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Fine-Grain Sparse Matrix Partitioning with Constraints

Fine-Grain Sparse Matrix Partitioning with Constraints. Erik Boman Sandia National Labs, Albuquerque, NM Umit Catalyurek Ohio State University. Outline. Load-balancing & partitioning Sparse matrix-vector multiplication Models: bipartite graph, hypergraph 1d and 2d distributions

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Fine-Grain Sparse Matrix Partitioning with Constraints

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  1. Fine-Grain Sparse Matrix Partitioning with Constraints Erik Boman Sandia National Labs, Albuquerque, NM Umit Catalyurek Ohio State University

  2. Outline • Load-balancing & partitioning • Sparse matrix-vector multiplication • Models: bipartite graph, hypergraph • 1d and 2d distributions • Constrained matrix partitioning • Hypergraph with fixed vertices • Vertex cover model • Numerical results

  3. Proc 3 Proc 1 Proc 2 Load Balancing, Graph Partitioning • Load Balancing • Assign work to processors to distribute work evenly and minimize communication • Graph or hypergraph partitioning • Vertices (weighted) = computation • Edge (weighted) = data dependence

  4. Parallel Sparse Matrix-Vector Multiply • Compute y=Ax, where A is large and sparse • A is distributed among processors • Kernel in scientific computing • Iterative methods (Krylov) • Eigenvalue computations • PageRank • Nice model problem • Other applications have similar computation and communication pattern

  5. Parallel Sparse Matrix-vector Algorithm • Step 1: Expand (fan-out) • Send xj to processors with nonzero in column j • Step 2: Local multiply-add • yi = yi + Aij xj • Step 3: Fold (fan-in) • Send partial results of y to relevant processors • Step 4: Sum partial results Two communication phases (expand and fold).

  6. Row distribution Column distribution 2 1 1 4 3 x 2 1 1 4 3 x 3 1 6 4 1 3 1 9 6 5 9 2 4 1 22 9 6 5 3 5 9 2 41 22 5 8 9 6 5 3 64 41 5 8 9 64 y A y A Communication (p=2)

  7. 1d Models • 1d distribution is most common • Assign either rows or columns • Models to represent a sparse matrix • Graph (only symmetric) • Bipartite graph • Hypergraph

  8. C2 C4 C1 C3 C5 Bipartite Graph Model • G=(R,C,E), where • R are row vertices • C are column vertices • Partition both R and C • But only use R for row distribution • Works in nonsym. case • Edge cut approximates comm. volume • Is NOT exact

  9. 1d Hypergraph Model • Hypergraph • Hyperedge is a set of vertices (1 or more) • Rows = vertices • Columns = hyperedges • Partition • Minimize hyperedges cut • Edge cut is exactly comm volume • Aykanat & Catalyurek (’96)

  10. 2D partitioning methods Reduces communication volume further Many variations 2D Cartesian (checkerboard) First partition rows Then partition columns Via multiconstraint hgraph partitioning Catalyurek & Aykanat 2D Mondriaan Recursive hypergraph bisection Bisseling & Vastenhouw Sparse Matrix Partitioning Courtesy: Rob Bisseling

  11. Fine-Grain Matrix Partitioning • Fine-grain partitioning • Assign each nonzero in matrix separately • Ultimate flexibility • Fine-grain hypergraph model • Catalyurek & Aykanat (2001) • Each nonzero is a vertex • Each row and column is a hyperedge • Exact model for comm. volume

  12. Bipartite Fine-Grain Model • Bipartite graph model • Partition both • vertices (rows, columns) • edges (nonzeros) • Minimize #vertices with incident edge in other partition

  13. Constrained Matrix Partitioning • Given A, x, y • where x and y already have a prescribed parallel distribution • Find optimal parallel distribution of A • Application: Iterative solver for Ax=b • Preconditioner code requires specific layout of the vectors • We are free to choose distribution for A, which is only used for matrix-vector multiply

  14. Bisection • First consider bisection (p=2). • Suppose wlog the 2*2 block structure to the right • Can always permute to get this form • We only need solve for the two off-diagonal blocks • Independent subproblems

  15. Hypergraph with fixed vertices Add constraints to fine-grain hypergraph: • Each row-net should contain a special blue vertex • Each column-net should contain a special red vertex • These special vertices are fixed, cannot change partition • Partitioning with fixed vertices available in Patoh and Zoltan packages

  16. Cover nonzeros by rows/columns Columns (4) Rows (4) Both (3)

  17. Graph Model: Vertex Cover x x x x x x x x

  18. Bipartite Matching König’s Theorem: Let G be a bipartite graph, let C be a minimum vertex cover for G, and let M be a maximum-cardinality matching. Then |C| = |M|. Polynomial time algorithm for bipartite graphs.

  19. K-way Generalization • Partition into k sets • Decouples into k(k-1) independent subproblems

  20. Numerical Results • Bisection (k=2), split using natural order Matching code provided by Dobrian, Halappanavar, and Pothen.

  21. Conclusions • Two new algorithms for fine-grain matrix partitioning with constraints • Empirical results show similar quality • VC/matching usually faster than partitioning • More experiments needed • Hypergraph partitioning: • Exact model, heuristic solver • Vertex cover via matching • Exact algorithm, but no balance constraint • K-way partitioning decouples, fully parallel

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