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25
On twodimensional sparse matrix partitioning: Models, methods, and a recipe
 SIAM J. Sci. Comput
, 2010
"... Abstract. We consider twodimensional partitioning of general sparse matrices for parallel sparse matrixvector multiply operation. We present three hypergraphpartitioningbased methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces f ..."
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Cited by 35 (18 self)
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Abstract. We consider twodimensional partitioning of general sparse matrices for parallel sparse matrixvector multiply operation. We present three hypergraphpartitioningbased methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces finegrain partitions. The other two produce coarser partitions, where one of them imposes a limit on the number of messages sent and received by a single processor, and the other trades that limit for a lower communication volume. We also present a thorough experimental evaluation of the proposed twodimensional partitioning methods together with the hypergraphbased onedimensional partitioning methods, using an extensive set of public domain matrices. Furthermore, for the users of these partitioning methods, we present a partitioning recipe that chooses one of the partitioning methods according to some matrix characteristics.
Multilevel direct Kway hypergraph partitioning with multiple constraints and fixed vertices
, 2007
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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 13 (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.
On computing inverse entries of a sparse matrix in an outofcore environment
, 2010
"... Abstract. The inverse of an irreducible sparse matrix is structurally full, so that it is impractical to think of computing or storing it. However, there are several applications where a subset of the entries of the inverse is required. Given a factorization of the sparse matrix held in outofcore ..."
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Cited by 11 (5 self)
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Abstract. The inverse of an irreducible sparse matrix is structurally full, so that it is impractical to think of computing or storing it. However, there are several applications where a subset of the entries of the inverse is required. Given a factorization of the sparse matrix held in outofcore storage, we show how to compute such a subset e ciently, by accessing only parts of the factors. When there are many inverse entries to compute, we need to guarantee that the overall computation scheme has reasonable memory requirements, while minimizing the cost of loading the factors. This leads to a partitioning problem that we prove is NPcomplete. We also show that we cannot get a close approximation to the optimal solution in polynomial time. We thus need to develop heuristic algorithms, and we propose: (i) a lower bound on the cost of an optimum solution; (ii) an exact algorithm for a particular case; (iii) two other heuristics for a more general case; and (iv) hypergraph partitioning models for the most general setting. We illustrate the performance of our algorithms in practice using the MUMPS software package on a set of reallife problems as well as some standard test matrices. We show that our techniques can improve the execution time by a factor of 50. Key words. Sparse matrices, direct methods for linear systems and matrix inversion, multifrontal method, graphs and hypergraphs. AMS subject classi cations. 05C50, 05C65, 65F05, 65F50 1. Introduction. We
A Parallel Matrix Scaling Algorithm
"... We recently proposed an iterative procedure which asymptotically scales the rows and columns of a given matrix to one in a given norm. In this work, we briefly mention some of the properties of that algorithm and discuss its efficient parallelization. We report on a parallel performance study of ou ..."
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Cited by 8 (3 self)
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We recently proposed an iterative procedure which asymptotically scales the rows and columns of a given matrix to one in a given norm. In this work, we briefly mention some of the properties of that algorithm and discuss its efficient parallelization. We report on a parallel performance study of our implementation on a few computing environments.
On sharedmemory parallelization of a sparse matrix scaling algorithm
"... Abstract—We discuss efficient shared memory parallelization of sparse matrix computations whose main traits resemble to those of the sparse matrixvector multiply operation. Such computations are difficult to parallelize because of the relatively small computational granularity characterized by smal ..."
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Cited by 7 (3 self)
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Abstract—We discuss efficient shared memory parallelization of sparse matrix computations whose main traits resemble to those of the sparse matrixvector multiply operation. Such computations are difficult to parallelize because of the relatively small computational granularity characterized by small number of operations per each data access. Our main application is a sparse matrix scaling algorithm which is more memory bound than the sparse matrix vector multiplication operation. We take the application and parallelize it using the standard OpenMP programming principles. Apart from the common race condition avoiding constructs, we do not reorganize the algorithm. Rather, we identify associated performance metrics and describe models to optimize them. By using these models, we implement parallel matrix scaling algorithms for two wellknown sparse matrix storage formats. Experimental results show that simple parallelization attempts which leave data/work partitioning to the runtime scheduler can suffer from the overhead of avoiding race conditions especially when the number of threads increases. The proposed algorithms perform better than these algorithms by optimizing the identified performance metrics and reducing the overhead. KeywordsSharedmemory parallelization, sparse matrices, hypergraphs, matrix scaling I.
A LinkBased Storage Scheme for Efficient Aggregate Query Processing on Clustered Road Networks
 INFORMATION SYSTEMS
, 2009
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On analysis of partitioning models and metrics in parallel sparse matrixvector multiplication
, 2013
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A matrix partitioning interface to PaToH in MATLAB
, 2009
"... We present the PaToH MATLAB Matrix Partitioning Interface. The interface provides support for hypergraphbased sparse matrix partitioning methods which are used for efficient parallelization of sparse matrixvector multiplication operations. The interface also offers tools for visualizing and measur ..."
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We present the PaToH MATLAB Matrix Partitioning Interface. The interface provides support for hypergraphbased sparse matrix partitioning methods which are used for efficient parallelization of sparse matrixvector multiplication operations. The interface also offers tools for visualizing and measuring the quality of a given matrix partition. We propose a novel, multilevel, 2D coarseningbased 2D matrix partitioning method and implement it using the interface. We have performed extensive comparison of the proposed method against our implementation of orthogonal recursive bisection and finegrain methods on a large set of publicly available test matrices. The conclusion of the experiments is that the new method can compete with the finegrain method while also suggesting new research directions.
Scientific Computing Group
"... Abstract—We present a new hypergraphbased method, the mediumgrain method, for solving the sparse matrix partitioning problem. This problem arises when distributing data for parallel sparse matrixvector multiplication. In the mediumgrain method, each matrix nonzero is assigned to either a row gr ..."
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Abstract—We present a new hypergraphbased method, the mediumgrain method, for solving the sparse matrix partitioning problem. This problem arises when distributing data for parallel sparse matrixvector multiplication. In the mediumgrain method, each matrix nonzero is assigned to either a row group or a column group, and these groups are represented by vertices of the hypergraph. For an m × n sparse matrix, the resulting hypergraph has m+n vertices and m+n hyperedges. Furthermore, we present an iterative refinement procedure for improvement of a given partitioning, based on the mediumgrain method, which can be applied as a cheap but effective postprocessing step after any partitioning method. The mediumgrain method is able to produce fully twodimensional bipartitionings, but its computational complexity equals that of onedimensional methods. Experimental results for a large set of sparse test matrices show that the mediumgrain method with iterative refinement produces bipartitionings with lower communication volume compared to current stateoftheart methods, and is faster at producing them. Keywordsparallel computing, sparse matrixvector multiplication, hypergraph I.