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Permuting Sparse Rectangular Matrices into BlockDiagonal Form
 SIAM Journal on Scientific Computing
, 2002
"... We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. W ..."
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Cited by 57 (19 self)
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We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose bipartite graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using stateoftheart graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.
Encapsulating Multiple CommunicationCost Metrics in Partitioning Sparse Rectangular Matrices for Parallel MatrixVector Multiplies
"... This paper addresses the problem of onedimensional partitioning of structurally unsymmetricsquare and rectangular sparse matrices for parallel matrixvector and matrixtransposevector multiplies. The objective is to minimize the communication cost while maintaining the balance on computational load ..."
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Cited by 35 (22 self)
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This paper addresses the problem of onedimensional partitioning of structurally unsymmetricsquare and rectangular sparse matrices for parallel matrixvector and matrixtransposevector multiplies. The objective is to minimize the communication cost while maintaining the balance on computational loads of processors. Most of the existing partitioning models consider only the total message volume hoping that minimizing this communicationcost metric is likely to reduce other metrics. However, the total message latency (startup time) may be more important than the total message volume. Furthermore, the maximum message volume and latency handled by a single processor are also important metrics. We propose a twophase approach that encapsulates all these four communicationcost metrics. The objective in the first phase is to minimize the total message volume while maintainingthe computationalload balance. The objective in the second phase is to encapsulate the remaining three communicationcost metrics. We propose communicationhypergraph and partitioning models for the second phase. We then present several methods for partitioning communication hypergraphs. Experiments on a wide range of test matrices show that the proposed approach yields very effective partitioning results. A parallel implementation on a PC cluster verifies that the theoretical improvements shown by partitioning results hold in practice.
HYPERGRAPH PARTITIONINGBASED FILLREDUCING ORDERING
, 2009
"... A typical first step of a direct solver for linear system Mx = b is reordering of symmetric matrix M to improve execution time and space requirements of the solution process. In this work, we propose a novel nesteddissectionbased ordering approach that utilizes hypergraph partitioning. Our approac ..."
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Cited by 9 (5 self)
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A typical first step of a direct solver for linear system Mx = b is reordering of symmetric matrix M to improve execution time and space requirements of the solution process. In this work, we propose a novel nesteddissectionbased ordering approach that utilizes hypergraph partitioning. Our approach is based on formulation of graph partitioning by vertex separator (GPVS) problem as a hypergraph partitioning problem. This new formulation is immune to deficiency of GPVS in a multilevel framework hence enables better orderings. In matrix terms, our method relies on the existence of a structural factorization of the input M matrix in the form of M = AAT (or M = AD2AT). We show that the partitioning of the rownet hypergraph representation of rectangular matrix A induces a GPVS of the standard graph representation of matrix M. In the absence of such factorization, we also propose simple, yet effective structural factorization techniques that are based on finding an edge clique cover of the standard graph representation of matrix M, and hence applicable to any arbitrary symmetric matrix M. Our experimental evaluation has shown that the proposed method achieves better ordering in comparison to stateoftheart graphbased ordering tools even for symmetric matrices where structural M = AAT factorization is not provided as an input. For matrices coming from linear programming problems, our method enables even faster and better orderings.
HypergraphPartitioningBased Sparse Matrix Ordering
"... In this work we propose novel sparse matrix ordering approaches based on hypergraph partitioning. The significance of hypergraphpartitioningbased (HPbased) ordering is threefold. First, almost all of the successful nested dissection ..."
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Cited by 3 (2 self)
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In this work we propose novel sparse matrix ordering approaches based on hypergraph partitioning. The significance of hypergraphpartitioningbased (HPbased) ordering is threefold. First, almost all of the successful nested dissection
Permuting Sparse Rectangular Matrices into SinglyBordered BlockDiagonal Form for Parallel Solution of LP Problems
, 2002
"... Coarsegrain parallelism inherent in the solution of blockangular Linear Programming (LP) problems has been exploited in recent research works. The objective of this work is to enhance these successful decompositionbased approaches for coarsegrain parallel solution of general LP problems by trans ..."
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Cited by 2 (1 self)
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Coarsegrain parallelism inherent in the solution of blockangular Linear Programming (LP) problems has been exploited in recent research works. The objective of this work is to enhance these successful decompositionbased approaches for coarsegrain parallel solution of general LP problems by transforming them into blockangular forms. In matrix theoretical view, this problem can be described as permuting a sparse rectangular matrix A into a singlybordered blockdiagonal (SB) form A SB with minimum border size while maintaining a given balance criterion on the diagonal blocks. In the twophase approach proposed in the literature, matrix A is permuted into a doublybordered blockdiagonal (DB) form ADB as an intermediate form to be transformed into an SB form A SB through column splitting. We show that the AtoADB transformation problem can be described as a graph partitioning by vertex separator (GPVS) problem on the bipartite graph representation of A . In this work, we propose a o...