the PIM-g2 algorithm used. The
wavelength performance differences for the FE-EDA and PIM-g2 algorithms is
slightly high for UL-Uniform instances, where the FE-EDA used up to 10 more
wavelengths than the number of wavelengths the PIM-g2 algorithm used.
The large differences between the number of ADMs used by our algorithms
and the number of ADMs computed by the general lower bound relation in Section
10.1, ADM-LB, suggests that the general lower bound is either a very weak ADMs
lower bound or some instance types are potentially difficult to satisfy using minimum
number of ADMs. For example, for instances with BL-Uniform distribution
of requests where the FE-EDA algorithm has the best ADMs performance, Table
10.1 shows that the ADMs performance of the FE-EDA is up to 42% worse than
the lower bound number of ADMs ADM-LB. Also for instances with UL-Uniform
requests where the PIM-g2 algorithm has the best ADMs performance, Table 10.2
shows that the ADMs performance of the PIM-g2 is up to 71% worse than the
lower bound number of ADMs ADM-LB. For Gaussian instances, though, both
FE-EDA and PIM-g2 algorithms used number of ADMs that is rather close to
the lower bound number of ADMs ADM-LB, differing by up to 10%, see Table
10.3. Although the use of maximum weight matching algorithm improved the
182

10.5 Numerical results and performance evaluation
ADMs performance for the algorithms we considered, this came at an added cost
of running time as Table 10.5 shows, where algorithms FE-EDA, PIM-g2 and
A1-g2 (which apply the weighted matching algorithm) are relatively slower than
algorithm A2-g2 (that does not apply the weighted matching algorithm).
The wavelength performance of the FE-EDA and PIM-g2 algorithms compared
favorably with the lower bound number of wavelength W-LB as Table 10.4
shows, where FE-EDA and IPM-g2 algorithms used up to 10% and 6% more
wavelengths than the lower bound number of wavelengths W-LB.
We report additional numerical results for the grooming algorithms considered
by this chapter in Appendix A.5. We considered networks with 16, 32, 64 and 96
nodes. The differences between the ADMs performance of different algorithms
grow when the size of the instance increases for all instance types we considered
except for Gaussian instances where the ADMs performance of the algorithms
FE-EDA and PIM-g2 is similar (see Tables A.15, A.16, A.17 and A.18).
We summarize the performance results for the algorithms FE-EDA, PIM-g2,
A1-g2, and A2-g2 in the plots in Figures 10.1 and 10.2, for ring networks with 64
nodes and the BL-Uniform and UL-Uniform distribution of requests (note that
we do not show the ADMs and wavelengths performance for Gaussian instances
because for the different algorithms the performance differences are small for most
instances we considered). The plots show that algorithm FE-EDA has overall the
best ADMs performance, followed by the algorithms PIM-g2 and A2-g2. The
good performance of the FE-EDA could be attributed to its use of the EDA-SF
algorithm for path networks presented in Section 8.2.3.
183

10.5 Numerical results and performance evaluation
1.575
1.5
# ADMs/ADM-LB
1.425
1.35
1.275
1.2
1.125
FE-EDA
PIM-g2
A2-g2
A1-g2
1.05
151 301 451 601 751 901 1051 1201 1351 1501 1651
ADM-LB
(a)
# WAVELENGTHS/W-LB
1.55
1.5
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
FE-EDA
PIM-g2
A1-g2 & A2-g2
101 201 301 401
W-LB
(b)
Figure 10.1: ADMs and Wavelength performance for instances with BL-Uniform
distribution of requests on 64-node ring network (a) ADMs (b) Wavelengths.
184

10.5 Numerical results and performance evaluation
# ADMs/ADM-LB
1.875
1.8
1.725
1.65
1.575
1.5
1.425
1.35
1.275
1.2
1.125
1.05
FE-EDA
PIM-g2
A2-g2
A1-g2
151 301 451 601 751 901 1051 1201 1351 1501
ADM-LB
(a)
1.2
# WAVELENGTHS/W-LB
1.15
1.1
FE-EDA
PIM-g2
A1-g2 & A2-g2
1.05
101 201 301 401 501 601 701
W-LB
(b)
Figure 10.2: ADMs and Wavelength performance for instances with UL-Uniform
distribution of requests on 64-node ring network (a) ADMs (b) Wavelengths.
185

Chapter 11
Conclusion and Future work
In this thesis we studied the WA problem for optical networks under different
network constraints to minimize number of wavelengths requirement. The ring
network which is one of popular topologies was studied with single-fiber, conversion,
multi-fiber, and multi-granularity wavelength models respectively. Tree of
rings which are natural extension of ring networks were studied with single fiber
model. We also studied tree networks and path networks with multi-fiber and
multi-granularity wavelength models respectively.
For single-fiber rings without wavelength conversion capability we described
several greedy heuristics for the WA problem in Chapter 3. The heuristics were
the adaptation of the algorithm for the RAP problem for structured programs
which can be modeled by circular arc graphs. We implemented the heuristics and
experimentally compared their performance with the best performance heuristic
for this problem proposed in [Carpenter and Cosares (2002)]. We showed
a significant improvement over the previously proposed heuristics for the WA
problem for rings. The heuristics we presented were also shown to guarantee a
2-approximation factor. The best known factor is given by the randomized algorithm
in [Kumar (1998)] with a 1.37 + o(1)-approximation asymptotic bound.
We extended the heuristics for single-fiber rings without conversion to tree of
rings without conversion and rings with cluster conversion capability in Chapter
4 and 5, respectively. We proposed a heuristic for the WA problem for tree of
rings and experimentally compared with the approximation algorithms in [Bian
et al. (June 2004)] for performance. Our experiments showed that the approxi-
186

mation algorithms had worst wavelength performance compared to heuristics in
all instances considered. This leaves an important challenge to further study tree
of rings networks.
For rings with cluster wavelength conversion capability we proposed WA
heuristics that combine the method used by heuristics for rings without conversion
considered in Chapter 3, and the wavelength conversion framework used
by the heuristics for rings with cluster conversion in [Cavendish and Sengupta
(2002)]. We experimentally compared our algorithms with the best performance
algorithm for rings with cluster conversion in [Cavendish and Sengupta (2002)].
Our experiments showed a significant performance improvement for our heuristics
in both number of wavelength and number of wavelength conversion requirements.
We studied multi-fiber trees and multi-fiber rings in Chapter 6 and 7, respectively.
For multi-fiber tree networks we considered the problem of minimizing
the overall number of fibers in the network to satisfy a given WA instance. We
proposed a 2-approximation algorithm for spider trees, a 3-approximation for
trees where any path has at most two nodes of degrees greater than 2, and a 4-
approximation for trees where any path has at most three nodes of degrees greater
than 2. In addition, we extended to a log n-approximation and n-approximation
algorithm for k-ary trees and arbitrary trees respectively. We noted that the
results for arbitrary trees are improved to a 4-approximation in [Chekuri et al.
(2003)], and a 2-approximation algorithm for spider trees is independently given
in [Potika et al. (Nicosia, Cyprus, Nov 8–10, 2001)]. The problem of minimizing
the overall number of fibers in a multi-fiber network require further consideration
as network topologies other than trees and rings are yet to be considered.
It was shown in [Li and Simha (2000)] that the WA problem is NP hard even
for k-fiber rings with k ≥ 2. We extended the work in [Ramaswami and Sasaki
(December 1998)] for single-fiber rings with limited wavelength conversion capability
to k-fiber rings with limited wavelength conversion capability to optimally
satisfy any WA instance. We showed that for a single-fiber ring network if only
one node has a degree 2 wavelength conversion capabilities then there exists an
instance with load L which cannot be satisfied using L wavelengths. We proposed
a dual-fiber ring network with only one node equipped with degree 2 wavelength
conversion capability such that any WA instance with load L is satisfied using L
187

wavelengths. In addition, we proposed a dual-fiber ring network with two nodes
equipped with degree 1 wavelength conversion capabilities such that any WA instance
with load L is satisfied using L wavelengths. We extended both results to
k-fiber rings with arbitrary k > 2.
In contrast to the preceding chapters, in Chapters 8, 9, and 10 we considered
networks with multi-granularity wavelengths in which more than one connection
requests may share a wavelength. We studied the traffic grooming problem for
unidirectional path networks with grooming ratio 2 in Chapter 10, and proposed
two heuristic algorithms for arbitrary instances with unit requests using interval
partition and graph edge decomposition methods. We experimentally compared
the two algorithms for their grooming performance and showed that the graph
edge decomposition method leads to the overall better grooming performance. It
also achieved optimal grooming results for instances with fixed length requests.
We extended our heuristics to unidirectional path networks with grooming ratio
g > 2 through recursion.
For unidirectional path networks with all-to-all instances of unit requests we
established a lower bound for grooming ratio 3 and 4 respectively, and an upper
bound for the case of grooming ratio 3 in Chapter 9. The lower and upper
bounds for unidirectional paths with grooming ratio 3 imply an asymptotic ratio
of 1.07. We considered the grooming problem for unidirectional ring networks
with grooming ratio 2 in Chapter 10. We proposed an algorithm for grooming
arbitrary instances with unit requests, that combined greedy and graph edge
decomposition methods. Two related algorithms were described and adapted
for grooming arbitrary instances with unit requests on unidirectional rings with
grooming ratio 2. We experimentally compared the two algorithms with our
heuristic for performance. Our experiments showed improved performance for
our algorithm for most instances, particularly, achieved best performance on all
instances with uniform bounded length requests - which correspond to uniform
instances for bidirectional ring networks. These algorithms can be extended to
unidirectional rings with arbitrary grooming ratios by adapting the extension
method used in Chapter 8. We remark that extending the heuristics for traffic
grooming on networks with grooming ratio 2 to high grooming ratios may lead
to worse solution as the grooming ratio increases. Future work in this direction
188

is to seek new approaches for grooming arbitrary instances on networks with
higher grooming ratios. In addition, unidirectional path networks with higher
grooming ratios and all-to-all instances require further consideration since unlike
unidirectional ring networks with all-to-all instances where exact methods are
know for several higher grooming ratio constraints [Bermond and Ceroi (2003);
Bermond et al. (2003); Hu (January, 2002); Bermond and Coudert (May 2003);
Dutta and Rouskas (January, 2002); Bermond et al. (2005)], there are no known
exact methods for the former for higher grooming ratio constraints other than 2.
189

Appendix A
Appendix
In this appendix we report more experimental results for different algorithms
considered in this thesis. The source code for our experiments is posted at
http://www.dcs.kcl.ac.uk/staff/radzik/PhDstudents/Justo/thesisweb.html, where
you can download and experiment yourself. In Appendix A.1 we report numerical
results for WA algorithms for ring networks considered in Chapter 3. Appendix
A.2 gives numerical results for WA algorithms for tree of rings networks
discussed in Chapter 4. And in Appendix A.3 we report numerical results for
WA algorithms for ring networks with cluster conversion capabilities considered
in Chapter 5.
Furthermore, we report numerical results for traffic grooming algorithms considered
in Chapter 8 and 10 for path and ring networks with general instances
of unit size requests. For path networks we report numerical results in Appendix
A.4, and for ring networks we report the results in Appendix A.5.
The numerical results show the average number (of wavelengths, ADMs and
wavelength conversions) taken from ten random instances.
190

A.1 Undirected rings
A.1 Undirected rings
In this section we report numerical results for WA algorithms for ring networks
considered in Chapter 3. We consider ring networks with 16 to 128 nodes using
instances with Uniform and Gaussian distribution of requests described in
Section 2.4.1. The instances are routed using the shortest path and the load
balancing routing decisions described in Section 2.4.2. Tables A.1, A.2, A.3,
and A.4 show the numerical results for networks with different number of nodes.
We show the average number of wavelengths used by the algorithms LPA, RPA,
minFPA, maxFPA, IFPA, min2P, I2P, FCV, IFCV, LG, and RG, and the lower
bound number of wavelengths W-LB for instances with Uniform-SP, Uniform-LB,
Gaussian-SP and Gaussian-LB distribution of requests.
191

A.1 Undirected rings
Distribution W-LB LPA RPA minFPA maxFPA IFPA min2P I2P FCV IFCV LG RG
41 41 41 41 41 41 41 41 41 41 41 43
81 81 82 81 82 81 82 81 82 81 82 83
154 154 154 154 154 154 154 154 154 154 154 157
Uniform-SP 301 301 301 301 301 301 301 301 301 301 301 305
368 368 368 369 368 368 369 368 369 368 369 374
446 446 446 446 447 446 446 446 446 446 447 449
509 509 509 509 510 509 509 509 510 509 512 515
577 577 577 577 577 577 581 577 577 577 579 584
42 42 43 42 43 42 44 42 43 42 43 46
81 82 82 82 82 81 84 82 82 81 82 87
159 160 161 161 161 159 165 162 161 159 161 170
Uniform-LB 320 320 321 320 321 320 330 324 321 320 322 337
395 395 396 397 396 395 407 400 397 395 396 415
477 477 478 479 477 477 490 485 479 477 479 500
554 554 554 556 556 554 570 562 557 554 556 578
637 637 637 638 639 637 651 645 639 637 640 663
95 95 95 95 95 95 95 95 95 95 95 95
187 187 187 187 187 187 187 187 187 187 187 187
371 371 371 371 371 371 371 371 371 371 371 371
Gaussian-SP 743 743 743 743 743 743 743 743 743 743 743 743
925 925 925 925 925 925 925 925 925 925 925 925
1109 1109 1109 1109 1109 1109 1109 1109 1109 1109 1109 1109
1288 1288 1288 1288 1288 1288 1288 1288 1288 1288 1288 1288
1482 1482 1482 1482 1482 1482 1482 1482 1482 1482 1482 1482
55 55 57 55 56 55 56 55 56 55 55 60
106 107 111 107 107 107 110 108 107 107 107 114
214 214 225 215 216 214 217 215 215 214 214 227
Gaussian-LB 426 426 452 429 432 426 434 429 429 426 426 448
527 527 553 534 535 527 541 532 532 527 527 550
639 640 667 644 648 640 653 648 644 640 640 666
751 751 793 759 762 751 767 760 757 751 751 781
847 847 887 856 856 847 861 856 853 847 847 879
Table A.1: Average number of wavelengths for 16-node ring network. The instances
have 128, 256, 512, 1024, 1280, 1536, 1792 and 2048 requests.
192

A.1 Undirected rings
Distribution W-LB LPA RPA minFPA maxFPA IFPA min2P I2P FCV IFCV LG RG
43 44 44 44 44 43 44 43 44 43 44 46
78 79 79 79 78 78 80 78 79 78 80 83
147 148 148 148 148 147 150 148 150 148 150 156
Uniform-SP 292 292 292 292 294 292 295 292 294 292 296 304
356 356 357 356 360 356 367 359 360 357 361 374
430 430 431 431 434 430 439 431 432 430 434 444
497 497 498 498 499 497 507 500 500 498 502 516
570 571 571 572 572 570 583 572 574 571 578 590
41 42 42 43 43 41 43 42 43 42 42 46
81 83 85 85 84 82 87 85 86 84 84 90
158 159 161 163 163 160 169 165 163 161 162 173
Uniform-LB 313 315 317 319 320 315 334 329 321 316 318 340
388 390 392 397 396 390 413 407 397 393 395 420
464 466 468 470 470 466 491 486 473 468 472 500
542 543 546 549 551 544 573 565 551 546 550 582
616 618 620 625 623 618 654 643 626 621 623 660
95 95 95 95 95 95 95 95 95 95 95 95
187 187 187 187 187 187 187 187 187 187 187 187
372 372 372 372 372 372 372 372 372 372 372 372
Gaussian-SP 744 744 744 744 744 744 744 744 744 744 744 744
935 935 935 935 935 935 935 935 935 935 935 935
1116 1116 1116 1116 1116 1116 1116 1116 1116 1116 1116 1116
1308 1308 1308 1308 1308 1308 1308 1308 1308 1308 1308 1308
1487 1487 1487 1487 1487 1487 1487 1487 1487 1487 1487 1487
54 54 57 55 54 54 55 54 54 54 54 59
107 107 111 108 108 107 109 107 108 107 107 115
213 214 223 215 216 214 219 215 215 214 214 227
Gaussian-LB 425 426 450 427 427 426 434 430 427 426 426 447
538 538 564 546 538 538 549 544 544 538 538 561
637 637 671 645 644 637 648 644 643 637 637 664
760 761 797 761 773 761 782 767 761 761 761 789
852 852 891 854 867 852 865 857 855 852 852 880
Table A.2: Average number of wavelengths for 32-node ring network. The instances
have 128, 256, 512, 1024, 1280, 1536, 1792 and 2048 requests.
193

A.1 Undirected rings
Distribution |R| W-LB LPA RPA minFPA maxFPA min2P FCV LG RG
128 43 44 44 44 44 44 44 44 46
256 78 79 79 80 81 80 81 81 83
512 150 151 153 152 153 153 155 155 159
Uniform-SP 1024 285 287 288 290 293 295 295 294 303
1280 349 352 351 351 358 363 359 359 374
1536 422 425 426 428 433 441 434 432 447
1792 488 490 492 493 501 511 498 502 518
2048 557 559 559 560 564 575 569 570 589
128 41 43 44 43 43 44 44 43 46
256 79 82 83 84 84 85 85 83 89
512 153 157 159 163 162 166 165 159 171
Uniform-LB 1024 306 310 314 320 320 332 323 315 338
1280 381 387 391 398 397 415 400 392 421
1536 458 464 466 475 477 501 480 470 502
1792 533 538 543 550 553 578 556 546 583
2048 604 609 613 624 623 654 626 617 659
128 95 95 95 95 95 95 95 95 95
256 189 189 189 189 189 189 189 189 189
512 374 374 374 374 374 374 374 374 374
Gaussian-SP 1024 745 745 745 745 745 745 745 745 745
1280 925 925 925 925 925 925 925 925 925
1536 1120 1120 1120 1120 1120 1120 1120 1120 1120
1792 1308 1308 1308 1308 1308 1308 1308 1308 1308
2048 1491 1491 1491 1491 1491 1491 1491 1491 1491
128 55 55 57 55 55 56 55 55 60
256 107 107 113 109 107 109 108 107 115
512 214 214 227 215 219 221 215 214 227
Gaussian-LB 1024 426 427 447 428 437 435 428 427 444
1280 542 542 571 549 545 553 547 542 563
1536 637 638 670 644 653 650 642 638 653
1792 768 768 810 778 780 783 776 768 796
2048 850 851 894 858 863 863 856 850 881
Table A.3: Average number of wavelengths for 64-node ring network. The instances
have 128, 256, 512, 1024, 1280, 1536, 1792 and 2048 requests.
194

A.1 Undirected rings
n Distribution |R| W-LB LPA RPA minFPA maxFPA min2P FCV LG RG
512 143 147 147 151 153 153 152 149 157
Uniform-SP 1024 282 286 286 290 294 298 295 292 304
1536 422 425 427 426 437 441 436 433 449
2048 557 560 564 570 571 592 578 571 593
512 157 163 165 168 168 173 169 165 177
Uniform-LB 1024 307 314 318 324 323 340 329 318 341
1536 454 462 467 477 478 498 482 468 499
96 2048 608 618 623 637 635 665 642 625 667
512 372 372 372 372 372 372 372 372 372
Gaussian-SP 1024 746 746 746 746 746 746 746 746 746
1536 1114 1114 1114 1114 1114 1114 1114 1114 1114
2048 1486 1486 1486 1486 1486 1486 1486 1486 1486
512 212 212 223 213 213 214 213 212 224
Gaussian-LB 1024 424 424 445 426 426 435 425 424 444
1536 663 663 693 664 668 679 664 663 690
2048 879 880 920 888 891 893 884 880 914
512 152 154 156 154 158 155 157 157 162
Uniform-SP 1024 290 293 294 296 302 304 303 299 312
1536 420 424 426 428 437 444 438 433 449
2048 553 557 560 568 574 584 580 570 591
512 155 162 163 166 167 169 168 162 174
Uniform-LB 1024 307 316 319 326 330 336 330 319 342
1536 452 462 466 481 480 503 484 469 501
128 2048 607 618 624 639 640 667 648 625 667
512 374 374 374 374 374 374 374 374 374
Gaussian-SP 1024 750 750 750 750 750 750 750 750 750
1536 1121 1121 1121 1121 1121 1121 1121 1121 1121
2048 1489 1489 1489 1489 1489 1489 1489 1489 1489
512 214 214 223 215 214 219 215 214 226
Gaussian-LB 1024 427 427 446 429 433 435 429 427 446
1536 638 638 668 638 642 652 639 638 664
2048 861 862 903 863 862 875 863 862 894
Table A.4: Average number of wavelengths for 96-node and 128-node ring networks.
The instances have 512, 1024, 1536, and 2048 requests.
195

A.2 Undirected tree of rings
A.2 Undirected tree of rings
In this section we report numerical results for WA algorithms for tree of rings
networks considered in Chapter 4. We consider tree of rings with 10 rings, where
number of nodes on each ring is chosen uniformly at random from the intervals
8 − 16 and 16 − 32. We use instances with Uniform and Gaussian distribution of
requests described in Section 2.4.1. The instances are routed using the shortest
path and the load balancing routing decisions described in Section 2.4.2. Tables
A.5 and A.6 show the numerical results for networks where the number of nodes
for each ring is chosen uniformly at random from the interval 8 − 16 and 16 − 32
respectively. We show the average number of wavelengths used by the algorithms
LP-DFS, APX, LG, and RG, and the lower bound number of wavelengths W-
LB, for instances with Uniform-SP, Uniform-LB, Gaussian-SP and Gaussian-LB
distribution of requests.
196

A.2 Undirected tree of rings
Distribution |R| W-LB LP-DFS APX LG RG
128 51 53 77 53 53
256 81 82 124 82 83
Uniform-SP 512 178 182 266 182 182
1024 339 342 506 342 342
2048 728 731 1080 731 731
2560 944 999 1263 999 999
3072 963 1064 1202 1064 1064
4096 1214 1345 1519 1345 1345
128 45 46 72 46 46
256 83 85 127 84 85
Uniform-LB 512 178 180 268 180 180
1024 321 325 475 324 325
2048 812 815 1194 815 815
2560 788 808 1045 807 809
3072 1134 1250 1391 1250 1250
4096 1333 1475 1660 1475 1476
128 17 18 31 18 19
256 34 37 67 36 36
Gaussian-SP 512 66 68 131 68 67
1024 123 127 253 125 126
2048 252 260 512 258 257
2560 471 471 673 472 472
3072 354 367 600 370 370
4096 464 482 796 484 486
128 16 17 31 17 17
256 29 30 57 31 31
Gaussian-LB 512 57 61 120 60 60
1024 111 116 233 117 114
2048 205 219 448 219 212
2560 439 445 615 444 444
3072 333 355 549 356 360
4096 439 468 722 473 471
Table A.5: Average number of wavelengths for tree of rings where the number of nodes
for each ring is chosen uniformly at random from the interval 8 − 16. The instances
have 128, 256, 512, 1024, 2048, 2560, 3072, and 4096 requests.
197

A.2 Undirected tree of rings
Distribution |R| W-LB LP-DFS APX LG RG
512 170 178 205 178 178
1024 364 379 432 378 379
1536 542 556 639 556 557
Uniform-SP 2048 795 819 941 819 820
2560 880 911 1048 910 912
3072 1149 1201 1420 1201 1201
4096 1330 1489 1700 1488 1488
512 177 185 213 184 186
1024 351 362 417 362 362
Uniform-LB 1536 522 543 627 542 543
2048 669 730 836 729 730
2560 1043 1083 1227 1083 1084
3072 1101 1155 1362 1154 1155
4096 1303 1458 1686 1456 1458
512 72 74 105 76 76
1024 127 133 207 143 137
Gaussian-SP 1536 192 201 311 209 202
2048 251 261 403 279 264
2560 323 345 520 363 346
3072 490 506 820 508 508
4096 663 673 1116 674 674
512 58 66 94 66 67
1024 115 130 182 133 132
Gaussian-LB 1536 168 185 272 189 188
2048 222 246 350 254 249
2560 268 306 423 310 307
3072 427 443 693 450 444
4096 567 585 916 595 587
Table A.6: Average number of wavelengths for tree of rings where the number of nodes
for each ring is chosen uniformly at random from the interval 16 − 32. The instances
have 512, 1024, 1536, 2048, 2560, 3072, and 4096 requests
198

A.3 Undirected rings with cluster wavelength conversion capabilities
A.3 Undirected rings with cluster wavelength
conversion capabilities
In this section we report numerical results for WA algorithms for ring networks
with cluster wavelength conversion capabilities considered in Chapter 5. We
consider ring networks with 16 to 96 nodes using instances with Uniform and
Gaussian distribution of requests described in Section 2.4.1. The instances are
routed using the shortest path and the load balancing routing decisions described
in Section 2.4.2. Tables A.7, A.8, A.9, and A.10 show the numerical results
for networks with different number of nodes. We show the average number of
wavelengths (W) and wavelength conversions (C) used by the algorithms SPR,
minFP-WC, LP-WC, and IFP-WC, and the lower bound number of wavelengths
W-LB, for instances with Uniform-SP, Uniform-LB, Gaussian-SP and Gaussian-
LB distribution of requests.
199

A.3 Undirected rings with cluster wavelength conversion capabilities
SPR minFP-WC LP-WC IFP-WC
Distribution |R| W-LB W C W C W C W C
128 42 42 13 42 0 42 0 42 0
256 84 84 27 84 0 84 0 84 0
512 153 154 56 154 15 154 0 153 3
Uniform-SP 1024 295 295 102 295 20 295 0 295 7
1536 446 446 142 446 23 446 0 446 13
2048 580 580 200 580 22 580 0 580 11
2560 731 731 202 731 0 731 0 731 0
3072 852 852 283 855 0 852 0 852 0
128 42 42 16 43 0 42 0 42 0
256 81 82 34 82 0 81 0 81 0
512 160 161 66 161 6 161 0 160 0
Uniform-LB 1024 314 314 114 316 6 314 0 314 1
1536 479 480 189 482 5 480 0 479 0
2048 631 631 226 632 4 631 0 631 0
2560 793 793 307 794 0 793 0 793 0
3072 949 949 358 952 0 949 0 949 0
128 95 95 2 95 0 95 0 95 0
256 187 187 2 187 0 187 0 187 0
512 377 377 2 377 98 377 0 377 1
Gaussian-SP 1024 741 741 3 741 206 741 0 741 0
1536 1113 1113 3 1113 328 1113 0 1113 0
2048 1475 1475 2 1475 404 1475 0 1475 0
2560 1847 1847 4 1847 0 1847 0 1847 0
3072 2213 2213 3 2213 0 2213 0 2213 0
128 52 53 11 52 0 52 0 52 0
256 108 110 26 109 0 109 0 109 0
512 216 221 55 217 2 217 0 217 0
Gaussian-LB 1024 432 442 112 436 6 432 0 432 0
1536 637 649 162 642 8 637 0 637 0
2048 842 858 203 846 14 842 0 842 0
2560 1094 1119 294 1099 0 1094 0 1094 0
3072 1265 1292 306 1281 0 1265 0 1265 0
Table A.7: Average number of wavelengths (W) and conversions (C) for 16-node ring
network. The instances have 128, 256, 512, 1024, 1536, 2048, 2560 and 3072 requests.
200

A.3 Undirected rings with cluster wavelength conversion capabilities
SPR minFP-WC LP-WC IFP-WC
Distribution |R| W-LB W C W C W C W C
128 41 42 14 42 0 42 0 41 0
256 78 79 35 79 0 79 0 78 0
512 156 157 66 156 0 157 0 156 0
Uniform-SP 1024 288 291 143 290 1 288 0 288 0
1536 425 427 215 426 2 425 0 425 0
2048 565 569 306 567 2 566 0 565 0
2560 703 707 352 705 2 703 0 703 0
3072 835 840 430 837 2 836 0 835 0
128 41 42 20 43 1 42 0 42 0
256 79 81 40 82 1 80 0 79 1
512 158 161 82 162 1 160 0 159 0
Uniform-LB 1024 309 315 173 316 1 311 0 311 0
1536 462 469 246 470 1 465 0 464 0
2048 617 628 347 625 0 618 0 619 0
2560 777 788 425 786 1 778 0 780 0
3072 925 936 523 936 1 927 0 927 0
128 95 95 1 95 0 95 0 95 0
256 187 187 1 187 0 187 0 187 0
512 375 375 3 375 0 375 0 375 0
Gaussian-SP 1024 749 749 3 749 0 749 0 749 0
1536 1113 1113 1 1113 0 1113 0 1113 0
2048 1486 1486 3 1486 0 1486 0 1486 0
2560 1856 1856 1 1856 0 1856 0 1856 0
3072 2225 2225 3 2225 0 2225 0 2225 0
128 53 54 12 54 0 54 0 54 0
256 109 111 26 110 0 110 0 110 0
512 210 213 46 212 0 210 0 210 0
Gaussian-LB 1024 427 436 106 432 0 428 0 428 0
1536 646 657 164 651 0 646 0 646 0
2048 850 867 214 863 0 851 0 850 0
2560 1049 1068 242 1057 0 1050 0 1050 0
3072 1278 1305 316 1287 0 1279 0 1279 0
Table A.8: Average number of wavelengths (W) and conversions (C) for 32-node ring
network. The instances have 128, 256, 512, 1024, 1536, 2048, 2560 and 3072 requests.
201

A.3 Undirected rings with cluster wavelength conversion capabilities
SPR minFP-WC LP-WC IFP-WC
Distribution |R| W-LB W C W C W C W C
128 42 43 18 43 1 43 0 42 1
256 77 78 37 78 1 78 0 77 0
512 153 156 84 155 1 154 0 153 1
Uniform-SP 1024 282 288 167 286 2 284 0 283 1
1536 417 424 255 421 2 418 0 417 1
2048 559 568 343 564 4 561 0 560 2
2560 694 703 436 696 2 695 0 694 1
3072 829 840 536 832 2 831 0 829 2
128 41 43 22 44 0 42 0 42 1
256 79 83 42 85 1 83 0 81 1
512 154 161 94 162 2 158 0 159 1
Uniform-LB 1024 309 319 192 323 2 313 0 316 1
1536 456 469 290 471 2 461 0 465 1
2048 610 627 378 631 2 615 0 622 0
2560 759 780 486 780 2 765 0 772 1
3072 914 935 586 935 3 919 0 927 1
128 93 93 2 93 0 93 0 93 0
256 189 189 1 189 0 189 0 189 0
512 377 377 1 377 0 377 0 377 0
Gaussian-SP 1024 746 746 2 746 0 746 0 746 0
1536 1121 1121 2 1121 0 1121 0 1121 0
2048 1486 1486 2 1486 0 1486 0 1486 0
2560 1859 1859 2 1559 0 1859 0 1859 0
3072 2228 2228 2 2228 0 2228 0 2228 0
128 56 57 13 57 0 57 0 56 0
256 108 110 26 109 0 109 0 108 0
512 214 218 49 216 0 215 0 215 0
Gaussian-LB 1024 422 428 102 424 0 422 0 422 0
1536 643 657 165 647 0 644 0 644 0
2048 838 856 204 843 0 838 0 838 0
2560 1048 1072 254 1061 0 1048 0 1048 0
3072 1277 1306 318 1287 0 1277 0 1277 0
Table A.9: Average number of wavelengths (W) and conversions (C) for 64-node ring
network. The instances have 128, 256, 512, 1024, 1536, 2048, 2560 and 3072 requests.
202

A.3 Undirected rings with cluster wavelength conversion capabilities
SPR minFP-WC LP-WC IFP-WC
Distribution |R| W-LB W C W C W C W C
128 42 43 18 43 1 43 0 42 0
256 78 80 37 81 1 79 0 78 1
512 147 151 82 152 1 149 0 147 0
Uniform-SP 1024 282 289 173 286 2 285 0 283 2
1536 419 428 269 425 3 422 0 420 1
2048 555 565 353 562 3 558 0 555 2
2560 693 707 468 699 3 697 0 695 2
3072 819 835 551 826 4 823 0 819 2
128 42 44 23 46 2 46 0 44 1
256 78 81 43 84 1 81 0 80 1
512 156 164 95 167 1 161 0 162 1
Uniform-LB 1024 308 320 195 325 2 315 0 320 2
1536 457 472 295 478 3 465 1 473 1
2048 608 630 397 636 3 618 0 627 2
2560 751 774 499 780 4 760 1 772 2
3072 907 934 600 939 4 916 1 931 2
128 94 94 1 94 0 94 0 94 0
256 189 189 1 189 0 189 0 189 0
512 375 375 1 375 0 375 0 375 0
Gaussian-SP 1024 749 749 2 749 0 749 0 749 0
1536 1124 1124 2 1124 0 1124 0 1124 0
2048 1486 1486 3 1486 0 1486 0 1486 0
2560 1861 1861 2 1861 0 1861 0 1861 0
3072 2234 2234 3 2234 0 2234 0 2234 0
128 55 55 11 55 0 55 0 55 0
256 107 109 24 108 0 107 0 107 0
512 211 215 48 213 0 212 0 212 0
Gaussian-LB 1024 425 432 101 426 0 425 0 425 0
1536 642 655 162 647 0 642 0 642 0
2048 847 863 207 858 0 847 0 847 0
2560 1050 1069 254 1063 0 1050 0 1050 0
3072 1286 1314 330 1306 0 1287 0 1287 0
Table A.10: Average number of wavelengths (W) and conversions (C) for 96-node ring
network. The instances have 128, 256, 512, 1024, 1536, 2048, 2560 and 3072 requests.
203

A.4 Traffic grooming on path networks - General Instances
A.4 Traffic grooming on path networks - General
Instances
In this section we report numerical results for traffic grooming algorithms for
unidirectional path networks with grooming ratio 2 and instances with unit size
requests considered in Chapter 8. We consider path networks with 16 to 96
nodes using instances with BL-Uniform, UL-Uniform and Gaussian distribution
of requests described in Section 2.4.1. No routing decisions are needed for path
networks since every pair of nodes has a unique path. Tables A.11, A.12, A.13,
and A.14 show the numerical results for path networks with different number of
nodes. We show the average number of ADMs (ADM) and wavelengths (W) used
by the algorithms EDA-SF, EDA-LF, IPA-MWM, IPA-LWM, and IPA-SGM, and
the lower bound number of ADMs ADM-LB and wavelengths W-LB, for instances
with UL-Uniform, BL-Uniform, and Gaussian distribution of requests.
EDA
IPA
EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W ADM W
128 34 102 119 34 123 35 125 34 129 34 156 34
256 70 203 221 71 230 70 239 70 248 70 308 70
512 141 401 421 141 450 142 456 141 471 141 610 141
UL-Uniform 1024 274 790 809 275 859 275 870 274 895 274 1206 274
1536 416 1187 1205 416 1280 416 1284 416 1318 416 1812 416
2048 560 1588 1608 561 1689 561 1701 560 1743 560 2421 560
128 37 103 109 37 112 37 114 37 117 37 144 37
256 73 204 212 73 219 73 220 73 226 73 289 73
512 145 404 413 145 421 145 427 144 434 144 580 144
Gaussian 1024 288 803 812 288 827 288 833 288 846 287 1149 288
1536 432 1204 1215 433 1239 432 1240 432 1253 432 1701 432
2048 576 1604 1616 576 1648 576 1645 576 1664 576 2286 576
128 30 98 109 30 112 30 122 30 126 30 148 30
256 58 190 202 58 214 58 227 58 234 58 291 58
512 111 372 384 111 409 111 430 111 442 111 570 111
BL-Uniform 1024 226 741 754 226 798 226 826 226 851 226 1140 226
1536 334 1105 1119 334 1181 334 1214 334 1243 334 1699 334
2048 439 1466 1480 439 1548 439 1593 439 1634 439 2254 439
Table A.11: Average number of ADMs (ADM) and wavelengths (W) for 16-node
path network. The instances have 128, 256, 512, 1024, 1536 and 2048 requests.
204

A.4 Traffic grooming on path networks - General Instances
EDA
IPA
EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W ADM W
128 34 108 137 35 139 35 149 34 154 34 174 34
256 66 203 251 67 256 68 273 66 279 66 328 66
512 136 401 465 137 484 138 515 136 531 136 646 136
UL-Uniform 1024 268 788 865 269 915 270 969 268 999 268 1263 268
1536 397 1173 1256 398 1334 399 1418 397 1463 397 1877 397
2048 598 1759 1840 600 1948 600 2083 598 2144 598 2811 598
128 36 104 109 37 113 36 115 36 118 36 145 36
256 72 203 211 73 217 72 219 72 224 72 288 72
512 144 403 413 144 423 144 426 144 433 144 577 144
Gaussian 1024 290 805 816 290 834 290 838 290 850 290 1143 290
1536 433 1205 1216 433 1242 433 1241 433 1257 433 1730 433
2048 645 1801 1814 646 1839 646 1847 645 1866 645 2566 645
128 29 103 129 29 130 29 142 29 144 29 165 29
256 54 192 228 55 232 55 260 54 266 54 310 54
512 107 373 422 108 437 108 492 107 505 107 610 107
BL-Uniform 1024 209 732 786 210 825 211 932 209 959 209 1193 209
1536 314 1090 1146 314 1212 314 1359 314 1397 313 1779 313
2048 469 1631 1686 470 1782 470 1988 469 2045 469 2663 469
Table A.12: Average number of ADMs (ADM) and wavelengths (W) for 32-node
path network. The instances have 128, 256, 512, 1024, 1536 and 2048 requests.
205

A.4 Traffic grooming on path networks - General Instances
EDA
IPA
EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W ADM W
128 33 117 158 34 157 34 169 33 173 33 189 33
256 69 217 291 71 290 72 315 69 320 69 353 69
512 130 408 525 132 532 132 584 130 594 130 677 130
UL-Uniform 1024 263 794 983 266 1005 266 1095 263 1122 263 1313 263
1536 396 1184 1409 400 1455 400 1583 396 1629 396 1951 396
2048 531 1574 1832 535 1900 535 2059 531 2127 531 2579 531
128 37 104 111 37 113 37 115 37 119 37 147 37
256 74 206 212 74 218 74 220 74 226 74 290 74
512 144 403 412 145 422 144 425 144 434 144 577 144
Gaussian 1024 288 803 815 289 833 289 835 288 847 288 1143 288
1536 431 1203 1215 432 1239 432 1240 431 1255 431 1725 431
2048 580 1607 1619 580 1645 580 1645 580 1663 580 2285 580
128 28 115 151 29 151 29 166 28 168 28 185 28
256 55 204 269 57 269 56 302 55 308 55 341 55
512 105 387 488 108 494 107 565 105 575 105 654 105
BL-Uniform 1024 202 735 885 205 907 204 1048 202 1075 202 1252 202
1536 302 1094 1273 306 1316 305 1524 302 1567 302 1852 302
2048 403 1450 1641 406 1705 407 1996 403 2043 403 2458 403
Table A.13: Average number of ADMs (ADM) and wavelengths (W) for 64-node
path network. The instances have 128, 256, 512, 1024, 1536 and 2048 requests.
206

A.4 Traffic grooming on path networks - General Instances
EDA
IPA
EDA-SF EDA-LF IPA-MWM IPA-LWM IPA-SGM
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W ADM W
128 34 129 171 37 171 37 180 34 185 34 198 34
256 67 226 313 70 313 70 338 67 343 67 374 67
512 131 418 566 135 572 134 626 131 636 131 702 131
UL-Uniform 1024 264 805 1052 267 1065 268 1170 264 1191 264 1343 264
1536 390 1191 1516 395 1542 395 1697 390 1733 390 1991 390
2048 525 1578 1960 529 2015 528 2209 525 2264 525 2635 525
128 38 105 112 38 115 38 115 38 118 38 147 38
256 73 204 213 73 218 73 220 73 225 73 287 73
512 142 401 411 143 421 143 423 142 431 142 563 142
Gaussian 1024 289 805 815 290 831 290 835 289 847 289 1136 289
1536 430 1202 1212 430 1233 431 1229 430 1256 430 1716 430
2048 577 1604 1616 577 1643 577 1645 577 1663 577 2263 577
128 28 126 166 30 166 30 178 28 182 28 200 28
256 53 216 295 57 294 56 327 53 332 53 364 53
512 101 392 526 104 528 104 605 101 615 101 683 101
BL-Uniform 1024 202 749 963 207 972 206 1122 202 1145 202 1291 202
1536 302 1103 1377 306 1400 306 1623 302 1659 302 1906 302
2048 397 1457 1767 402 1812 402 2117 397 2172 397 2514 397
Table A.14: Average number of ADMs (ADM) and wavelengths (W) for 96-node
path network. The instances have 128, 256, 512, 1024, 1536 and 2048 requests.
207

A.5 Traffic Grooming on ring networks - General Instances
A.5 Traffic Grooming on ring networks - General
Instances
In this section we report numerical results for traffic grooming algorithms for
unidirectional ring networks with grooming ratio 2 and instances with unit size
requests considered in Chapter 10. We consider ring networks with 16 to 96
nodes using instances with BL-Uniform, UL-Uniform and Gaussian distribution
of requests described in Section 2.4.1. No routing decisions are needed for unidirectional
ring networks since every pair of nodes has a unique path. Tables
A.15, A.16, A.17, and A.18 show the numerical results for ring networks with
different number of nodes. We show the average number of ADMs (ADM) and
wavelengths (W) used by the algorithms FE-EDA, PIM-g2, A2-g2, and A1-g2,
and the lower bound number of ADMs ADM-LB and wavelengths W-LB, for
instances with UL-Uniform, BL-Uniform, and Gaussian distribution of requests.
208

A.5 Traffic Grooming on ring networks - General Instances
PIM-g2 FE-EDA A1-g2 A2-g2
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W
128 35 79 111 36 114 37 123 37 117 37
256 68 150 191 70 196 70 219 72 205 72
512 136 285 341 140 346 141 386 142 362 142
UL-Uniform 1024 271 555 625 276 631 277 698 279 658 279
1536 403 822 896 407 902 409 984 414 942 414
2048 526 1083 1155 531 1162 533 1248 538 1207 538
2560 661 1343 1434 669 1440 670 1536 675 1494 675
3072 789 1608 1690 795 1697 797 1795 804 1751 804
128 35 74 87 35 88 35 98 35 91 35
256 67 142 159 68 160 68 179 69 165 69
512 133 275 299 134 301 134 323 135 309 135
Gaussian 1024 262 537 573 265 576 266 607 269 591 269
1536 390 789 824 393 828 393 858 394 839 394
2048 518 1050 1089 521 1091 522 1127 524 1110 524
2560 647 1307 1348 650 1351 651 1391 652 1370 652
3072 775 1568 1614 780 1618 781 1661 784 1638 784
128 23 82 101 23 100 23 111 26 105 26
256 42 164 186 43 182 42 197 52 189 52
512 83 330 356 84 351 83 377 110 361 110
BL-Uniform 1024 159 593 631 159 625 159 666 189 646 189
1536 231 848 893 233 886 232 935 263 913 263
2048 306 1132 1176 310 1167 307 1236 351 1213 351
2560 385 1522 1559 392 1551 393 1619 493 1600 493
3072 469 1903 1934 490 1928 495 2013 634 1994 634
Table A.15: Average ADMs and wavelengths for 16-node ring network. The instances
have 128, 256, 512, 1024, 1536, 2048, 2560 and 3072 requests.
209

A.5 Traffic Grooming on ring networks - General Instances
PIM-g2 FE-EDA A1-g2 A2-g2
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W
128 38 89 144 40 147 41 153 41 148 41
256 69 162 249 72 254 74 271 74 260 74
512 137 301 443 144 452 146 491 146 470 146
UL-Uniform 1024 268 575 766 277 780 280 873 283 827 283
1536 401 841 1059 412 1076 416 1211 417 1134 417
2048 466 975 1205 479 1226 485 1381 488 1291 488
2560 662 1375 1622 674 1641 680 1854 685 1733 685
128 34 75 89 35 90 35 99 35 92 35
256 68 142 160 68 161 69 172 69 164 69
512 133 274 297 134 300 134 319 135 306 135
Gaussian 1024 263 536 568 265 570 265 596 267 580 267
1536 390 796 834 392 836 393 868 395 850 395
2048 455 937 979 460 983 460 1024 463 1004 463
2560 650 1313 1362 656 1366 656 1407 659 1386 659
128 23 93 125 23 124 24 134 30 130 30
256 41 160 219 41 217 42 242 47 235 47
512 79 331 417 81 403 80 450 107 425 107
BL-Uniform 1024 152 630 749 159 731 157 792 201 760 201
1536 224 967 1109 232 1074 228 1158 315 1110 315
2048 257 1094 1259 261 1219 261 1306 349 1252 349
2560 365 1542 1726 380 1679 370 1775 488 1717 488
Table A.16: Average ADMs and wavelengths for 32-node ring network. The instances
have 128, 256, 512, 1024, 1536, 2048 and 2560 requests.
210

A.5 Traffic Grooming on ring networks - General Instances
PIM-g2 FE-EDA A1-g2 A2-g2
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W
128 36 104 165 39 166 42 171 43 165 43
256 71 180 304 77 310 80 321 81 312 81
512 135 321 549 146 560 149 590 150 575 150
UL-Uniform 1024 269 604 284 982 1005 291 1082 290 1045 290
1536 400 875 1369 420 1397 429 1520 426 1460 426
2048 530 1141 1738 561 1772 570 1946 569 1860 569
2560 659 1409 2069 691 2105 698 2341 699 2231 699
128 36 77 91 37 93 37 101 37 95 37
256 67 141 159 68 161 68 175 68 165 68
512 132 275 298 133 301 134 323 135 310 135
Gaussian 1024 263 536 571 266 573 267 602 268 586 268
1536 390 795 828 392 832 393 862 395 845 395
2048 520 1056 1094 522 1097 523 1135 525 1116 525
2560 648 1311 1348 650 1352 651 1390 654 1369 654
128 22 105 148 22 148 24 155 31 150 31
256 40 187 269 42 265 42 289 58 277 58
512 75 335 472 76 462 79 517 102 498 102
BL-Uniform 1024 146 671 901 149 870 152 969 214 926 214
1536 216 1017 1301 224 1241 224 1371 334 1309 334
2048 280 1313 1686 291 1606 289 1778 424 1699 424
2560 351 1524 1989 358 1909 358 2125 460 2035 460
Table A.17: Average ADMs and wavelengths for 64-node ring network. The instances
have 128, 256, 512, 1024, 1536, 2048 and 2560 requests.
211

A.5 Traffic Grooming on ring networks - General Instances
PIM-g2 FE-EDA A1-g2 A2-g2
Distribution |R| W-LB ADM-LB ADM W ADM W ADM W ADM W
128 35 118 172 39 174 42 178 45 169 45
256 70 195 326 78 328 81 338 83 328 83
512 137 339 601 150 612 154 636 157 621 157
UL-Uniform 1024 269 618 1099 290 1118 298 1183 297 1155 297
1536 397 895 1561 431 1590 440 1697 439 1652 439
2048 533 1179 1992 570 2035 582 2188 577 2117 577
2560 663 1446 2399 709 2449 723 2643 718 2553 718
128 34 75 89 35 90 35 98 36 92 36
256 68 142 161 69 162 69 177 70 166 70
512 133 274 300 134 302 135 323 136 310 136
Gaussian 1024 261 533 564 263 567 265 591 265 576 265
1536 389 795 829 391 831 392 867 394 847 394
2048 520 1063 1102 523 1104 524 1144 526 1125 526
2560 656 1340 1391 660 1394 661 1433 663 1411 663
128 21 118 162 22 160 24 165 34 159 34
256 39 195 287 42 288 43 302 57 294 57
512 75 342 515 77 513 80 564 100 550 100
BL-Uniform 1024 144 629 941 145 927 151 1052 183 1021 183
1536 213 901 1333 213 1304 218 1506 260 1455 260
2048 279 1172 1712 280 1665 283 1945 335 1872 335
2560 345 1444 2085 346 2020 350 2363 408 2277 408
Table A.18: Average ADMs and wavelengths for 96-node ring network. The instances
have 128, 256, 512, 1024, 1536, 2048 and 2560 requests.
212

Bibliography
V. Auletta, I. Caragiannis, L. Gargano, C. Kaklamanis, and P. Persiano. Sparse
and limited wavelength conversion in all-optical tree networks. Theoretical
Computer Science, 270(2):361–399, 2002. 2.5
V. Auletta, I. Caragiannis, C. Kaklamanis, and P. Persiano. Randomized path
colouring on binary trees. In Lecture notes in Computer Science, editor, 3rd
International Workshop on Approximation Algorithms for Combinatorial Optimization
Problems(APPROX’00), 1913:60–71, Springer-Verlag, 2000. 2.5
Y. Aumann and Y. Rabani. Improved bounds for all optical routing. In Proc. of
the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’95),
pages 567–576, January, 1995. 2.5
J. Bermond, L. Braud, and D. Coudert. Traffic grooming on the path. SIROCO,
June 2005. 2.5, 2.6, 8, 8.3, 9, 9.1, 9.2
J. Bermond and S. Ceroi. Minimizing sonet adms in unidirectional wdm rings
with grooming ratio 3. Networks, 41:83–86, 2003. 2.5, 11
J. Bermond, C. Colbourn, D. Coudert, G. Ge, A. Ling, and X. Muñoz.
Traffic grooming in unidirectional wdm rings with grooming ratio c = 6.
SIAM/SIDMA, 19(2):523–542, 2005. 2.5, 11
J. Bermond, C. Colbourn, A. Ling, and M. Yu. Grooming in unidirectional rings:
k 4 − e designs. Discrete Mathematics (to appear). 2.5
213

BIBLIOGRAPHY
J. Bermond and D. Coudert. Traffic grooming in unidirectional wdm ring networks
using design theory. IEEE ICC, Anchorage, Alaska, May 2003. 2.5,
11
J. Bermond, D. Coudert, and X. Muñoz. Traffic grooming in unidirectional wdm
ring networks: the all-to-all unitary case. In the 7th IFIP Working Conference
on Optical Network Design and Modelling (ONDM’03), pages 1135–1153, 2003.
2.5, 11
J. Bermond and J. Schönheim. g-decomposition of k n where g has four vertices
or less. Discrete Mathematics, 19:113–120, 1977. 9.1, 9.3, 9, 10, 18
Z. Bian, Q. Gu, and X. Zhou. Wavelength assignment on bounded degree trees
of rings. Technical Report TR 2004-01, School of Computing Science, Simon
Fraser University, January 2004. 2.5, 4
Z. Bian, Q Gu, and X. Zhou. Tight bounds for wavelength assignment on tree
of rings. Technical Report TR 2004-05, School of Computing Science, Simon
Fraser University, June 2004. 2.5, 2.6, 4, 4.4, 4.4, 11
I. Caragiannis, C. Kaklamanis, A. Ferreira, S. Pérennes, and H. Rivano. Fractional
path colouring with applications to wdm networks. In 28th International
Colloquium on Automata, Languages and Programming (ICALP’01), Lecture
notes in Computer Science, pages 732–743, Springer-Verlag, 2001. 2.5
I. Caragiannis, C. Kaklamanis, and P. Persiano. Symmetric communication in
all-optical tree networks. Parallel Processing Letters, 10(4):305–313, 2000. 2.5
I. Caragiannis, C. Kaklamanis, and P. Persiano. Wavelength routing in all-optical
tree networks: A survey. Bulletin of the European Association for Theoretical
Computer Science, 76, 2002. 2.5, 2.5
T. Carpenter and S. Cosares. Demand routing and slotting on ring networks.
DIMACS Technical Report 97-02, 1997. 3.4
T. Carpenter and S. Cosares. Comparing heuristics for demand routing and slot
assignment on ring networks. Telecommunication Systems, 21(2–4):319–337,
2002. 2.5, 2.6, 3, 3.3.2, 3.4, 3.4.1, 3.5, 5.2.2, 11
214

BIBLIOGRAPHY
D. Cavendish. On the complexity of routing and wavelength assignment in wdm
rings with limited wavelength conversion. Proceedings of Asian-Pacific Optical
and Wireless Communications Conference and Exhibit, Nov 12-16, 2001,
Beijing, China. 2.5, 5.1
D. Cavendish and B. Sengupta. Routing and wavelength assignment in wdm
ring networks with heterogeneous wavelength conversion capabilities. IEEE
Infocom, 2002. 2.5, 2.6, 5, 5.1, 5.3, 11
G. Călinescu, O. Frieder, and P. Wan. Minimizing electronic line terminals for
automatic ring protection in general wdm optical networks. IEEE Journal of
Selected area on Communications, 20(1):183–189, January 2002. 2.5
G. Călinescu and P. Wan. Traffic partition in wdm/sonet rings to minimize sonet
adms. J. Comb. Optim., 6(4):425–453, 2002. 2.5, 2.6, 10, 10.1, 10.2, 10.4, 10.4.2
C. Chekuri, M. Mydlarz, and F. Shepherd. Multicommodity demand flow in a
tree (extended abstract). To appear in Proceedings of the 30th ICALP, 2003.
2.5, 6, 11
C. Cheng. Improved approximation algorithm for the demand routing and slotting
problem with unit demands on rings. SIAM Journal on Discrete Mathematics,
17(3):384–402, 2004. 2.5
A. Chiu and E. Modiano. Traffic grooming algorithms for reducing electronic
multiplexing costs in wdm ring networks. IEEE/OSA Journal of Lightwave
Technology, 18:2–12, January 2000. 2.5
V. Chvátal. A greedy heuristic for the set covering problem. Mathematics of
Operation Research, 4:233–235, 1979. 2.5
C. Colbourn and J. Dinitz. editors. the crc handbook of combinatorial designs.
CRC Press, 1996. 2.5
C. Colbourn and P. Wan. Minimizing drop cost for sonet/wdm networks with 1 8
wavelength requirements. Networks, 37:107–116, 2001. 2.5
215

BIBLIOGRAPHY
R. Cole, K. Ost, and S. Schirra. Edge-coloring bipartite multigraphs in o(e·log d)
time. Combinatorica, 21(1):5–12, 2001. 6.1, 2
S. Cosares and I. Saniee. An optimization problem related to balancing loads on
sonet rings. Telecommunication Systems, 3, 1994. 2.4.2
X. Deng, G. Li, W. Zang, and Y. Zhou. A 2-approximation algorithm for path
colouring on a restricted class of tree of rings. Journal on Algorithms, 47(1):
1–13, 2003. 2.5, 4
D. Dor and M. Tarsi. Graph decomposition is np-complete: A complete proof of
holyer’s conjecture. SIAM J. Comput., 26(4):1166–1187, 1997. 8.2
R. Dutta, S. Huang, and G. Rouskas. Traffic grooming in path, star, and tree
networks: Complexity, bounds, and algorithms. SIGMETRICS’03, June, 2003.
2.5, 8, 10
R. Dutta and G. Rouskas. On optimal traffic grooming in wdm rings. IEEE
Journal on selected areas in communications, 20(1), January, 2002. 2.5, 11
J. Edmonds. Maximum matching and a polyhedron with 0,1 - vertices. Journal
of Research of the National Bureau of Standards, 69(B):125–130, 1965a. 8.2,
8.2.2, 8.2.2, 2, 3, 8.2.3.2, 8.4, 10.3, 10.3.3, 10.4.2
J. Edmonds. Paths, trees, and flowers. Canadian Journal on Mathematics, pages
449–467, 1965b. 8.2, 8.2.2, 8.2.2, 2, 3, 8.2.3.2, 8.4, 10.3, 10.3.3, 10.4.2
T. Eilam, S. Moran, and S. Zaks. Lightpath arrangement in survivable rings to
minimize the switching cost. IEEE Journal of Selected Area on Communications,
20(1):172–182, January 2002. 2.5
L. Epstein and A. Levin. Better bounds for minimizing sonet adms. In 2nd Workshop
on Approximation and Online algorithms, Bergen, Norway, September
2004. 2.5
T. Erlebach. Approximation algorithms and complexity results for path colouring
problems in trees of rings. In Proceedings of the 26th International Symposium
216

BIBLIOGRAPHY
on Mathematical Foundations of Computer Scince (MFCS 2001), LNCS, 2136:
351–362, Springer-Verlag, 2001. 2.5
T. Erlebach and K. Jansen. Call scheduling in trees, rings and meshes. In Proc. of
30th Hawaii International Conference on System and Sciences, pages 221–222,
1997a. 2.2.2, 2.5
T. Erlebach and K. Jansen. Scheduling of virtual connections in fast networks. In
Proc. of 4th Workshop on parallel Systems and Algorithms (PASA ’96), World
Scientific, pages 13–32, 1997b. 2.2.2
T. Erlebach, K. Jansen, C. Kaklamanis, M. Mihail, and P. Persiano. Optimal
wavelength routing on directed fiber trees. Theoretical computer science, 221(1-
2):119–137, 1999. 2.5, 2.5
T. Erlebach, A. Pagourtzis, K. Potika, and S. Stefanakos. Resource allocation
problems in multifiber wdm tree networks. To appear in: Proceedings of the
29th Workshop on Graph Theoretic Concepts in Computer Science (WG 2003),
Elspeet, The Netherlands, June 2003. c○Springer Verlag, 2003. 2.5, 6
T. Erlebach and S. Stefanakos. Wavelength conversion in networks of bounded
treewidth. TIK-Report Nr. 132, ETH Zurich, pages 1–9, April, 2002. 2.5
A. Ferreira, S. Pérennes, A. Richa, H. Rivano, and N. Stier. On the design of
multifiber wdm networks. In AlgoTel’02, Mze, France, pages 25–32, May 2002.
2.2.2.1
M. Flammini, L. Moscardelli, M. Shalom, and S. Zaks. Approximating the traffic
grooming problem. 16th Annual International Symposium on Algorithms and
Computation (ISAAC), Sanya, Hainan, China, December 2005. 2.5
M. Garey, D. Johnson, G. Miller, and C. Papadimitriou. The complexity of
colouring circular arcs and chords. SIAM J.Alg. Disc. Math., 1(2):216–227,
1980. 2.5, 3.1
217

BIBLIOGRAPHY
L. Gargano and U. Vaccaro. Routing in all-optical networks: Algorithmic and
graph-theoretic problems. In Numbers, Information and Complexity, I. Althofer
et al. (Eds.), Kluwer Academic Publisher, pages 555–578, Feb. 2000.
1
O. Gerstel, P. Lin, and G. Sasaki. Cost effective traffic grroming in wdm rings. In
INFOCOM’98, Seventeenth Annual Joint Conference of the IEEE Computer
and Communications Societies, 1998a. 2.5, 10.1
O. Gerstel, P. Lin, and G. Sasaki. Wavelength assignment in a wdm ring to
minimize cost of embedded sonet rings. In INFOCOM’98, Seventeenth Annual
Joint Conference of the IEEE Computer and Communications Societies, pages
69–77, 1998b. 2.5
M. Golumbic. Algorithmic graph theory and perfect graphs. Academic Press,
Inc, San Diego, Canada, 1980. 2.5, 3.1, 3, 3.3.1, 2, 3.3.2, 3, 7.1, 8.2, 1
M. Gurusamy and C. Siva Ram Murthy. WDM optical networks concepts, design
and algorithms. Prentice Hall PTR Upper Saddle River, New Jersey 07458,
ISBN 0-13-060637-5, 2002. 1, 2.2.2.2
L. Hendren, G. Gao, E. Altman, and C. Mukerji. A register allocation framework
based on hierarchical cyclic interval graphs. ACAPS Technical Memo 33
(Revised), 1993. 3, 3.1, 1, 3.2, 3.2
J. Hu. Optimal traffic grooming for wavelength-division-multplexing rings with
all-to-all uniform traffic. OSA Journal of Optical networking, 1(1):32–42, January,
2002. 2.5, 11
A. Itai and M. Rodeh. Finding a minimum circuit in a graph. Proc. 9th ACM
Symposium in Theory of Computing, pages 1–10, 1977. 8.2
G. Justo. Path multi-colouring in branched trees. Technical Report TR-03-02,
Department of Computer Science, King’s College London, 2003. 2.6
G. Justo and T. Radzik. Wavelength assignment in multifiber rings with limited
conversion. Technical Report TR-04-06 Department of Computer Science -
KCL, July, 2004. 2.6
218

BIBLIOGRAPHY
G. Justo and T. Radzik. Improved heuristics for static wavelength assignment in
undirected rings with heterogeneous wavelength conversion capability. Technical
Report TR-05-05 Department of Computer Science - KCL, July, 2005a.
2.6
G. Justo and T. Radzik. Improved heuristics for wavelength assignment in undirected
rings and tree of rings. Technical Report TR-05-04 Department of Computer
Science - KCL, July, 2005b. 2.6
G. Justo and T. Radzik. Traffic grooming on ring networks. Technical Report
TR-06-06 Department of Computer Science - KCL, June, 2006. 2.6
G. Justo and T. Radzik. Traffic grooming on path network. Technical Report
TR-06-02 Department of Computer Science - KCL, March, 2006. 2.6
C. Kaklamanis and P. Persiano. Efficient wavelength routing on directed fiber
trees. In Proceedings of the 4th Annual European Symposium on Algorithms
ESA ’96, LNCS, 1136:460–470, Springer-Verlag, 1996. 2.5
I. Karapetian. On colouring of arc graphs. Dokladi of the Academy of Science of
the Armenian SSR, 70:306–311, 1980. 2.5
R. Klasing. Methods and problems of wavelength-routing in all-optical networks.
In Proc. of the MFCS’98 Workshop on Communication, Brno, Czech Republic,
pages 1–9, August, 1998. 2.2.1
J. Kleinberg and A. Kumar. Wavelength conversion in optical networks. In Proc.
of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
’99), pages 566–575, January, 1999. 2.5
V. Kumar. Approximating circular arc colouring and bandwidth allocation in alloptical
ring networks. In Proc. of the 1st International Workshop on Approximation
Algorithms for Combinatorial Optimization Problems (APPROX’98),
pages 147–158, 1998. 2.5, 11
V. Kumar and E. Schwabe. Improved access to optical bandwidth in trees. Proceedings
of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA ’97), pages 437–444, January, 1997. 2.5
219

BIBLIOGRAPHY
T. Lee, K. Lee, and S. Park. Optimal routing and wavelength assignment in wdm
ring networks. IEEE Journal on Selected Areas in Communications, 18(10),
2000. 2.5
G. Li and R. Simha. On the wavelength assignment problem in multifiber wdm
star and ring networks. In Proceedings of ICALP, 2000. 2.5, 7, 11
H. Liu and F. Tobagi. A novel efficient technique for traffic grooming in wdm sonet
with multiple line speeds. IEEE International Conference on Communications,
3:1694–1698, June, 2004. 2.5
L. Margara and J. Simon. Wavelength assignment problem on all-optical networks
with k fibers per link. In Proceedings of ICALP, pages 768–779, 2000. 2.5, 7
K. Mehlhorn and S. Näher. Leda: A platform for combinatorial and geometric
computing. Cambridge University Press, 1999. 2.4.3
M. Mihail, C. Kaklamanis, and S. Rao. Efficient access to optical bandwidth. In
proceedings of the 36th Annual Symposium of Foundations of Computer Science
(FOCUS’95), pages 548–557, 1995. 2.5
T. Nishizeki and K. Kashiwagi. On the 1.1 edge-colouring of multigraphs. SIAM
Journal on Discrete Mathematics, 3(3):391–410, August 1990. 2.5, 4, 3, 4.6
C. Nomikos, A. Pagourtzis, and S. Zachos. Routing and path multicolouring. In
Information Processing Letters, volume 80(5), pages 249–256, 2001. 2.5, 2.6, 6,
6.1, 6.2
E. Potika, C. Nomikos, A. Pagourtzis, and S. Zachos. Path multi-colouring in
weighted graphs. Proc. 8th Panhellenic Conference on Informatics, I:178–186,
Nicosia, Cyprus, Nov 8–10, 2001. 2.6, 6, 11
W. Press, S. Teukolsky, W. Vetterrling, and B. Flannery. Numerical recipes in C.
The Art of Scientific Computing, Second Edition ISBN 0-521-43108-5, 1992.
2.4.3
220

BIBLIOGRAPHY
Y. Rabani. Path colouring on the mesh. In Proc. of the 37th IEEE Symposium on
Foundations of Computer Science (FOCS 96), pages 400–409, October, 1996.
2.5
P. Raghavan and E. Upfal. Efficient routing in all-optical networks. In AIAA
Journal. In proceedings of the 26th Annual ACM Symposium on the Theory of
Computing, Montreal, Quebec, Canada, pages 134–143, 1994. 2.2.1, 2.5
R. Ramaswami and G. Sasaki. Multiwavelength optical networks with limited
wavelength conversion. In ACM/IEEE Trans. Networking, 6(6):744–754, December
1998. 2.5, 2.6, 7, 7, 7.1, 5, 7.1, 6, 7, 7.1, 1, 11
A. Schrijver, P. Seymour, and P. Winkler. The ring loading problem. Siam
Journal of Discrete Mathematics, 11(1):1–14, February 1998. 2.4.2
M. Shalom and S. Zaks. A 10/7 + ɛ approximation scheme for minimizing the
number of adms in sonet rings. In First Annual International Conference on
Broadband Networks, San-Jose, california, USA, October 2004. 2.5
C. Shannon. A theorem on colouring the lines of a network. J. Math. Phys,, 28:
148–151, 1949. 2.5, 4, 2, 4.6
S. Stefanakos and T. Erlebach. Routing in all-optical ring networks revisited.
IEEE, 2004. 2.4.2
A. Tucker. Colouring a family of circular arc graphs. SIAM Journal on Applied
Mathematics, 29:493–502, 1975. 3.4
G. Wilfong and P. Winkler. Ring routing and wavelength translation. In Proc. of
the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’98),
pages 333–341, January, 1998. 2.5, 2.5
X. Yang and B. Ramamurthy. Dynamic routing in translucent wdm optical networks.
IEEE, 2002. 1
X. Yang and B. Ramamurthy. Sparse regeneration in a translucent wdm optical
network. In Proceedings, SPIE Asia-Pacific Optical and Wireless Communications,
Beijing China, Nov. 2001. 1
221

BIBLIOGRAPHY
S. Zaks. On the complexity of the traffic grooming problem. Workshop on Graph
Classes, Width parameters and Optimization, Prague, 2005. 2.5
X. Zhang and C. Qiao. An effective and comprehensive approach for traffic
grooming and wavelength assignment in sonet/wdm rings. IEEE/ACM Trans.
On Networking, 8(5), 2000. 2.5, 2.6, 10, 10.2, 10.4, 10.4.2, 10.5
222