T.G. Kolda, "Partitioning Sparse Rectangular Matrices for Parallel Processing," Proceedings of 5th International Symposium on Solving Irregularly Structured Problems in Parallel, Irregular'98 (Lecture Notes in Computer Science, vol. 1457), pp. 473--482, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....number can also be obtained by adding the weights of the vertices in each part. 2.3 Generalized Graph Model for Structurally Symmetric Nonsymmetric Square Matrices The standard graph model is not suitable for the partitioning of nonsymmetric matrices. A recently proposed bipartite graph model [17, 26] enables the partitioning of rectangular as well as structurally symmetric nonsymmetric square matrices. In this model, each row and column is represented by a vertex, and the sets of vertices representing the rows and columns form the bipartition, i.e. V = VR [ V C . There exists an edge between ....

T. G. Kolda, "Partitioning sparse rectangular matrices for parallel processing," Lecture Notes in Computer Science, vol. 1457, pp. 68--79, 1998.

....Fig. 2a displays the bipartite graph representation of the sample matrix given in Fig. 1. In addition to the work of Ferris and Horn [12] the bipartite graph model is also used in the literature for partitioning rectangular as well as nonsymmetric square matrices. Hendrickson and Kolda [25, 27, 28, 41] showed that the matrix partitioning problem for parallel matrix vector multiplication can be described as a bipartite GPES problem in which unweighted edge cut (w ij =1 in (9) relates to parallel communication. Rashid and Davis [55] exploited the bipartite graph model for a coarse grain parallel ....

T.G. Kolda, "Partitioning Sparse Rectangular Matrices for Parallel Processing," Proceedings of 5th International Symposium on Solving Irregularly Structured Problems in Parallel, Irregular'98 (Lecture Notes in Computer Science, vol. 1457), pp. 473--482, 1998.

....calculation instead of just the matrix vector product. Unfortunately, the standard graph model has no capacity to consider the larger problem. 5 A Better Combinatorial Model Some of the limitations discussed in x4.2 can be addressed by a more expressive model proposed by Kolda and Hendrickson [3, 4, 8]. The model uses a bipartite graph to represent the matrix. In this graph, there is a vertex for each row and another vertex for each column. An edge connects row vertex i to column vertex j if there is a nonzero in matrix location [i; j] The structure of nonsymmetric and nonsquare matrices can ....

T. G. Kolda, Partitioning sparse rectangular matrices for parallel processing. In Proc. Irregular'98, 1998.

....[39] Kernighan Lin [31] Fiduccia Mattheyses [12] and multilevel [6, 26, 29, 30] methods are given for the bipartite graph model. The modification of the spectral method was previously introduced by Berry, Hendrickson, and Raghavan [5] Further, the alternating partitioning method of Kolda [33] is presented; this method is specific to the bipartite case. Finally in section 6, we measure the performance of various methods for partitioning rectangular or structurally unsymmetric matrices. We compare di#erent methods on a collection of matrices from least squares, linear programming, ....

....graph partitioning model to identify parallelism in sparse Gaussian elimination. Neither of these e#orts addresses the problem of parallelizing matrix vector products. Previous attempts to address the general matrix partitioning problem for matrix vector multiplication include the work of Kolda [33] and an earlier report on this research [23] The authors have recently become aware of a closely related work by C atalyurek and Aykanat [7] In their approach, the structure of the unsymmetric matrix is represented by a hypergraph in which rows are vertices and columns are hyperedges. A ....

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T. G. Kolda, Partitioning sparse rectangular matrices for parallel processing, in Solving Irregularly Stuctured Problems in Parallel: 5th International Symposium, Irregular '98, Berkeley, CA, 1998, A. Ferreira et al., eds., in Lecture Notes in Comput. Sci. 1457, Springer-Verlag, New York, 1998, pp. 68--79.

.... [14] solving the normal equations that arise in interior point methods using CG [15] or computing the truncated SVD of hypertext matrices in information retrieval [2] Although matrix partitioning has been well studied in the symmetric case, little work has been done in the rectangular case [13]. The symmetric problem is commonly phrased in terms of partitioning graphs. We will show that the rectangular problem can be conveniently phrased in terms of partitioning bipartite graphs. We will extend the work of Berry, Hendrickson, and Raghavan [2] and Kolda [13] for partitioning rectangular ....

....done in the rectangular case [13] The symmetric problem is commonly phrased in terms of partitioning graphs. We will show that the rectangular problem can be conveniently phrased in terms of partitioning bipartite graphs. We will extend the work of Berry, Hendrickson, and Raghavan [2] and Kolda [13] for partitioning rectangular matrices by incorporating multilevel schemes which are already popular in the symmetric case [1, 8, 10, 11] Multilevel methods, described in Sect. 3, have three phases: coarsening, base level partitioning, and un coarsening with refinement. In Sect. 4, we will ....

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Tamara G. Kolda. Partitioning sparse rectangular matrices for parallel processing. In Proc. 5th Intl. Symposium on Solving Irregularly Structured Problems in Parallel (Irregular '98), to appear.

....encode symmetric data dependencies and symmetric partitions. These limitations are particularly problematic for iterative solvers on unsymmetric or non square matrices. When using preconditioners, the inability of the standard model to capture multiple phase calculations are also problematic. In [11,12,23] Kolda and Hendrickson propose a bipartite graph model for describing matrixvector multiplication which addresses some of these shortcomings. The bipartite model can also be applied to other applications involving unsymmetric dependencies and multiple phases. A bipartite graph, G = V 1 ; V 2 ; ....

T. G. Kolda. Partitioning sparse rectangular matrices for parallel processing. In A. Ferreira et al., editors, Solving Irregularly Structured Problems in Parallel: 5th International Symposium, Irregular '98, Berkeley, California, USA, August 9--11, 1998, number 1457 in Lecture Notes in Computer Science, pages 68--79. Springer-Verlag, 1998.

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T.G. Kolda, "Partitioning Sparse Rectangular Matrices for Parallel Processing," Proceedings of 5th International Symposium on Solving Irregularly Structured Problems in Parallel, Irregular'98 (Lecture Notes in Computer Science, vol. 1457), pp. 473--482, 1998.

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