Parallel Graph Partitioning for Complex Networks

Transcription

1 Parallel Graph Partitioning for Complex Networks Henning Meyerhenke, Peter Sanders, Christian Schulz High-performance Graph Algorithms and Applications in Computational Science Dagstuhl 1 Christian Schulz: Institute for Theoretical Computer Science

2 ɛ-balanced Graph Partitioning Partition graph G = (V, E, c : V R >0, ω : E R >0 ) into k disjoint blocks s.t. total node weight of each block 1 + ɛ total node weight k total weight of cut edges as small as possible Applications: FEM, linear equation systems, VLSI design, route planning,... 2 Christian Schulz:

3 Multilevel Graph Partitioning contraction phase contract input graph local improvement initial partitioning uncontract output partition uncoarsening phase Successful in many systems (using matchings or AMG for coarsening) 3 Christian Schulz:

4 KaHIP Karlsruhe High Quality Partitioning buffoon [ALENEX12] social [SEA14] separators distr. evol. alg. [ALENEX12] [DIMACS12] highly balanced: [SEA13] A 0 B A 0 B A C C C B V F W [ESA11] cycles a la multigrid input graph Output Partition [IPDPS10]... flows etc. [ESA11]... edge local improvement ratings match + [SEA12/14] contract uncontract initial partitioning parallel [IPDPS10] Multilevel Graph Partitioning 4 Christian Schulz:

5 Matching-based Coarsening A a B b a+b A+B 5 Christian Schulz:

6 Matching-based Coarsening A B a b a+b A+B 5 Christian Schulz:

7 Matching-based Coarsening Problem bad for networks that are highly irregular substantial reduction is hard using matchings may contract wrong edges! 6 Christian Schulz:

8 Matching-based Coarsening Problem bad for networks that are highly irregular substantial reduction is hard using matchings may contract wrong edges! 6 Christian Schulz:

9 Basic Idea aggressive contraction / simple and fast local search main idea: contract clusterings clustering paradigm: internally dense and externally sparse 7 Christian Schulz:

10 Basic Idea Contraction of Clusterings a b c A C B a+b+c A+B+C contraction: respect balance and cut avoid large blocks: size constraint U construct coarse graph using hashing recurse until graph is small 8 Christian Schulz:

11 Basic Idea Contraction of Clusterings a b c A C B a+b+c A+B+C contraction: respect balance and cut avoid large blocks: size constraint U construct coarse graph using hashing recurse until graph is small 8 Christian Schulz:

12 Label Propagation Cut-based, Linear Time Clustering Algorithm [Raghavan et. al] cut-based clustering using size-constraint label propagation start with singletons traverse nodes in random order or smallest degree first move node to cluster having strongest eligible connection modification eligible: w.r.t size constraint U Scan 9 Christian Schulz:

13 Label Propagation Iteration Cut [%] Christian Schulz:

14 Label Propagation Iteration Cut [%] Christian Schulz:

15 Label Propagation Iteration Cut [%] Christian Schulz:

16 Label Propagation Iteration Cut [%] Christian Schulz:

17 Label Propagation Iteration Cut [%] Christian Schulz:

18 Label Propagation Iteration Cut [%] Christian Schulz:

19 Label Propagation Iteration Cut [%] Christian Schulz:

20 Label Propagation Iteration Cut [%] Christian Schulz:

21 Label Propagation Iteration Cut [%] Christian Schulz:

22 Label Propagation Iteration Cut [%] Christian Schulz:

23 Label Propagation Simple Local Search Greedy Local Search: start with partition from coarser level traverse nodes in random order move node to cluster having strongest eligible connection eligible: w.r.t size constraint U := (1 + ɛ) V k Scan Augmentations of the partitioning algorithm paper 11 Christian Schulz:

24 Parallelization 12 Christian Schulz:

25 Graph Distribution over PEs Graph Distribution: a PE receives n/p vertices and their edges Processor I Processor II Communication ghost nodes: adjacent nodes on other processor (communication!) interface nodes: nodes adjacent to ghost nodes 13 Christian Schulz:

26 Label Propagation Distributed Memory each PE has a static part of the graph, only block IDs can change Overlap Computation and Communication (PE centric view): V Phase i 1 Phase i Phase i+1 Scan At the end of phase i: send block ID updates of phase i to neighboring PEs receive block ID updates from neighboring PEs from phase i 1 * while scanning in phase i, messages are routed through the network * algorithm converged nothing will be communicated + algorithms to maintain balance of clusters/blocks 14 Christian Schulz:

27 Parallel Augmentations input graph local improvement... local improvement... contract initial uncontract contract uncontract partitioning V-cycles: don t contract cut edges modify label propagation to respect partition Node Ordering for LP: perform smallest degree first only locally 15 Christian Schulz:

28 Experiments 16 Christian Schulz:

29 Instances graph n m graph n m Large Graphs p2p-gnutella citationciteseer M wordassociation coauthorsdblp PGPgiantcompo cnr M -euall web-google M as-22july copapersciteseer M soc-slashdot copapersdblp M loc-brightkite as-skitter M enron amazon M loc-gowalla eu M coauthorsciteseer in M 13.6M wiki-talk M Huge Graphs uk M 262M sk M 1.8G arabic M 553M uk M 3.3G ɛ = 3%, k {2, 4, 8, 16, 32, 64}, geom. mean of averages, 10 seeds 17 Christian Schulz:

30 Experimental Results Selected Algorithm impr. cut t [s] UFast 51.7% 1.5 UFastV 54.7% 3.0 UEcoV/B 60.9% 11.5 UStrong 75.1% KaFFPaEco 22.2% 36.2 KaFFPaStrong 66.2% kmetis 45.8% 0.4 hmetis 60.5% Scotch baseline 10.6 UStrong best quality, running time KaFFPaStrong best cut of hmetis 6% worse than avg. cut of UStrong UEcoV/B outperforms hmetis, order of magnitute less time UFast better quality but slower 18 Christian Schulz:

31 Experimental Results Selected Algorithm impr. cut t [s] UFast 51.7% 1.5 UFastV 54.7% 3.0 UEcoV/B 60.9% 11.5 UStrong 75.1% KaFFPaEco 22.2% 36.2 KaFFPaStrong 66.2% kmetis 45.8% 0.4 hmetis 60.5% Scotch baseline 10.6 UStrong best quality, running time KaFFPaStrong best cut of hmetis 6% worse than avg. cut of UStrong UEcoV/B outperforms hmetis, order of magnitute less time UFast better quality but slower 18 Christian Schulz:

32 Experimental Results Selected Algorithm impr. cut t [s] UFast 51.7% 1.5 UFastV 54.7% 3.0 UEcoV/B 60.9% 11.5 UStrong 75.1% KaFFPaEco 22.2% 36.2 KaFFPaStrong 66.2% kmetis 45.8% 0.4 hmetis 60.5% Scotch baseline 10.6 UStrong best quality, running time KaFFPaStrong best cut of hmetis 6% worse than avg. cut of UStrong UEcoV/B outperforms hmetis, order of magnitute less time UFast better quality but slower 18 Christian Schulz:

33 Parallel Solution Quality Performance k = 2 blocks, 32PEs instances: meshes and social networks/web graphs ParMetis has ineffective coarsening (matching-based) due to memory consumption of coarsest graph (dist. among PEs) could not solve arabic-2005, sk-2005 and uk-2007 solved instances (ParMetis): fast and eco yield 19.2% and 27.4% improvement fast and eco slower on average social networks/web graphs: fast: 38% less cut edges and > 2 faster eco: 45% less cut edges and slower best instance: 18 faster and 61.6% less cut edges impr. quality over FB or matching-based algorithms (other inst.) 19 Christian Schulz:

34 Strong Scaling Social Networks 1000 Fast sk-2007 Fast arabic-2005 Fast uk-2002 Fast uk-2007 Minimal uk-2007 total time [s] K 2K number of PEs p uk-2007 can be partitioned in 15.2 seconds (seq. 10.5min) more scaling results in the paper 20 Christian Schulz:

35 Conclusion cluster contraction improves running time and solution quality efficient parallelization dominates ParMetis on social networks (semi-)external variant of the algorithm much more results in the papers 21 Christian Schulz:

36 References Partitioning Complex Networks via Size-constrained Clustering H. Meyerhenke, P. Sanders and C. Schulz ArXiv: Parallel Graph Partitioning for Complex Networks H. Meyerhenke, P. Sanders and C. Schulz ArXiv: (Semi-)External Algorithms for Graph Partitioning and Clustering Y. Akhremtsev, P. Sanders and C. Schulz ArXiv: Recent Advances in Graph Partitioning A. Buluc, H. Meyerhenke, I. Safro, P. Sanders and C. Schulz ArXiv: Christian Schulz:

37 Open Problems Mostly from Recent Advances in GP large values of k: multilevel looses attractiveness multi-k-recursive parallel partitioning w.r.t. structure of cluster other metrics: max. comm volume, max. quotient graph degree, pareto objectives real performance benchmarks fixed k questionable if application tolerates node failures repartitioning with changed k mapping blocks onto communication networks using GP for graph tools such as Pregel exact ML algorithms for node separators using GP has a hammer for optimization problems, e.g. independent set graph coloring max clique 23 Christian Schulz:

38 Thank you! KaHIP: on Twitter: algo2.iti.kit.edu/documents/kahip twitter.com/projectkahip 24 Christian Schulz: