Engineering Multilevel Graph Partitioning Algorithms
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1 Engineering Multilevel Graph Partitioning Algorithms Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II 1 Nov. 10, 2011 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 Multilevel Algorithms Advanced Techniques Evolutionary Graph Partitioning Summary 2 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
3 Simulation - FEM Simulation space is discretized into a mesh Solution of partial differential equations are approximated by linear equations Number of vertices can become quite large time and memory Parallel processing required 3 Nov. 10, 2011 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 Nov. 10, 2011 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: linear equation systems, VLSI design, route planning,... 5 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
6 Multi-Level Graph Partitioning Successful in existing systems: Metis, Scotch, Jostle,..., KaPPa, KaSPar, KaFFPa, KaFFPaE 6 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
7 Why Yet Another MGP Algorithms? 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 embedding into global search strategies 7 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
8 Graph Partitioning 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 8 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
9 Graph Partitioning Edge Ratings 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}) 9 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
10 Global Paths Algorithm Approx. Weighted Matching [Maue Sanders 2007] Sort edges according to their weight/rating Grow a set of paths and even-length cycles Find optimum matching for every path and cycle Running time O(m + sort(m)) Approximation 1 2 OPT outperforms HEM, SHEM 10 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
11 Graph Partitioning Initial Partitioning KaPPa Compared PARTY, PMETIS, and SCOTCH SCOTCH yielded best results input graph output partition local improvement match contract initial uncontract partitioning 11 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
12 Graph Partitioning Initial Partitioning - One Possibility if we know how to bipartition, we can compute a k-way partition BIPART(G): 1. search for pseudo-peripheral nodes 2. perform two-sided BFS 3. post improvement using local search repeat (few times) and take best 12 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
13 Graph Partitioning Initial Partitioning - One Possibility if we know how to bipartition, we can compute a k-way partition BIPART(G): 1. search for pseudo-peripheral nodes 2. perform two-sided BFS 3. post improvement using local search repeat (few times) and take best 13 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
14 Graph Partitioning Initial Partitioning - One Possibility if we know how to bipartition, we can compute a k-way partition BIPART(G): 1. search for pseudo-peripheral nodes 2. perform two-sided BFS 3. post improvement using local search repeat (few times) and take best 14 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
15 Graph Partitioning Initial Partitioning - One Possibility if we know how to bipartition, we can compute a k-way partition BIPART(G): 1. search for pseudo-peripheral nodes 2. perform two-sided BFS 3. post improvement using local search repeat (few times) and take best 15 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
16 Local Improvement input graph output partition... local improvement... match contract initial uncontract partitioning 16 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
17 Graph Partitioning Local Improvement Methods compute gain v V 17 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
18 Graph Partitioning Local Improvement Methods compute gain v V g(v) = d ext (v) d int (v) 17 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
19 Graph Partitioning Local Improvement Methods compute gain v V g(v) = d ext (v) d int (v) store gain of boundary nodes (e.g. in a heap) 17 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
20 Graph Partitioning Local Improvement Methods move highest gain vertices to opposite block each node at most once update gain of neighbors Step: 0 Edge Cut: 5 17 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
21 Graph Partitioning Local Improvement Methods move highest gain vertices to opposite block each node at most once update gain of neighbors Step: 0 1 Edge Cut: Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
22 Graph Partitioning Local Improvement Methods move highest gain vertices to opposite block each node at most once update gain of neighbors Step: Edge Cut: Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
23 Graph Partitioning Local Improvement Methods move highest gain vertices to opposite block each node at most once update gain of neighbors Step: Edge Cut: Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
24 Graph Partitioning Local Improvement Methods stop after limit take best edge cut within balance constraint edge cut Step: Edge Cut: steps 17 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
25 Graph Partitioning Local Improvement Methods stop after limit take best edge cut within balance constraint edge cut Step: Edge Cut: steps 17 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
26 Advanced Graph Partitioning Karlsruhe Fast Flow Partitioner Focus: Local and Global Search Local Search: 1. Flow Based Improvement Methods 2. More Localized FM Algorithm (inspired by KaSPar) Global Search: 1. V-cycles and F-cycles similar to methods used in multigrid input graph local improvement match... local improvement... G s 1B 2B t V 1 V B 2 match contract initial partitioning uncontract contract uncontract 18 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
27 Flows as Local Improvement Two Blocks G s 1 B 2 B V 1 V 2 B t area B, such that each (s, t)-min cut is ɛ-balanced cut in G e.g. 2 times BFS (left, right) stop the BFS, if size would exceed (1 + ɛ) n 2 c(v 2) c(v 2new ) c(v 2 ) + (1 + ɛ) n 2 c(v 2) 19 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
28 Flows as Local Improvement Two Blocks s t G V 1 V B 2 obtain optimal cut in B since each cut in B yields a feasible partition improved two-partition 20 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
29 Flows as Local Improvement Adaptive Search search in larger areas for feasible cuts adaptively control the size of corridor B heuristic for Most Balanced Minimum Cuts [Picard et al. 1980] We need: SCC s, DAGs, Closed Vertex Sets, Topological Sorting G s t V 2 G s t V 1 B V 1 V B 2 the maximal upper bound factor is called α 21 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
30 Local Improvement for k-partitions Using Flows? on each pair of blocks input graph output partition... local improvement... match contract initial uncontract partitioning 22 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
31 Local Improvement for k-partitions Using Flows? on each pair of blocks 23 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
32 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
33 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
34 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
35 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
36 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
37 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
38 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
39 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
40 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
41 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
42 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
43 More Localized Local Search Idea: KaPPa, KaSPar more local searches are better Typical: k-way local search initialized with complete boundary Localization: 1. complete boundary maintained todo list T 2. initialize search with single node v rnd T 3. iterate until T = each node moved at most once 24 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
44 Global Search Iterated Multilevel [Walshaw 2004] input graph local improvement match... local improvement... match contract initial uncontract contract uncontract partitioning don t contract cut edges no new initial partitioning needed random tie breaking needed local improvement has to assure cuts 25 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
45 Global Search V-Cycles Coarsening Uncoarsening Graph partitioned Graph not partitioned 26 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
46 Global Search W-Cycles Coarsening Uncoarsening Graph partitioned Graph not partitioned 27 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
47 Global Search F-Cycles Coarsening Uncoarsening Graph partitioned Graph not partitioned 28 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
48 Experiments Testset Medium sized instances graph n m rgg rgg Delaunay Delaunay bcsstk elt fesphere cti memplus cs pwt bcsstk body t60k wing brack finan bel nld af shell Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
49 Experimental Evaluation Flows Var. (+F, -MB, -FM ) (+F, +MB, -FM) (+F, -MB, +FM) (+F, +MB, +FM) α cut % t[s] cut % t[s] cut % t[s] cut % t[s] Ref. (-F, -MB, +FM) final score of different configurations α flow region upper bound factor all value are improvements rel. to Ref. +/- F +/- Flow +/- MB +/- Most Bal. H. +/- FM +/- FM Algorithm 30 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
50 Experimental Evaluation Flows - Effectiveness Effectiveness (+F, +MB, -FM) (+F,-MB, +FM) (+F,+MB,+FM) α cut % cut % cut % (-F, -MB, +FM) each configuration has the same amount of time all value are improvements rel. to Ref. +/- F +/- Flow +/- MB +/- Most Bal. H. +/- FM +/- FM Algorithm 31 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
51 Experimental Evaluation Global Search Algorithm cut % t[s] Eff. Avg. 2 F-cycle V-cycle W-cycle W-cycle F-cycle V-cycle V-cycle relative fast basic configuration only global search is varied red significantly < blue red and blue significantly < reference 32 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
52 Experiments Remove Components Remove components step by step Repetition Avg. t Eff. Avg. KaFFPa Strong KWay MoreLocalizedSearch FCycle Flow Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
53 Experimental Results 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 % KaFFPa eco % 3.82 Scotch % 3.55 KaFFa fast % 0.98 kmetis % 0.83 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 34 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
54 KaFFPaE Distributed Evolutionary Graph Partitioning Goal: Integrate KaFFPa in an Evolutionary Strategy Evolutionary Algorithms: highly inspired by biology population of individuals selection, mutation, recombination,... Evolutionary Graph Partitioning: individuals partitions fitness edge cut Parallelization records in a few minutes for small graphs 35 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
55 Evolutionary Algorithms General Structure procedure steady-state-ea create initial population P while stopping criterion not fulfilled select parents P 1, P 2 from P combine P 1 with P 2 to create offspring o mutate offspring o evict individual in population using o return the fittest individual that occurred 36 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
56 KaFFPaE Recombination match contract two individuals P 1, P 2 : don t contract cut edges of P 1 or P 2 until no matchable edge is left coarsest graph Q-graph of overlay exchanging good parts is easy 37 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
57 KaFFPaE Exchanging good parts is easy G v 2 G v 2 v 1 v 3 v 1 v 3 v 4 v 4 curved lines have a large cut start with the green partition move v 2 to the opposite block 38 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
58 KaFFPaE Recombination - Generalization recombine a partition with any clustering of the graph e.g. : k = k partition with larger imbalances natural cuts: sparse cuts close to dense areas [Delling et al. 11] v plug and play: use the clustering that fits your domain 39 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
59 KaFFPaE Parallelization each PE has its own island (a local population) locally: perform combine and mutation operations communicate analog to randomized rumor spreading 1. rumor currently best local partition 2. local best partition changed send it to O(log P) random PEs communicate asynchronously 3. changed received a better one or improved locally 40 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
60 Quality k/algo. Reps. KaFFPaE Avg. impr. % % % % % % % % % overall % Middlesized testset, 2h time, 32 cores per graph and k 41 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
61 Quality k=64 mean min cut Repetitions KaFFPaE normalized time t n 42 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
62 Scalability mean min cut (mmc) p = 1 p = 2 p = 4 p = 8 p = 16 p = normalized time t n p = 64 p = 128 p = Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
63 Scalability pseudo speedup p = 2 p = 4 p = 8 p = 16 p = 32 p = 64 p = 128 p = normalized time t n 44 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
64 Experimental Results Example Street network Europe V = 18M, E = 44M, k = 64 Buffoon kmetis 45 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
65 Outlook Walshaw Benchmark runtime is not an issue 614 instances (ɛ {1%, 3%, 5%}) focus on partition quality Algorithm < KaPPa KaSPar KaFFPa KaFFPaE Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
66 Summary distr. evol. Alg. [ALENEX12] V F W [ESA11] Cycles a la multigrid input graph Output Partition [IPDPS10] edge ratings match + contract todo... flows etc. [ESA11]... local improvement initial partitioning uncontract todo parallel [IPDPS10] n level [ESA10] Multilevel Graphpartitioning 47 Nov. 10, 2011 Peter Sanders, Christian Schulz Institute for Theoretical Computer Science, Algorithmics II
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