Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure

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1 Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure SEA 17 June 23, 217 Tobias Heuer and Sebastian Schlag I NSTITUTE OF T HEORETICAL I NFORMATICS A LGORITHMICS G ROUP KIT University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association

2 Hypergraphs Generalization of graphs hyperedges connect 2 nodes Graphs dyadic (2-ary) relationships Hypergraphs (d-ary) relationships Hypergraph H = (V, E, c, ω) Vertex set V = {1,..., n} Edge set E P (V ) \ v11 e4 e3 v1 v13 v8 v9 v6 1 e1 v3 v1 Node weights c : V R 1 Edge weights ω : E R 1 v12 v2 v7 e5 v5 v4 e2

3 Hypergraphs Generalization of graphs hyperedges connect 2 nodes Graphs dyadic (2-ary) relationships Hypergraphs (d-ary) relationships pin net Hypergraph H = (V, E, c, ω) Vertex set V = {1,..., n} Edge set E P (V ) \ v11 P = 1 P e E e = P v V e4 e3 d(v ) e1 v3 v1 Node weights c : V R 1 Edge weights ω : E R 1 v12 v2 v1 v13 v8 v9 v6 v7 e5 v5 v4 e2

4 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V1,..., Vk } such that: blocks Vi are roughly equal-sized: c(vi ) (1 + ε) c(v ) k connectivity objective is minimized: V1 V4 2 V2 V3

5 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V1,..., Vk } such that: blocks Vi are roughly equal-sized: c(vi ) (1 + ε) imbalance parameter c(v ) k connectivity objective is minimized: V1 V4 2 V2 V3

6 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V1,..., Vk } such that: blocks Vi are roughly equal-sized: c(vi ) (1 + ε) imbalance parameter c(v ) k connectivity objective is minimized: V1 V4 2 V2 V3

7 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V1,..., Vk } such that: blocks Vi are roughly equal-sized: c(vi ) (1 + ε) imbalance parameter c(v ) k connectivity objective is minimized: P e cut (λ 1) V1 ω(e) = 6 V2 connectivity: # blocks connected by net e V4 2 V3

8 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V1,..., Vk } such that: blocks Vi are roughly equal-sized: c(vi ) (1 + ε) imbalance parameter c(v ) k connectivity objective is minimized: P e cut (λ 1) V1 ω(e) = 6 V2 connectivity: # blocks connected by net e V4 2 V3

9 Applications VLSI Design Scientific Computing Application Domain Hypergraph Model facilitate floorplanning & placement 3 Goal minimize communication

10 The Multilevel Framework input hypergraph output partition match / cluster local search uncontract contract initial partitioning 4

11 This Talk: Coarsening Phase input hypergraph output partition match / cluster local search uncontract contract initial partitioning 5

12 Clustering-based Coarsening Common Strategy: avoid global decisions local, greedy algorithms Objective: identify highly connected vertices using foreach vertex v do cluster[v ] := argmax rating(v,u) neighbor u 6

13 Clustering-based Coarsening Common Strategy: avoid global decisions local, greedy algorithms Objective: identify highly connected vertices using foreach vertex v do cluster[v ] := argmax rating(v,u) neighbor u Main Design Goals:[Karypis,Kumar 99] 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning 3: maintain structural similarity 6 good coarse solutions

14 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets 7 easier initial partitioning

15 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo functions: hypergraph-tailored rating P r (u, v ) := net e containing u,v 7 ω(e) e 1

16 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo functions: hypergraph-tailored rating P r (u, v ) := large number... 7 net e containing u,v ω(e) e 1

17 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo functions: hypergraph-tailored rating P r (u, v ) := large number... 7 net e containing u,v ω(e) e 1 of heavy nets...

18 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets easier local search 2: reduce number of nets easier initial partitioning foo functions: hypergraph-tailored rating P r (u, v ) := large number... 7 net e containing u,v ω(e) e 1 of heavy nets with small size

19 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets X X easier local search 2: reduce number of nets easier initial partitioning foo functions: hypergraph-tailored rating P r (u, v ) := large number... 7 net e containing u,v ω(e) e 1 of heavy nets with small size

20 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets X X easier local search 2: reduce number of nets easier initial partitioning foo functions: hypergraph-tailored rating P r (u, v ) := large number... net e containing u,v ω(e) e 1 of heavy nets with small size foo 3: maintain structural similarity good coarse solutions fooprefer clustering over matching ensure balanced vertex weights 7

21 Clustering-based Coarsening Main Design Goals: 1: reduce size of nets X X easier local search 2: reduce number of nets easier initial partitioning foo functions: hypergraph-tailored rating P r (u, v ) := large number... net e containing u,v ω(e) e 1 of heavy nets with small size ensure balanced vertex weights 7 en ou gh? foo 3: maintain structural similarity good coarse solutions fooprefer clustering over matching

22 What could possibly go wrong?... a lot: u n1 v input 8 {u, v } obscured clusters

23 What could possibly go wrong?... a lot: u n1 v {u, v } input obscured clusters maximal matching random tie-breaking ISSUES prefer unclustered 8 heavy neighbors

24 What could possibly go wrong?... a lot: u n1 v {u, v } input obscured clusters maximal matching random tie-breaking ISSUES foo prefer unclustered heavy neighbors Problem: relying only on local information! 8

25 Our Approach: Community-aware Coarsening u n1 v input 9 {u, v } obscured clusters

26 Our Approach: Community-aware Coarsening u n1 v {u, v } input obscured clusters SOLUTION community detection 9

27 Our Approach: Community-aware Coarsening u n1 v {u, v } input obscured clusters SOLUTION community detection Framework: preprocessing: determine community structure only allow intra-community contractions 9

28 Our Approach: Community-aware Coarsening u n1 v {u, v } input obscured clusters SOLUTION community detection Framework: preprocessing: determine community structure How? only allow intra-community contractions 9

29 Detecting Community Structure Goal: partition graph into natural groups C Community: internally dense, externally sparse subgraph 1

30 Detecting Community Structure Goal: partition graph into natural groups C Community: internally dense, externally sparse subgraph (One) Formalization: [Newman,Girvan 4] fraction of intra-cluster edges Modularity mod(g, C ) := cov(g, C ) E[cov(G, C )] Coverage cov(g, C ) := P C C 1 E(C) E

31 Detecting Community Structure Goal: partition graph into natural groups C Community: internally dense, externally sparse subgraph (One) Formalization: [Newman,Girvan 4] fraction of intra-cluster edges Modularity mod(g, C ) := cov(g, C ) E[cov(G, C )] Coverage cov(g, C ) := P C C E(C) E Efficient Heuristic: Louvain Method (multilevel, local, greedy) [Blondel et al. 8] repeatedly move nodes to neighbor communities coarsen graph & repeat 1

32 Graph Representations of Hypergraphs Hypergraph 11

33 Graph Representations of Hypergraphs Hypergraph Clique Graph natural sparsity: edges per net e -destroys -exaggerates importance of large nets e 2 11

34 Graph Representations of Hypergraphs Hypergraph Clique Graph natural sparsity: edges per net e -destroys -exaggerates importance of large nets e 2 Bipartite (Star) Graph + compact representation: O( P ) space - nets become nodes & part of clustering 11

35 Bipartite Graphs: Modeling Pecularities Density: d := 12 m n = P /n P /m = d(v ) e

36 Bipartite Graphs: Modeling Pecularities Density: d := m n P /n P /m = = d(v ) e d 1 V ' E Hypernodes (>-nodes) Hyperedges ( -nodes) d (v ) ' e 12

37 Bipartite Graphs: Modeling Pecularities Density: d := m n P /n P /m = = d(v ) e d 1 d 1 V ' E V E d (v ) ' e d (v ) e Hypernodes (>-nodes) Hyperedges ( -nodes) 12

38 Bipartite Graphs: Modeling Pecularities Density: d := m n P /n P /m = = d(v ) e d 1 d 1 d 1 V E V ' E V E d (v ) e d (v ) ' e d (v ) e Hypernodes (>-nodes) Hyperedges ( -nodes) 12

39 Bipartite Graphs: Modeling Pecularities d 1 d 1 d 1 V E V ' E V E d (v ) e d (v ) ' e d (v ) e Hypernodes (>-nodes) Hyperedges ( -nodes) foo addressed via edge weights: 13

40 Bipartite Graphs: Modeling Pecularities d 1 d 1 d 1 V E V ' E V E d (v ) e d (v ) ' e d (v ) e Hypernodes (>-nodes) Hyperedges ( -nodes) foo addressed via edge weights: ω (v, e) := 1 13 baseline

41 Bipartite Graphs: Modeling Pecularities d 1 d 1 d 1 V E V ' E V E d (v ) e d (v ) ' e d (v ) e Hypernodes (>-nodes) Hyperedges ( -nodes) foo addressed via edge weights: ω (v, e) := 1 ωe (v, e) := 13 1 e baseline small nets higher influence

42 Bipartite Graphs: Modeling Pecularities d 1 d 1 d 1 V E V ' E V E d (v ) e d (v ) ' e d (v ) e Hypernodes (>-nodes) Hyperedges ( -nodes) foo addressed via edge weights: ω (v, e) := 1 ωe (v, e) := ωde (v, e) := 13 baseline 1 e d(v ) e small nets + high degree higher influence higher influence

43 Experiments Benchmark Setup System: 1 core of 2 Intel Xeon 2.6 Ghz, 64 GB RAM Hypergraphs:1 Application Benchmark Set Representation Density Class Community Str. # Hypergraphs # in Subset VLSI ISPD98 DAC212 direct direct d 1 d 1 X X Sparse Matrix UF-SPM row-net d 1, d 1, d 1 some instances SAT Solving SAT14 literal primal dual d 1 d 1 d 1 X X X k {2, 4, 8, 16, 32, 64, 128} with imbalance: ε = 3% 8 hours time limit / instance Comparing KaHyPar-CA with: KaHyPar-K hmetis-r & hmetis-k PaToH-Default & PaToH-Quality

44 Initial Cut Improv. [%] Avg. Cut Improv. [%] Min Cut Improv. [%] Comparison of Edge Weighting Schemes d 1 2 d CA(ω) CA(ωe ) CA(ωde ) CA(ω) CA(ωe ) CA(ωde ) d 1 CA(ω) CA(ωe ) CA(ωde )

45 Initial Cut Improv. [%] Avg. Cut Improv. [%] Min Cut Improv. [%] Comparison of Edge Weighting Schemes d 1 2 d CA(ω) CA(ωe ) CA(ωde ) CA(ω) CA(ωe ) CA(ωde ) d 1 CA(ω) CA(ωe ) CA(ωde )

46 Initial Cut Improv. [%] Avg. Cut Improv. [%] Min Cut Improv. [%] Comparison of Edge Weighting Schemes d 1 ωde (v, e) := 2 d(v ) e d CA(ω) CA(ωe ) CA(ωde ) CA(ω) CA(ωe ) CA(ωde ) d 1 CA(ω) CA(ωe ) CA(ωde )

47 Initial Cut Improv. [%] Avg. Cut Improv. [%] Min Cut Improv. [%] Comparison of Edge Weighting Schemes d 1 ωde (v, e) := 2 d(v ) e d CA(ω) CA(ωe ) CA(ωde ) CA(ω) CA(ωe ) CA(ωde ) d 1 CA(ω) CA(ωe ) CA(ωde )

48 Initial Cut Improv. [%] Avg. Cut Improv. [%] Min Cut Improv. [%] Comparison of Edge Weighting Schemes d 1 2 d ωde (v, e) := d(v ) e ω (v, e) := CA(ω) CA(ωe ) CA(ωde ) d 1 CA(ω) CA(ωe ) CA(ωde ) CA(ω) CA(ωe ) CA(ωde )

49 Initial Cut Improv. [%] Avg. Cut Improv. [%] Min Cut Improv. [%] Comparison of Edge Weighting Schemes d 1 2 d ωde (v, e) := d(v ) e ω (v, e) := CA(ω) CA(ωe ) CA(ωde ) d 1 CA(ω) CA(ωe ) CA(ωde ) CA(ω) CA(ωe ) CA(ωde )

50 Initial Cut Improv. [%] Avg. Cut Improv. [%] Min Cut Improv. [%] Comparison of Edge Weighting Schemes d 1 2 d ωde (v, e) := d(v ) e ω (v, e) := CA(ω) CA(ωe ) CA(ωde ) d 1 CA(ω) CA(ωe ) CA(ωde ) ω (v, e) := 1 CA(ω) CA(ωe ) CA(ωde )

51 Experimental Results Partitioning Quality 1 Best Algorithm Example Algorithm 1 Algorithm 2 Algorithm # Instances 16

56 Experimental Results - Quality d 1: d 1: 17 d 1:

57 Experimental Results - Quality d 1: 18 d 1:

58 Experimental Results - Running Time Algorithm KaHyPar KaHyPar-CA hmetis-r hmetis-k PaToH-Q PaToH-D 19 All DAC212 ISPD Running Time [s] Primal Literal Dual SPM WebSocial

59 Experimental Results - Running Time Algorithm KaHyPar KaHyPar-CA hmetis-r hmetis-k PaToH-Q PaToH-D 19 All DAC212 ISPD Running Time [s] Primal Literal Dual SPM WebSocial

60 Conclusion & Discussion KaHyPar-CA - Community-aware Coarsening Community Detection via: modularity maximization Louvain Method (LM) bipartite, weighted graph Future Work: speedup preprocessing: parallel LM resolution limit multi-resolution modularity other formalizations: Infomap Surprise 2 KaHyPar-Framework Open-Source on Github:

61 References [Newman, Girvan 4] M. E. J. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical Review E, 69:26113, Feb 24. [Blondel et al. 8] V. D. Blondel, J. Guillaume, R. Lambiotte, and E. Lefebvre. Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 28(1):P18, 28. [Karypis, Kumar 99] G. Karypis and V. Kumar. Multilevel K-way Hypergraph Partitioning. In Proceedings of the 36th ACM/IEEE Design Automation Conference, pages ACM,

62 Taxonomy of Hypergraph Partitioning Tools Recursive Bisection Direct k-way 1998 MLPart PaToH Sparse Matrices Mondriaan hmetis-r VLSI Zoltan 1999 hmetis-k parallel Parkway kpatoh multi-constraint UMPa multi-objective KaHyPar-R n-level KaHyPar-K n-level