Hypergraph Models for Sparse Matrix Partitioning and Reordering

HYPERGRAPH MODELS FOR SPARSE MATRIX PARTITIONING AND REORDERING  Umit V. C ataly urek Ph.D. in Computer Engineering and Information Science Supervisor: Assoc. Prof. Cevdet Aykanat November, 1999 Graphs have been widely used to represent sparse matrices for various scienti c applications including one-dimensional (1D) decomposition of sparse matrices for parallel sparse-matrix vector multiplication (SpMxV) and sparse matrix reordering for low ll factorization. The standard graph-partitioning based 1D decomposition of sparse matrices does not re ect the actual communication volume requirement for parallel SpMxV.We propose two computational hypergraph models which avoid this crucial de ciency of the graph model on 1D decomposition. The proposed models reduce the 1D decomposition problem to the well-known hypergraph partitioning problem. In the literature, there is a lack of 2D decomposition heuristic which directly minimizes the communication requirements for parallel SpMxV computations. Three novel hypergraph models are introduced for 2D decomposition of sparse matrices for minimizing the communication volume requirement. The rst hypergraph model is proposed for ne-grain 2D decomposition of the sparse matrices for parallel SpMxV. The second hypergraph model for 2D decomposition is proposed to produce jagged-like decomposition of the sparse matrix. The checkerboard decomposition based parallel matrix-vector multiplication algorithms are widely encountered in the literature. However, only the load balancing problem is addressed in those works. Here, we propose a new hypergraph model which aims the minimization of communication volume while maintaining the load balance among the processors for checkerboard decomposition, as the third model for 2D decomposition. The proposed model reduces the decomposition problem to the multi-constraint hypergraph partitioning problem. The notion of multi-constraint partitioning has recently become popular in graph partitioning. We applied the multi-constraint partitioning to the hypergraph partitioning problem for solving checkerboard partitioning. Graph partitioning by vertex separator (GPVS) is widely used for nested dissection based low ll ordering of sparse matrices for direct solution of linear systems. In this work,

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