Engineering Multilevel Graph Partitioning Algorithms

Transcription

1 Engineering Multilevel Graph Partitioning Algorithms Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II 1 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II KIT die Kooperation vonengineering Forschungszentrum Multilevel Graph Karlsruhe Partitioning GmbH Algorithms und Universität Karlsruhe (TH)

2 Overview Introduction Parallel Graph Partitioning (KaPPa) n-level Graph Partitioning (KaSPar) Experimental Evaluation Future Work 2 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

3 Simulation - FEM Simulation space is discretized into a mesh Solution of partitial differential equations are approximated by linear equations Number of vertices can become quite large time and memory Parallel processing required 3 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

4 The Common Parallel Approach Mesh partitioned via dual graph 1. Each volume (data, calculation) is represented by a vertex 2. Interdependencies are represented by edges All PE s get same amount of work Communication is expensive Graph Partitioning Problem: Partition a graph into (almost) equally sized blocks, such that the number of edges connecting vertices from different blocks is minimal. 4 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

5 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: sparse matrix reordering, VLSI design, graph compression,... 5 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

6 Multi-Level Graph Partitioning Successful in existing systems: DiBaP, Chaco, Jostle, Metis, Scotch,... 6 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

7 Multi-Level Graph Partitioning Edge Contraction w w u, v x ω({x, z}) = ω({u, z}) + ω({v, z}). c(x) = c(u) + c(v) u v x 7 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

8 Why Yet Another MGP Algorithms? Karlsruhe Parallel Partitioner Karlsruhe Sequential Partitioner scalable parallel algorithm for the case k =# of processors large inputs high quality (better (even parallel) than other seq. algorithms) understand / engineer multi level method bridge gaps theory practice 8 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

9 Matching Selection Goals: 1. large edge weights sparsify 2. large #edges few levels 3. uniform node weights represent input 4. small node degrees represent input unclear objective gap to approx. weighted matching which only considers 1.,2. match contract input graph... Our Solution: Apply approx. weighted matching to general edge rating function 9 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

10 Edge Ratings Tried ω({u, v}) ω({u, v}) expansion({u, v}) := c(u) + c(v) expansion ω({u, v}) ({u, v}) := c(u)c(v) expansion 2 ω({u, v})2 ({u, v}) := c(u)c(v) ω({u, v}) innerouter({u, v}) := Out(v) + Out(u) 2ω(u, v) where c = node weight, ω =edge weight, Out(u) := {u,v} E ω({u, v}) match contract input graph Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

11 Approx. Weighted Matching Use parallel prepartitioner (optional) currently geometric recursive bipartitioning if coords. available Global Path Algorithm [MaueSanders 2007] locally parallel [ManneBisseling 2007] for border nodes 11 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

12 Initial Partitioning Every PE can perform initial partitioning using different seeds Compared PARTY, PMETIS, and SCOTCH SCOTCH yielded best results input graph output partition local improvement match contract initial uncontract partitioning 12 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

13 Local Improvement Basic Approach Use linear time local search [Fiduccia Mattheyses 82] on multiple pairs of blocks in parallel input graph output partition... local improvement... match contract initial uncontract partitioning 13 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

14 Finding Pairs of Blocks Color edges of quotient graph Q several parallel algorithms tried Each color matching in Q independent block pairs 14 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

15 Pairwise Local Search exchange boundaries two local searches adopt best New queue selection strategy topgain 15 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

16 Karlsruhe Sequential Partitioner 1. Contraction contract a single edge between two levels possibly n levels finegrained contraction consequtive levels are very similar no matching algorithm required use edge rating expansion uniform distribution of node weights addressable priority queue: defines order of edges to be contracted dynamic graph data structure 2. Local Search efficent stopping criterion avoid quadratic runtime 3. General Trial Trees improve quality by independent trials 16 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

17 Local Search Nodes unmarked, active, marked active nodes u compute gains of moving from one block to another choose the maximum gain when moved active marked can t become active anymore unmarked neighbours of marked active v u v 17 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

18 Local Search - Stopping Criteria touching each node on each level leads to Ω(n 2 ) runtime need flexible stopping criteria gains in each step identically distributed, independent random variables expectation µ variance σ 2 compute µ and σ 2 from previous steps stop after p steps if pµ 2 > ασ 2 + β u v u v 18 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

19 Results Example Street network Europe V = 18M, E = 44M, k = 16, KaSPar ParMetis 19 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

20 Scalablity total time [s] KaPPa Strong KaPPa Fast KaPPa Minimal scotch kmetis parmetis k Street network Europe ( V = 18M, E = 44M) 20 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

21 Quality Comparison with Other Systems Geometric mean, imbalance ɛ = 0.03: 11 graphs (78K 18M nodes) k {2, 4, 8, 16, 64} algorithm large graphs best avg. t[s] KaFFPa strong KaSPar strong % KaPPa strong % Scotch % 3.83 kmetis % 0.71 Repeating scotch as long as KasPar strong run and choosing the best result 12.1% larger cuts Walshaw instances, road networks, Florida Sparse Matrix Collection, random Delaunay triangulations, random geometric graphs, social networks. 21 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

22 Sources of Quality Improvement 1. edge ratings largest effect by far 2. more localized refinement. Surprise! previous parallelizations caused worse quality 3. better queue selection 4. better matching algorithms 5. more time invested 22 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

23 Future Work KaFFPa 1. try other pairwise improvers (MaxFlow-MinCut) 2. try F-Cycles graph partitioning 3. better initial partitioning for large k 4. integration into Meta-heuristic 5. exploit shared memory parallelism δ l G s lb rb t b 1 b B 2 δr input graph match local improvement match... local improvement... contract initial uncontract contract uncontract partitioning 23 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

24 10th DIMACS Implementation Challenge Announcement Graph Partitioning and Clustering Organizers: 1. David Bader Georgia Institute of Technology 2. Peter Sanders and Dorothea Wagner, KIT 3. Web: 1st phase: call for instances and applications deadline soon 2nd phase: paper submission 3rd phase: challenge workshop 4th phase: proceedings Deadline for expressing interest in participation: 31. March 24 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II

25 Thank you! Contact: 25 Mar. 3, 2011 Manuel Holtgrewe, Vitaly Osipov, Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II