Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering

Transcription

1 Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering George Karypis and Vipin Kumar Brian Shi CSci /09/2017

2 Outline Introduction Graph Partitioning Problem Multilevel Graph Partitioning Algorithm Parallelizing The Multilevel Graph Partitioning Algorithm Initial Partitioning Phase Uncoarsening Phase Parallel Refinement Algorithm Parallel Multilevel Sparse Matrix Ordering Algorithm Performance,Scaling Analysis, and Results

3 Graph Partitioning Problem The graph partitioning problem is NP-complete. Definition: Let G = (V, E) be a graph with V = n. A p-way partition of G consists of V 1,..., V p V such that V is a disjoint union of V 1,..., V p with V i n/p and the edge cuts are minimized.

4 Multilevel Graph Partitioning Algorithm Consists of three main phases. 1. Coarsening Phase: A matching of G is constructed and edges are contracted. 2. Initial Partitioning Phase: A partition of a smaller coarsened graph is performed. 3. Uncoarsening Phase: The partitioned graph is uncontracted and map back to the original graphs.

5 Coarsening Phase Sequence of smaller graphs {G l } are constructed by finding maximal matchings starting from G 0 = G. Graph Matching Let G = (V, E) be an undirected graph, then a matching of G is a subset of edges, M E, such that no vertex in V is incident to more than one edge in M. Vertices in matched edges contracted Random Matching Heavy Edge Matching

6 Partitioning and Uncoarsening 1. Initial Partitioning Phase: A partition of a smaller coarsened graph is performed. 2. Serial algorithms Greedy Graph Growing Partition (GGGP). 3. Partitioned graph is projected back to the original graph. 4. Refinement algorithms used to minimize edge-cuts and load balancing.

7 Parallelizing The Multilevel Graph Partitioning Algorithm Assumption: p-way partitioning with p processors Parallelization exploited in the recursize nature of the bisection algorithm p processors perform one bisection. Then p/2 processors perform bisections of each half. Goal: Parallel algorithm for graph bisection.

8 Coarsening Phase Main Idea Assume p = 2 2r processors arranged in a 2-D array, and (V 0, E 0 ) = (V, E) Distribute the verticesinto p subsets: V0 0,, V0 1 p 1,..., V0 Processor P ij will contain subset of edges in E 0 with incident vertices in V i 0 and V j 0

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10 Coarsening Phase The edge matchings, M0 i, will be done along the diagonal processors. After completion each P ii does two broadcasts along its row and columns to distribute M i 0. E 1 = M 0 = i Mi 0 Each P ij contains edges with incident vertices in V i 1 and V j 1 Once G k is coarse enough we reduce number of working processors by folding.

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12 Initial Partitioning Phase Coarsest graph will be contained in a single processor. Can be done sequentially using GGGP. Several runs of GGGP are run using different random starting vertices. We can copy the coarsest graph to multiple processors and run these trials concurrently. We keep the partition giving the smallest edge-cut.

13 Uncoarsening Phase Project coarse graphs back to original graphs. Refinements are made during each step of the projection Serial algorithms (Kernighan-Lin variants) used when coarse graph resides in one processor. The processor folding is reversed.

14 Parallel Refinement Algorithm Each P ij computes local gain lg v for each v V j 0. Total gain of v, g v, is computed via a sum reduction along the columns. Each P ii selects vertices, U i V0 i, with positive gain. Broadcast U i row and column-wise. Recompute the local gains and repeat. Balancing partitions Start vertex swaps from heavier parts of partition Use a load balancing iteration if there is more than 2% imbalance.

15 Parallel Multilevel Sparse Matrix Ordering Algorithm Assume a bisection is already constructed. A, B be the boundary vertices. Need to construct vertex separator.

16 Parallel Vertex Cover Algorithm Processors are arranged in a 2-D array. A i = A V i 0 and B i = B V i 0 Each P ij stores A i and B j and computes minimal cover of edges between A i, B j locally. Denote A ij c A and Bc ij B and A ij c Bc ij Union of A ij c, Bc ij the minimal cover across all processors forms a cover.

17 Parallel Vertex Cover Algorithm Constructing Minimal Cover Broadcast Bc i = j Bc ij to processors in same column Each P ij removes vertices from A ij c whose edges are covered in Bc j A ij c A ij c Broadcast A i c = j A ij c to processors in same row Each P ij removes vertices from B j c whose union is B j c Then a minimal cover is got via S = ( p 1 i=0 ) ( p 1 A i c j=0 B j c )

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21 Scaling Analysis Time to broadcast is n = V 0 and m = E 0 Time for bisection is T broadcast = O T bisection = O ( n p ) ( ) ( ) m n + O p p Overall runtime of the p-way partitioning algorithm is ( ( ) ( )) m n T partition = O + O p log(p) p

22 Test Cases

23 Results

24 References G. Karypis and V. Kumar, A parallel algorithm for multilevel graph partitioning and sparse matrix ordering, Journal of Parallel and Distributed Computing, vol. 48, no. 1, p , G. Karypis and V. Kumar, Multilevel graph partitioning schemes, ICPP (3), pp , A. Grama, G. Karypis, V. Kumar, and A. Gupta, Introduction to Parallel Computing. Addison-Wesley, second ed., 2003.