Advanced Coarsening Schemes for Graph Partitioning - Ilya ... - KIT

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Advanced Coarsening Schemes 

for Graph Partitioning 

Ilya Safro, Peter Sanders, Christian Schulz 

Argonne National Laboratory and Karlsruhe Institute of Technology (KIT) 

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4 

LEVEL 0 

1 KIT Ilya – Universität Safro, des Peter LandesSanders, Baden-Württemberg Christian und Schulz: 

Advanced Coarsening Schemes for Graph Partititoning 

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LEVEL 1 

Department of Informatics 

Institute for Theoretical Computer Science, Algorithmics II 

nationales Großforschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu

Overview 

Introduction 

Multilevel Algorithms 

Algebraic Distance 

Advanced Coarsening 

Experiments 

Summary 

2 Ilya Safro, Peter Sanders, Christian Schulz: 



Institute for Theoretical Computer Science, Algorithmics II

Motivation 

Social Networks 

SNWs are dense (Facebook > 900M, Google+ > 100M) 

too large for a single machine 

analysis required 





Motivation 


distribute graph over multiple machines 

analysis: edges between PEs ↔ communication 

vertices ↔ computation 





Motivation 


Graph Partitioning Problem: 

Partition a graph into (almost) equally sized blocks, such that the 

number of edges connecting vertices from different blocks is minimal. 





ɛ-Balanced Graph Partitioning 

Partition graph G = (V , E, c : V → R>0, ω : E → R>0) 

into k disjoint blocks s.t. 

1 + ɛ 

total node weight of each block ≤ total node weight 

k 

total weight of cut edges as small as possible 

Applications: 

FEM, linear equation systems, VLSI design, route planning, . . . 





Multi-Level Graph Partitioning 

Successful in existing systems: 

Metis, Scotch, Jostle,. . . , KaPPa, KaSPar, KaFFPa, KaFFPaE 





This Talk 

Advanced Coarsening Schemes 

algebraic multigrid 

contract 



input 

graph 

... 

algebraic distance 




Relaxation Process 

Goal: distinguish local and non-local edges 

∀ i ∈ V : x i := rand() 

Do k times 

∀ i ∈ V : x k 

i := (1 − α)x k−1 

i 



+ α ∑j ωijx k−1 

j / ∑ij ωij Conjecture 

If |x k 

i − xk 

j | > |xk u − xk v | then the local connectivity between u and v is 

stronger than that between i and j. 


Institute for Theoretical Computer Science, Algorithmics II 

40 

20


Example 



40 

goal: distinguish local and non-local edges 

20 




Example - random initialization 

assign random values 

relaxation steps to equalize 






Example - 10 relaxation steps 

relaxation steps to equalize 

local differences equalize faster 





Stationary iterative relaxation 

rewrite process as x (k+1) = H JORx (k) , where 



H JOR = (1 − α) I + αD −1 W 

⇒ basically the JOR method to solve Lx = 0 

L = D − W graph laplacian, α relaxation paramter 

algebraic distance ↔ JOR smoothing property 







H JOR = (1 − α) I + αD −1 W 


L = D − W graph laplacian, α relaxation paramter 

H JOR lazy random walk matrix 







H JOR = (1 − α) I + αD −1 W 


Definition 

Extended 2-normed algebraic distance between i and j after k 

iterations x (k+1) = Hx (k) on R random initializations 

if {i, j} ∈ E, otherwise ρ (k) 

ij 

ρ (k) 

 

R 

ij := ∑ |x 

r=1 

(k,r) 

i 

= 0 

− x (k,r) 

j | 2 

1/2 Department of Informatics 


Weighted Aggregation 

inspired by AMG 

each node may be divided into fractions 

different fractions form a coarse node 

1. how to select seed nodes 

2. how is classical likelihood expressed 

3. modification for graph partitioning 



contract 

input 

graph 

... 



Coarse Nodes 

C-points Selection 

find dominating set C ⊂ V s.t. F = V \ C “strongly coupled” to C 

each node in C is a seed of a coarse aggregate 

SEEDS 





Coarse Nodes 

C-points Selection 

find dominating set C ⊂ V s.t. F = V \ C “strongly coupled” to C 

each node in C is a seed of a coarse aggregate 

C = ∅, F = V 

until 

∀u ∈ F : ∑v∈C 1/ρu,v ≥ θ ∑v∈N(u) 1/ρu,v 

transfer node from F to C 

θ ∈ (0, 1), usually θ = 0.5 



SEEDS 



Coarse Nodes 

classical AMG interpolation matrix P 

express likelihood of v belonging to c ∈ C → P ∈ R |V |x|C| 

Pv,c = 

⎧ 

⎪⎨ 

⎪⎩ 



 

Belong to several aggregates 

1/ρv,c 

∑ 

k∈N(v)∩C 

1/ρ v,k 

v ∈ F , c ∈ N(v) ∩ C 

1 v ∈ C, v = c 

0 otherwise 



Coarse Nodes 

classical AMG interpolation matrix P 





Lc = P t L f P 

volume of c(q ∈ C) := ∑v c(v)Pv,q (sum of weights constant) 

wc 1,c 2 = ∑k=l P k,c1 w k,lP l,c2 



Coarse Nodes 

modified AMG interpolation matrix P 


previous (matching based) work: 




vertices should not become too heavy 

coarse levels should stay sparse 

→ modified algorithm to compute likelihood P 



Modified Algorithm 

Likelihood for v ∈ V 

restrict to at most two strongest connections → sparsity 

avoid overloaded vertices 







if v ∈ C then Pv,v = 1 










else 

find max pair (1/ρe 1 + 

1/ρe 2 ) 

s.t. coarse vertices not 

overloaded when 

splitting v 










else 

... 

if no such pair then 

find max edge (1/ρe 1 ) 

s.t. coarse vertices not 

overloaded when 

aggregating v 










else 

... 

... 

if no such edge then 

move v to C 









N c v ← selected C-nodes 

Pv,c ← 

1/ρvc 

∑ k∈N c v 1/ρ vk , c ∈ N c v 







Experiments 

Algorithm Configurations 

ECO KaFFPaEco, a good trade-off of quality and runtime 

ECO-ALG matching based coarsening, refinement as in ECO, rating 

function ex alg(e) := expansion ∗2 (e)/ρe 

STRONG matching based coarsening, strong refinement scheme 

AMG-ECO AMG coarsening based, refinement as in ECO 

AMG AMG coarsening based, refinement as in STRONG 

expansion ∗2 ({u, v}) := ω({u,v})2 

c(u)c(v) 





Experimental Results 

Social Networks - Algebraic Distance Helps! 



ECO 

ECO-ALG 

ECO 

ECO-ALG 

k quality full time 

2 1.38 0.77 

4 1.24 1.32 

8 1.15 1.29 

16 1.09 1.27 

32 1.06 1.18 

64 1.06 1.13 




Social Networks - AMG helps! 

ECO-ALG 

AMG-ECO 



ECO-ALG 

AMG-ECO 

k quality uncoar. time 

2 1.16 3.62 

4 1.11 2.14 

8 1.07 1.94 

16 1.06 1.69 

32 1.00 1.60 

64 1.01 2.99 




Potentially Hard Graphs 

potentially hard mixtures (star-like structures) 

mixtures of snw, fem graphs, VLSI, ... 

S 0, . . . , St connected center S 0 by ≤ 3% random edges 

ECO 

ECO-ALG 



ECO 

ECO-ALG 

k quality full time 

2 1.42 0.51 

4 1.15 0.88 

8 1.12 1.08 




Potentially Hard Graphs 

potentially hard mixtures (star-like structures) 

mixtures of snw, fem graphs, VLSI, ... 

S 0, . . . , St connected center S 0 by ≤ 3% random edges 

ECO-ALG 

AMG-ECO 

ECO-ALG 

AMG-ECO 



STRONG 

AMG 

STRONG 

AMG 

k quality uncoar. time quality uncoar. time 

2 1.18 0.55 1.15 2.11 

4 1.23 0.64 1.13 1.69 

8 1.08 0.98 1.05 1.37 



Conclusion 

Experiments 

algebraic distance and AMG help on irregular instances 

on regular instances (FEM, ...) no significant difference 

Further Material in the Paper 

more details 

more experiments 

Future Work 

apply AMG and algebraic distance to other multilevel algorithms 

implementation within KaFFPa 

Lunch 







Thank you! 

Contact: christian.schulz@kit.edu 

sanders@kit.edu 

safro@mcs.anl.gov

Advanced Coarsening Schemes for Graph Partitioning - Ilya ... - KIT

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