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Two improved algorithms for envelope and wavefront reduction

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Abstract

Two algorithms for reordering sparse, symmetric matrices or undirected graphs to reduce envelope and wavefront are considered. The first is a combinatorial algorithm introduced by Sloan and further developed by Duff, Reid, and Scott; we describe enhancements to the Sloan algorithm that improve its quality and reduce its run time. Our test problems fall into two classes with differing asymptotic behavior of their envelope parameters as a function of the weights in the Sloan algorithm. We describe an efficientO(nlogn+m) time implementation of the Sloan algorithm, wheren is the number of rows (vertices), andm is the number of nonzeros (edges). On a collection of test problems, the improved Sloan algorithm required, on the average, only twice the time required by the simpler RCM algorithm while improving the mean square wavefront by a factor of three. The second algorithm is a hybrid that combines a spectral algorithm for envelope and wavefront reduction with a refinement step that uses a modified Sloan algorithm. The hybrid algorithm reduces the envelope size and mean square wavefront obtained from the Sloan algorithm at the cost of greater running times. We illustrate how these reductions translate into tangible benefits for frontal Cholesky factorization and incomplete factorization preconditioning.

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Authors and Affiliations

  1. Department of Computer Science, Old Dominion University, 23529-0162, Norfolk, VA, USA

    Gary Kumfert & Alex Pothen

  2. NASA Langley Research Center, ICASE, 23681-0001, Hampton, VA, USA

    Alex Pothen

Authors
  1. Gary Kumfert
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  2. Alex Pothen
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Additional information

This work was partially supported by the U. S. National Science Foundation grants CCR-9412698, DMS-9505110, and ECS-9527169, by U. S. Department of Energy grant DE-FG05-94ER25216, and by the National Aeronautics and Space Administration under NASA Contract NAS1-19480 while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA.

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Kumfert, G., Pothen, A. Two improved algorithms for envelope and wavefront reduction. Bit Numer Math 37, 559–590 (1997). https://doi.org/10.1007/BF02510240

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  • DOI: https://doi.org/10.1007/BF02510240

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