An algorithm for graph optimal monomorphism - Systems, Man and ...

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628 1FEE TRANSAC‘TIONS ON SYSTFMS, MAN, ANI) C’YBEKNETIC’S, VOI.. 20. NO. 3. MAY/JUNE 1990
An Algorithm for Graph Optimal
Monomorphism
A. K. C. WONG, M. YOU, AND S. C. CHAN
Abstract -An algorithm for finding the optimal monomorphism between
two attributed graphs is proposed. The problem is formulated as a
tree search problem. To guide the search, the branch-and-bound heuristic
approach is adopted, using an efficient consistent lower bounded
estimate for the evaluation function of the cost associated with the
optimal solution path in the search tree. This algorithm is a generalization
of an algorithm for graph optimal isomorphism which has been
compared with another algorithm for its performance. The present
algorithm’s potential of engineering application is demonstrated by a
simple structural pattern recognition problem and a plant allocation
and distribution problem.
I. INTRODUCTION
HE graph monomorphism problem [3],[4], also known
T as the subgraph isomorphism problem [5], [61, is one
that finds whether or not a one-to-one incidence preserving
vertex mapping between a graph and a subgraph of
another larger graph exists. The notion of optimality
becomes important in graph monomorphism when a perfect
matching between the attributed values of the two
graphs may not be available. Notable methods that deal
with the generalization of the graph monomorphism problem
include Ghahraman, Wong, and Au’s graph optimal
monomorphism algorithms [71, Shapiro and Haralick’s
forward-checking algorithm for inexact matching [8], and
Tsai and Fu’s subgraph error-correcting isomorphism algorithm
[9]. Various applications of the graph optimal
monomorphism problem have been outlined in [7]. A
recent experiment on image understanding by inexact
graph-matching can be found in [lo]. Approaches to solve
the general graph isomorphism/monomorphism problem
are discussed in [4], [6]. A further generalization of the
graph monomorphism problem is the largest common
subgraph isomorphism problem [ill, [121.
In this paper, we propose an algorithm for finding the
optimal monomorphism between two attributed graphs.
The optimal monomorphism is the best matching as it
minimizes the “cost” or “error” incurred in the mapping.
Such minimization can also be applied directly to the cost
of graph transformation [13], or to the increment of
entropy in a random graph synthesis process [14]-[18],
Manuscript received October 16, 1988; revised November 6, 1989.
The authors are with the PAMI Group, Department of Systems
Design Engineering, University of Waterloo, Waterloo, ON, Canada,
N2L 3G1.
IEEE Log Number 8933540.
without the need of extending the order of the smaller
graph to that of the larger graph in order to get an
optimal isomorphism between them.
Following [7], we call one of the two graphs the pattern
graph and the other one the base graph. Let the orders of
the pattern graph and the base graph be m and n,
respectively, with m G n. In the particular case where
m = n, the problem is to find the optimal isomorphism
between the two graphs. The decision tree for the graph
monomorphism problem between the pattern graph and
the base graph has height m, and n - p sons for each
node at level p = 0,l; . . , m - 1. Searching through the
decision tree by the branch-and-bound technique, we can
identify the optimal monomorphism between the pattern
graph and the base graph.
11. THE EVALUATION FUNCTION IN THE
BRANCH-AND-BOUND ALGORITHM
In general, solving a branch-and-bound problem requires
an evaluation function that assigns a cost to the
branch incident to each node N of the tree. Each node N
at level p > 0 is described by a collection of pairs N =
{(i, si)}, where i = 1,2,. .., p correspond to the vertex
indices of the pattern graph and the 4;’s are the distinct
vertex indices of the base graph matched with the i’s.
Assume that N = ((1, q,),(2, q2); . .,(p, qp)} indicates
the unique path from the root to N. Then the cost
k(p,q,) assigned to the branch incident to N is defined
as
where
P-1
k(p,q,)=c(p,q,)+ C c’((.j9P)~(qj~qp)) (1)
c’( ( a 9 b) , ( c, 4 ) = .((a 2 b)
j= 1
3 ( c 7 4 ) + c( ( b 3 a) 3 ( d, c))
(2)
and c(x,y) is the mapping cost between the graph elements
x and y in the pattern graph and base graph,
respectively.
The choice of the evaluation function critically determines
the search result. In this section, we first define the
evaluation function f* and then propose a consistent
lower bounded estimate f of f* with reference to [l].
The evaluation function f* is defined in such a way
that its value f*(N) at a node N={(i,q,)li=1,2;..,p)
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WONO et 01.: AN ALGORITHM FOR GRAPH OPTIMAI. MONOMORPHISM
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of level p is the cost g*(N) of an optimal path from the
root to node N plus the cost h*(N) of an optimal path
from node N to a leaf T = {(i, q,)li = 1,2; . ., m), where m
is the height of the tree, i.e.,
f*( N) = g*( N) + h*( N) (3)
P
g*(N) = c k(i,q,) (4)
1 = 1
m
h*(N)=min ( c k(i,q,,)) (5)
,=p+l
where k(x, y) is as defined in (1) and t denotes a feasible
path from N to T.
The value f*(N) is the cost of an optimal path from
the root R to a leaf T that is constrained to go through
node N. It is used to order the nodes in the tree for
expansion. In application, an estimate, f(N), called a
heuristic function of f*(N), is used instead, in which
f(N)=g*(N)+h(N) (6)
where h(N) is a “consistent lower bounded estimate” of
h*(N) using any available heuristic information. As far as
h(N) G h*(N), using f(N) as the lower bounded estimate
for each node N, the branch-and-bound algorithm would
1) expand fewer nodes as compared with a search algorithm
using no heuristic information (such as the breathfirst
algorithm, which assumes h(N) = 0 for all N), and 2)
always guarantee a minimum cost solution path in the
decision tree.
Let N={(l,q,),(2,q2);. .,(p,q,)) be the current node
of the decision tree at level p. Then we have the sets NI
and N2 of vertices, which have already been matched
between the pattern graph and the base graph, i.e.,
NI = (192,. . .,PI
N2 = (q19q2,. . ., 4,l
where i is matched with q1 for i = 1,2; . * , p. We also
have the sets M, and M, of vertices which have not yet
been matched between the pattern graph and the base
graph as follows:
MI = {p+l,p+2;..,m}
M, = { q I q E I 17 2,. . . 3 n) \ 4 ).
Let k’(i, q) be the cost of adding a pair of vertices (i, q)
to N, where i E M, and q E M,. We define
P
k’(i,q) =c(i,q)+ C c’((i7j),(q9q,)). (7)
1-1
Then, for each unmatched vertex i E M,, we find a corresponding
vertex q E M, such that the cost k’(i,q) is
minimized. Note that such a mapping H: M, -+ M, may
be many-to-one. Let a(N) be the total cost of H. We
have
m
Similarly, we can define b(N) as the total cost of an
optimal mapping H’: MI x M , + M, x M,. That is, for
each arc (i, j) with endpoints i,j E MI, we find a minimum
cost mapping H‘: (i, j) -+(q, r), such that (q, r) is an
arc with endpoints q, r E M,. H‘ can also be many-to-one.
Hence
With a(N) and b(N) defined, we then use their sum as
the lower bounded estimate of h*(N), i.e.,
h( N) =a( N) + b( N). ( 10)
Property: For any node N of the decision tree, h(N) is
a consistent lower bounded estimate of h*(N), i.e.,
The heuristic function for graph optimal monomorphism
given by (6)-(10) is a generalization of that used in
[l], whose performance has been compared with the
heuristic function used in [21.
In [2], by defining the cost of mapping a vertex i E MI
to a corresponding vertex q E M, to be
the total cost h’(N) of an “error-correcting’’ homomorphism
between MI and M, is taken as
h’(~) = min k”(i,q). (13)
EM, qEMZ
Since h‘(N) < h*(N) is always true, h’(N) is used as the
lower bounded estimate of h*(N) for finding the subgraph
error-correcting isomorphism between two graphs.
An algorithm is said to be more informed than another
if it uses more accurate heuristic information, i.e., using
estimated values closer to h*(N). Since h*(N) > h(N) >
h’(N) always holds, the algorithm which uses h(N) as a
lower bounded estimate of h*(N) is more informed than
the algorithm using h’(N). Experimental results of graph
optimal isomorphism reported in [l] indeed show that the
algorithm that uses h(N) expands fewer number of nodes
and requires less CPU time than the algorithm that uses
h’(N). It is evident that our algorithm, that uses h(N) for
graph optimal monomorphism, will be an efficient one.
Example: 1: To illustrate the computation of f(N)
given by (61, let G be the pattern graph with m = 4 and H
be the base graph with n = 5 as shown in Fig. 1, in which
the attributed values of the vertices and edges of both
graphs are enclosed in brackets. For simplicity, we set
c(x, y) = 1s - tl, where s and t are the attributed values of
x and y, respectively, and set c((a, b), (c, d)) =
c((b, a),(d, c)) such that c’((a, b),(c, d)) = 2c((a, b),(c, d)).
ForN={(1,4),(2,5)),p=2,Ml={3,4),and M,=(1,2,3),
I

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND < YBERNETIC‘S, VOL. 20, NO. 3, MAY/JUNE 1990
Graph H
Fig. 1. Pattern graph G and base graph H.
we have
=5
A
a( N ) = min k’(i,q)
i=3 4EMz
111. THE ALGORITHM AND APPLICATIONS
The algorithm for graph optimal monomorphism based
on (6)-(10) is presented next.
Begin {hitiation)
Set upper-bound = infinity. (infinify is a vely large
number 1
Set tentative-solution = (empty).
Set N2 =(empty).
Set f (N) = 0.
Set active-node-list = {(N2, f(N))). {root node)
End {Initiation)
Begin {Main process}
While active-node-list is not empty do: (node expan-
sion}
Set current-node = first element of active-node
-list.
Set N; = first element of current-node. {i.e., N, in
(N*, f(”
Set p’ = number of elements in N;.
Set p = p’+ 1.
Set M, = { p + 1, p + 2, . 1 * , m) . (m = size of pattern
graph)
Set MI =(1,2;..,n)\N;. {n=size ofbasegraph)
For each element s in Mi do:
* Set qp=s.
* Set N2 = N2/ U qp. {append qp to N;}
* Set M,=M;\q,.
* Compute f(N)= g*(N)+ a(N)+ MN).
* If p#m, then
. Insert (N2, f(N)) in active-node-list in
ascending order according to the value of
f(N). (new node generated)
* Else if f(N) < upper-bound, then
. Set upper-bound = f(N).
* Set tentative-solution = {NJ.
* End
Remove current-node from active-node-list. (current-node
is replaced by its active sons)
Remove any element in active-node-list whose
f(N) value is greater than or equal to upper-bound.
{prune unpromising nodes)
Output optimal-solution = {(i, qi)(i = 1,2; * *, m)
where qi is the ith element of tentative-solution.
Output optimal-cost = upper-bound.
End {Main process)
1

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WONG et 01.: AN ALGORITHM FOR GRAPH OPTIMAI. MONOMORPHISM
Root
63 1
1
2 3
([ 1,41, (2.51,
(3,21,(4,1))
Fig. 2. Decision tree for G and H.
\
(( 1,41, (2,51,
(3.21, (4.31)
( Optimal solution 1
4
Fig. 3. Runway model A.
Fig. 4. Runway model B.
Example: 2: The complete search tree for the pattern
graph and the base graph shown in Fig. 1, based on the
graph optimal monomorphism algorithm, is shown in
Fig. 2.
Example: 3: The conspicuous features of an airport
field are its runways. Figs. 3-5 show three different
models of the open-V runway configuration. Airports that
have open-V runway configurations include Houston Intercontinental
Airport (model A), Jacksonville International
Airport (model B), and Mid-Continent Airport at
Wichita (model C). Fig. 6 shows the image graph of the
prominent runways of the Jacksonville International Airport
obtained from the photograph shown in Fig. 7 [19].
The matrix representations of the attributed graphs of
Figs. 3-6 are shown in Tables I-IV, respectively. An
entry aij of each matrix represents:
1) the attributed value of vertex i, i.e., the length of
runway i, for i = j; or otherwise,
2) the attributed value of edge (U), i.e., the center-tocenter
distance between runway i and runway j
plus the acute angle between them.'
In this example, we find optimal matching between the
image graph and each of the three models. The cost of
matching two graph elements is taken as the absolute
difference between the attributed values of the two elements.
Table V shows the output of our graph optimal
monomorphism algorithm, which indicates that the image
graph of the Jacksonville runways and runway model B
incur the lowest cost of matching.
Example: 4: Consider the site selection, plant allocation,
and distribution problem as described in the following.
'This measure is not perspective-invariant and is used here for illustrative
purposes only; other more stable measures may be used instead.
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632 IZEE TRANSACTIONS ON SYSTEMS, MAN. AND C'YHERNFTICS, VOIL 20, NO. 3, MAY/JUNF 1990
TABLE I1
ATTRIH~JTED VALUES OF FIG. 4
1 2 3 4 5 6 7
1
~ ~
70 6+0 45+60 49+60 22+78
2 70 40+60 45+60 17+78
3 72 6+0 23+20
4 72 27+20
5 (symmetrical) 30
6
7
~ ~~ ~
27+20 28+60
22+20 22+60
18+78 22+60
23+78 27+60
5+82 10+40
30 10+42
50
1 2 /
Fig. 5. Runway model C.
Fig. 6. Image graph of Jacksonville International Airport.
TABLE Ill
ATTRIBUTED VALUES OF FIG. 5
1 2 3 4 5 6 7 8
1 75 17+0 44+50 51+50 47+50 21+65 21+65 27+25
2 38 38+50 45+50 46+50 23+65 25+65 37+25
3 37 9+0 13+0 24+65 26+65 37+25
4 55 17+0 32+65 34+65 45+25
5 (symmetrical) 85 25+65 26+65 32+25
6 25 4+0 19+90
7 22 15+90
8 30
TABLE IV
ATTRIBUTED VALUES OF FIG. 6
1 2 3 4 5 6
1 62 5+0 54+82 27+70 31+12 29+41
2 59 49+82 22+70 26+12 24+41
3 60 27+12 23+70 25+41
4 25 4+58 7+29
5 (symmetrical) 26 7+29
6 54
TABLE V
RESULTS OF GRAPH OPTIMAL MONOMORPHISM
Pattern Graph (Fig. 6)
Base Matching
Graph cost Vertex Mapping ((I, 9,))
Fig. 3 983 ((1,6),(2,5),(3,1),(4,2),(5,4),(6,3))
Fig. 4 424 ((I, 1),(2,2),(3,3),(4,5),(5,6),(6,7))
Fig. 5 634 ((1,4),(2,5),(3,1),(4,2),(5,3),(6,8))
Fig. 7. Photograph of Jacksonville International Airport (reprinted
from [19]).
TABLE I
ATTRIBUTED VALUES OF FIG. 3
1 2 3 4 5 6
1 50 3+0 5+0 38+62 40+62 42+62
2 50 2+0 35+62 37+62 39+62
3 60 35+62 37+62 39+62
4 65 2+0 5+0
5 (symmetrical) 65 3+0
6 65
The main objective of this problem is to select five sites
out of ten potential ones based on some considerations.
The ten potential sites are shown in Fig. 8, in which the
demands for a product in the neighborhoods of the sites,
given in tons per day, are enclosed in brackets. The sites
selected will be used as sales outlets for the product, and
each of them will have its own packaging plant to do the
final packing of the product. The capacities of the packaging
plants are shown in Table VI. The first consideration
is that the capacity of the packaging plant allocated to a
site shall very closely match the demand for the product
in the neighborhood of the site. Out of the five selected
sites, one of them will be chosen as the central sales
outlet, which will be furnished with a processing plant to
process all the raw materials required for the product.
The processed raw materials will then be distributed to
the other sales outlets daily by individual fleets. The
second consideration is therefore to minimize the total
cost of transportation by selecting a suitable site as the
central sales outlet. The last consideration is that, while
the distance between any two sales outlets shall be as
short as possible to facilitate distribution and overall
management, it is also advisable to have the outlets kept
far away from each other to avoid competition amongst
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WONG et al.: AN ALGORITHM FOR GRAPH OPTIMAL MONOMORPHISM
633
2 (lot/d)
0
10 (7t/d)
0
5 (Ilt/d)
e
4 (14t/d)
e
C
Fig. 8.
Ten potential sites of sales outlets.
TABLE VI
CAPACITIES OF PACKAGING PLANTS
TABLE VI1
DESIRABLE DISTANCES BETWEEN PACKAGING PLANTS
Size Capacity Available Large Medium Small
Large 15 t/day 1 no.
Medium 10 t/day 1 no.
Small 5 t/day 3 nos.
Large 5 5 5
Medium 5 4 4
Small 5 4 3
IN km
them, especially when they will have their own policies of
operation. The desirable distances between different sizes
of sales outlets are given in Table VII.
The ten potential sites are represented by a matrix, as
shown in Table VIII, in which entry nrr gives the demand
for the product in tons per day in the neighborhood of
site i whereas entry n,, for i # j gives the distance from
site i to site j in kilometers. Tables IX, X, and XI show
the matrix representations of three different arrangements
of the processing plant and the packaging plants.
The processing plant may be installed with any size of
packaging plant. The entries m,, or m,, in Tables IX, X,
and XI indicate
1) the capacity of packaging plant i for i = j;
2) the capacity of packaging plant j for i = 1, i.e., the
ideal quantity of the product to be transported
daily from the processing plant (which is installed
at the same site as packaging plant 1) to packaging
plant j; and
3) the desirable distance between packaging plant i
and packaging plant j for 1 < i < j.
The problem is formulated as a graph optimal
monomorphism problem by taking Table VI11 as the base
graph and Tables IX, X, and XI as the pattern graphs.
The idea is to find the best match between the real-world
TABLE VI11
ATTRIBUTED VALUES OF TEN POTENTIAL SITES
1 2 3 4 5 6 7 8 9 10
1 5 5 91010 6 4 4 8 8
2 5 1 0 4 9 1 1 1 0 7 4 5 8
3 9 4 8 6 1 1 1 2 9 5 4 7
4 1 0 9 6 1 4 7 1 0 9 7 4 5
5101111 7 1 1 5 6 8 7 4
6 6 1 0 1 2 1 0 5 6 3 7 9 6
7 4 7 9 9 6 3 3 4 6 4
8 4 4 5 7 8 7 4 3 3 3
9 8 5 4 4 7 9 6 3 5 4
10 8 8 7 5 4 6 4 5 4 7
TABLE IX
ATTRIBUTED VALUES OF PLANT ARRANGEMENT A
1 2 3 4 5
1 1 5 1 0 5 5 5
2 10 10 4 4 4
3 5 4 5 3 3
4 5 4 3 5 3
5 5 4 3 3 5
situation of the sites given in Table VI11 and each of the
ideal situations as presented in Tables IX, X, and XI. The
costs for the mapping of the graph elements are
1) c(i, j) = a x(m,, - n,,) if mrr > n,,.
2) c(i,j) = b x(n,, - m,,) if m,, < rill.
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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND C'YBERNETICS, VOI.. 20, NO. 3, MAY/JUNE 1990
TABLE X
ATTRIBUTED VALUES OF PLANT ARRANGEMENT B
1 2 3 4 5
1 1 0 1 5 5 5 5
2 1 5 1 5 5 5 5
3 5 5 5 3 3
4 5 5 3 5 3
5 5 5 3 3 5
TABLE XI
ATRIBCITED VALUES OF PLANT ARRANGEMENT C
1 2 3 4 5
1 5 15 10 5 5
2 15 15 5 5 5
3 IO 5 1 0 4 4
4 5 5 4 5 3
5 5 5 4 3 5
2 (10t/d)
5 (llt/d)
8
Fig. 9. Plant arrangement A (matching cost = 181).
io (7t/d)
8
5 (llt/d)
8
4 (14t/d)
Fig. 10. Plant arrangement B (matching cost = 195).

WONG et al.: AN ALGORITHM FOR GRAPH OPTIMAL MONOMORPHISM 635
2 (iot/d)
7 (3t/d)
5 (llt/d)
Fig. 11. Plant arrangement C (matching cost = 186).
TABLE XI1
REsuL-rs OF GRAPH OFWMAL MONOMORPHISM
Base Graph (Table VIII)
Pattern Matching
Graph cost Vertex Mapping ((i, 9,))
Table IX 181 ((1,4),(2,3),(3,9),(4,10),(5,8))
Table X 195 ((1,2),(2,3),(3, I), (4,9), 68))
Table XI 186 ((1,9),(2,4),(3,3),(4,101,(58))
3) c‘((1, j),(q, r)) = c X quantity X nqr where
a) quantity = nji if mlj > njj
b) quantity = m , if m , < n . .
Jl
4) c’((i, j),(q, r)) = d x lmij - n,,l if 1 < i < j.
where U, b, c, and d are parameters.
For the purpose of this example, we take U = 10, b = 5,
and c = d = 1. The result of matching the graphs, based
on our algorithm for graph optimal monomorphism, is
summarized in Table XI1 and Figs. 9- 11.
IV. CONCLUSION
In this paper, we have proposed an algorithm for finding
the optimal monomorphism between attributed graphs
or random graphs. Here the graph optimal monomorphism
problem is formulated as a tree search problem;
then the branch-and-bound heuristic is adopted to guide
the tree search. The heuristic consists of a consistent
lower bounded estimate for the evaluation function of the
cost associated with the optimal solution in the decision
tree. Some illustrative examples have also been given.
REFERENCES
[l] M. You and A. K. C. Wong, “An algorithm for graph optimal
isomorphism,” in hoc. 7th. Int. Conf. on Part. Recogn., Montreal,
Canada, 1984, pp. 316-319.
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[I 11 A. K. C. Wong and F. A. Akinniyi, “A net based algorithm for
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M. You and A. K. C. Wong, “On distance of attributed graphs
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-, “Random graphs and their distance measure,” Pmc. I982
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approach,” in NATO AS1 Series, Vol. F30. Pattern Recognition
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[18]
[19]
Theory and Applications, P. A. Devijver and J. Kittler, Eds. Berlin,
Heidelberg: Springer-Verlag, 1987, pp. 323-345.
A. K. C. Wong, J. Constant, and M. You, “Random graphs,”
Syntactic and Structural Pattern Recognition - Fundamentals, Advances,
and Applications, H. Bunke and A. Sanfeliu, Eds. New
Jersey: World Scientific, 1989.
J. Stroud, Airports of The World. London: Putnam & Company,
1980.
Andrew K. C. Wong received the Ph.D. degree
in 1968 from Carnegie-Mellon University.
He is currently a Professor in the Department
of Systems Design Engineering, and Director of
the PAM1 (Pattern Analysis and Machine Intelligence)
Lab at the University of Waterloo. From
1984 to 1987, he assumed the responsibility of
Research Director, in charge of the Robotic
Vision and Knowledge Base Project. Since 1987
he has been heading two projects under two
Ontario Centres of Excellence, namely, “Flexible
Automation” under the Manufacturing Research Corporation of
Ontario, and “Knowledge Acquisition, Synthesis and Organization”
under the Information Technology Research Centre of Ontario. Since
then, he has authored and co-authored chapters/sections in a good
number of engineering books and published many articles in scientific
journals and conference proceedings. He is currently an Associate
Editor of the Journal of Computers in Biology and Medicine. Over the
years, he has also served as Consultant to a number of major industries
including IBM, NCR, Northern Telecom, GM, Hughes, and Spar
Aerospace.
Manlai You was born in Suao, Ilan, Taiwan,
Republic of China, on July 3, 1949. He grdduated
from Mingchi Institute of Technology, Taiwan,
in 1970. He received the M.S. degree in
product design from Illinois Institute of Technology,
Chicago, in 1977, and the Ph.D degree
in systems design from the University of Waterloo,
Waterloo, ON, Canadd, in 1983.
From 1972 to 1975 he was a Product Design
Engineer at Manya Plastics Corporation, Taiwan.
From 1977 to 1980, he lectured on product
design at Mingchi Institute of Technolom. He was a Research Assistant
andPostdoctGrate Researcher at the Department of Systems Design,
University of Waterloo, in the periods from 1980 to 1983 and 1983 to
1984, respectively. He is currently an Associate Professor of Product
Design at Mingchi Institute of Technology, Taiwan. His fields of interest
include pattern recognition, ergonomics, computer-aided design, and
design methodology.
S. C. Chan received the B S degree in civil
engineering from University of Hong Kong, in
1973 and the M S. degree in management science
(with specialization in operational research)
from Imperial College, U K , in 1984
He is currently with the Pattern Analysis and
Machine Intelligence Laboratory of the Department
of Systems Design Engineering of the University
of Waterloo while working toward the
Ph.D degree His research interests include
graph mapping, pattern recognition, and analysis
of macromolecular sequences
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