Partitioning sparse matrices with eigenvectors of graphs

Journal Article · Sun Jul 01 04:00:00 EDT 1990 · SIAM Journal on Matrix Analysis and Applications

DOI:https://doi.org/10.1137/0611030· OSTI ID:6696852

Pothen, A.; Simon, H. D.; Liou, K. ^[1]

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1. Pennsylvania State Univ., Univ. Park, PA (United States) NASA Ames Research Center, Moffett Field, CA (United States)

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorithms for computing separators. Finally, the time required to compute the Laplacian eigenvector is reported, and the accuracy with which the eigenvector must be computed to obtain good separators is considered. The spectral algorithm has the advantage that it can be implemented on a medium-size multiprocessor in a straightforward manner.

• Research Organization: National Aeronautics and Space Administration, Moffett Field, CA (United States). Ames Research Center
• Sponsoring Organization: USDOE
• OSTI ID: 6696852
• Journal Information: SIAM Journal on Matrix Analysis and Applications, Journal Name: SIAM Journal on Matrix Analysis and Applications Vol. 11:3; ISSN SJMAEL; ISSN 0895-4798
• Publisher: SIAM
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
990200* -- Mathematics & Computers
ALGORITHMS
FACTORIZATION
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PARALLEL PROCESSING
PROGRAMMING