k-way Hypergraph Partitioning via n-level Recursive Bisection

Transcription

1 k-way Hypergraph Partitioning via n-level Recursive Bisection Sebastian Schlag, Vitali Henne, Tobias Heuer, Henning Meyerhenke Peter Sanders, Christian Schulz January 10th, ALENEX 16 INSTITUTE OF THEORETICAL INFORMATICS ALGORITHMICS GROUP 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 ) \ Node weights c : V R 1 e 1 e 2 v 1 v 2 v 3 e 3 Edge weights ω : E R 1 v 6 e 4 v 5 v 6 v 7 e 5 v 4 v 7 1

3 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 ) \ Node weights c : V R 1 pin e 1 e 2 v 1 v 2 v 3 e 3 Edge weights ω : E R 1 v 6 e 4 v 5 v 6 v 7 e 5 v 4 v 7 1

4 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V 1,..., V k } such that: blocks V i are roughly equal-sized: c(v i ) (1 + ε) c(v ) total weight of cut hyperedges is minimized k V 1 V 2 v 7 v 6 2

5 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V 1,..., V k } such that: blocks V i are roughly equal-sized: c(v i ) (1 + ε) c(v ) total weight of cut hyperedges is minimized k imbalance parameter V 1 V 2 v 7 v 6 2

6 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V 1,..., V k } such that: blocks V i are roughly equal-sized: c(v i ) (1 + ε) c(v ) total weight of cut hyperedges is minimized k imbalance parameter V 1 V 2 v 7 v 6 2

7 Hypergraph Partitioning Problem Partition hypergraph H = (V, E, c, ω) into k disjoint blocks Π = {V 1,..., V k } such that: blocks V i are roughly equal-sized: c(v i ) (1 + ε) c(v ) total weight of cut hyperedges is minimized k imbalance parameter hyperedge connecting multiple blocks V 1 V 2 v 7 v 6 2

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

9 Multilevel Paradigm input hypergraph Coarsening match / contract cluster 4

10 Multilevel Paradigm input hypergraph Coarsening match / contract cluster initial partitioning 4

11 Multilevel Paradigm input hypergraph output partition Coarsening match / contract cluster local search uncontract Uncoarsening 4 initial partitioning

12 Taxonomy of Hypergraph Partitioning Tools Recursive Bisection MLPart Direct k-way 1998 PaToH hmetis-r VLSI hmetis-k 1999 Sparse Matrices Mondriaan 2005 Zoltan parallel 2006 Parkway 2008 UMPa multi-objective

13 Taxonomy of Hypergraph Partitioning Tools Recursive Bisection MLPart Direct k-way 1998 PaToH hmetis-r VLSI hmetis-k 1999 Sparse Matrices Mondriaan 2005 Zoltan parallel 2006 Parkway 2008 UMPa multi-objective 2013 KaHyPar n-level

14 Why Yet Another Multilevel Algorithm? input hypergraph output partition match / cluster local search contract uncontract 6 initial partitioning

15 Why Yet Another Multilevel Algorithm? match / cluster Tradeoff: input hypergraph # levels : + quality running time output partition local search contract uncontract 6 initial partitioning

16 Why Yet Another Multilevel Algorithm? input hypergraph # levels : + quality running time match / cluster Tradeoff: output partition Our Contribution: evade tradeoff n levels local search contract combine high quality with good performance uncontract initial partitioning 6

17 Why Yet Another Multilevel Algorithm? input hypergraph # levels : + quality running time match / cluster Tradeoff: output partition Our Contribution: evade tradeoff n levels local search contract combine high quality with good performance uncontract Motivation: KaSPar n-level graph partitioning initial partitioning 6

18 Coarsening 7

19 n-level Coarsening Phase contract only a single pair of vertices at each level 8

20 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair 8

21 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair r(u, v) := 1 c(v) c(u) hyperedge e containing u,v ω(e) e 1 8

22 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair r(u, v) := 1 c(v) c(u) hyperedge e containing u,v ω(e) e 1 large number... 8

23 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair r(u, v) := 1 c(v) c(u) hyperedge e containing u,v of heavy hyperedges... ω(e) e 1 large number... 8

24 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair r(u, v) := 1 c(v) c(u) large number... hyperedge e containing u,v of heavy hyperedges... ω(e) e 1... with small size 8

25 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair prefer light hypernodes r(u, v) := 1 c(v) c(u) large number... hyperedge e containing u,v of heavy hyperedges... ω(e) e 1... with small size 8

26 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair repeat until: t hypernodes remain no valid pair remains (size constraint on hypernodes) 8

27 n-level Coarsening Phase contract only a single pair of vertices at each level How to determine that pair? compute rating r for all pairs of adjacent hypernodes choose pair (u, v) with highest rating (priority queue) update ratings for neighbors of contracted pair repeat until: t hypernodes remain no valid pair remains (size constraint on hypernodes) update can be expensive! 8

28 n-level Coarsening Phase Problem: # neighbors potentially large high-degree hypernodes large hyperedges,,, update all pins of all hyperedges incident to contracted pair 9

29 n-level Coarsening Phase Problem: # neighbors potentially large high-degree hypernodes large hyperedges,,, update all pins of all hyperedges incident to contracted pair Solution: lazy updates invalidate neighboring hypernodes re-calculate rating on demand 9

30 Initial Partitioning 10

31 Initial Partitioning not affected by n-level paradigm use portfolio of algorithms diversification random partitioning breadth-first search greedy hypergraph growing size-constrained label propagation try all algorithms multiple times select partition with best cut & lowest imbalance as initial partition initial partition 11

32 Local Search 12

33 Localized Local Search Idea traditional multilevel algorithms uncontract one level local search around complete border n-level localized local search [KaSPar] uncontract a single pair of nodes local search around 2 nodes fine-grained optimization limit search to constant # of moves per level otherwise V 2 local search steps in total stop pass after x fruitless moves 13

34 Localized FM Local Search Outline hypernodes unmarked, active, marked start around uncontracted vertex pair 14

35 Localized FM Local Search Outline hypernodes unmarked, active, marked start around uncontracted vertex pair compute gain for{ move to other block: +ω(e) if # pins in source = 1 g(v) = ω(e) if # pins in target = 0 hyperedge e containing v border hypernodes become active 14

36 Localized FM Local Search Outline hypernodes unmarked, active, marked start around uncontracted vertex pair compute gain for{ move to other block: +ω(e) if # pins in source = 1 g(v) = ω(e) if # pins in target = 0 hyperedge e containing v border hypernodes become active move highest-gain node to opposite block node becomes marked 14

37 Localized FM Local Search Outline hypernodes unmarked, active, marked start around uncontracted vertex pair compute gain for{ move to other block: +ω(e) if # pins in source = 1 g(v) = ω(e) if # pins in target = 0 hyperedge e containing v border hypernodes become active move highest-gain node to opposite block node becomes marked unmarked neighbors active (if border node) active neighbors update gain 14

38 Localized FM Local Search Outline hypernodes unmarked, active, marked start around uncontracted vertex pair compute gain for{ move to other block: +ω(e) if # pins in source = 1 g(v) = ω(e) if # pins in target = 0 hyperedge e containing v border hypernodes become active move highest-gain node to opposite block node becomes marked unmarked neighbors active (if border node) active neighbors update gain update & activation can be expensive! 14

39 Localized FM Local Search Engineering Problem: # neighbors potentially large high-degree hypernodes large hyperedges,,, large number of activations & updates on each level 15

40 Localized FM Local Search Engineering Problem: # neighbors potentially large high-degree hypernodes large hyperedges,,, large number of activations & updates on each level Known solutions for updates: perform δ-gain updates [Papa, Markov] exclude locked hyperedges from gain update [Krishnamurthy] will remain cut 15

41 Localized FM Local Search Engineering Problem: # neighbors potentially large high-degree hypernodes large hyperedges,,, large number of activations & updates on each level Known solutions for updates: perform δ-gain updates [Papa, Markov] exclude locked hyperedges from gain update [Krishnamurthy] will remain cut New solution for activations: cache gain values compute gain g(v) at most once along the n-level hierarchy 15

42 Experiments Benchmark Setup System: 1 core of 2 Intel Xeon 2.6 Ghz, 64 GB RAM # Hypergraphs: [publicly available] -UF Sparse Matrix Collection 192 -SAT Competition 2014 Application Track 100 -ISPD98 VLSI Circuit Benchmark Suite 18 k {2, 4, 8, 16, 32, 64, 128} imbalance: ε = 3% 250 min time limit 2170 instances Comparison with: hmetis-r & hmetis-k PaToH-Default & PaToH-Quality 16

43 Experimental Results Partitioning Quality 1 Algorithm Best Algorithm 1 Algorithm 2 Algorithm 3 Example # Instances

44 Experimental Results Partitioning Quality 1 Algorithm Best Algorithm 1 Algorithm 2 Algorithm 3 Example # Instances

45 Experimental Results Partitioning Quality 1 Algorithm Best Algorithm 1 Algorithm 2 Algorithm 3 Example # Instances

46 Experimental Results Partitioning Quality 1 Algorithm Best Algorithm 1 Algorithm 2 Algorithm 3 Example # Instances

47 Experimental Results Partitioning Quality 1 Algorithm Best Algorithm 1 Algorithm 2 Algorithm 3 Example # Instances

48 Experimental Results Partitioning Quality All Instances Algorithm Best Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q # Instances

49 Experimental Results Partitioning Quality 1 Algorithm Best Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q Sparse Matrices # Instances

50 Experimental Results Partitioning Quality 0.20 ISPD98 VLSI 1 Algorithm Best Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q # Instances

51 Experimental Results Smaller Imbalance 1 Algorithm Best Subset of all Instances ( ε = 1%) Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q # Instances

52 Experimental Results Larger Imbalance 1 Algorithm Best Subset of all Instances ( ε = 10%) Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q # Instances

53 Experimental Results Running Time All Instances (ε = 3%) T (Best) T (Algorithm) Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q # Instances

54 Future Work improve running time: ignore large hyperedges [PaToH] stop local search if improvement becomes unlikely [KaSPar] aaa improve quality: introduce V-cycles evolutionary algorithm [KaHIP] aaa improve balancing: optimize locally - rebalance globally 24

55 Conclusion & Discussion evade running time / quality tradeoff of multilevel algorithms n-level hierarchy engineered coarsening phase portfolio-based approach to initial partitioning highly tuned local search algorithm All Instances Algorithm Best Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q # Instances

56 Coffee Break! 26

57 Benchmark Set Details 10 7 Hypernodes 10 7 Hyperedges 10 8 Pins SPM SAT ISPD SPM SAT ISPD SPM SAT ISPD 27

58 Benchmark Results Partitioning Quality 1-(Best/Algorithm) Algorithm hmetis-k hmetis-r KaHyPar PaToH-D PaToH-Q SAT # Instances