Hypergraph models for sparse matrix partitioning and reordering

Abstract

Graphs have been widely used to represent sparse matrices for various scientific applications including one-dimensional (ID) decomposition of sparse matrices for parallel sparse-matrix vector multiplication (SpMxV) and sparse matrix reordering for low fill factorization. The standard graph-partitioning based ID decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel SpMxV. We propose two computational hypergraph models which avoid this crucial deficiency of the graph model on ID decomposition. The proposed models reduce the ID 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 first hypergraph model is proposed for fine-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 fill ordering of sparse matrices for direct solution of linear systems. In this work, we also show that the GPVS problem can be formulated as hypergraph partitioning. We exploit this finding to develop a novel hypergraph partitioning-based nested dissection ordering. The recently proposed successful multilevel framework is exploited to develop a multilevel hypergraph partitioning tool PaToH for the experimental verification of our proposed hypergraph models. Experimental results on a wide range of realistic sparse test matrices confirm the validity of the proposed hypergraph models. In terms of communication volume, the proposed hypergraph models produce 30% and 59% better decompositions than the graph model in ID and 2D decompositions of sparse matrices for parallel SpMxV computations, respectively. The proposed hypergraph partitioning-based nested dissection produces 25% to 45% better orderings than the widely used multiple mimirnum degree ordering in the ordering of various test matrices arising from different applications.