Finding Nonoverlapping Substructures of a Sparse Matrix

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Many applications of scientific computing rely on computations on sparse matrices. The design of efficient implementations of sparse matrix kernels is crucial for the overall efficiency of these applications. Due to the high compute-to-memory ratio and irregular memory access patterns, the performance of sparse matrix kernels is often far away from the peak performance on a modern processor. Alternative data structures have been proposed, which split the original matrix A into A{sub d} and A{sub s}, so that A{sub d} contains all dense blocks of a specified size in the matrix, and A{sub s} contains the remaining entries. This enables ... continued below

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Pinar, Ali & Vassilevska, Virginia August 11, 2005.

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Many applications of scientific computing rely on computations on sparse matrices. The design of efficient implementations of sparse matrix kernels is crucial for the overall efficiency of these applications. Due to the high compute-to-memory ratio and irregular memory access patterns, the performance of sparse matrix kernels is often far away from the peak performance on a modern processor. Alternative data structures have been proposed, which split the original matrix A into A{sub d} and A{sub s}, so that A{sub d} contains all dense blocks of a specified size in the matrix, and A{sub s} contains the remaining entries. This enables the use of dense matrix kernels on the entries of A{sub d} producing better memory performance. In this work, we study the problem of finding a maximum number of nonoverlapping dense blocks in a sparse matrix, which is previously not studied in the sparse matrix community. We show that the maximum nonoverlapping dense blocks problem is NP-complete by using a reduction from the maximum independent set problem on cubic planar graphs. We also propose a 2/3-approximation algorithm that runs in linear time in the number of nonzeros in the matrix. This extended abstract focuses on our results for 2x2 dense blocks. However we show that our results can be generalized to arbitrary sized dense blocks, and many other oriented substructures, which can be exploited to improve the memory performance of sparse matrix operations.

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  • Journal Name: Electric Transactions Numerical Analysis; Journal Volume: 21; Related Information: Journal Publication Date: 2005

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  • Report No.: LBNL--54498
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 886815
  • Archival Resource Key: ark:/67531/metadc876767

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  • August 11, 2005

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  • Sept. 21, 2016, 2:29 a.m.

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  • Sept. 30, 2016, 2:41 p.m.

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Pinar, Ali & Vassilevska, Virginia. Finding Nonoverlapping Substructures of a Sparse Matrix, article, August 11, 2005; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc876767/: accessed September 21, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.