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Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings

  • Conference paper
Graph Theory and Sparse Matrix Computation

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 56))

Abstract

In the last two decades, many heuristics have been developed for finding good elimination orderings for sparse Cholesky factorization. These heuristics aim to find elimination orderings with either low fill, low operation count, or low elimination height. Though many heuristics seem to perform well in practice, there has been a marked absence of much theoretical analysis to back these heuristics. Indeed, few heuristics are known to provide any guarantee on the quality of the elimination ordering produced for arbitrary matrices.

In this work, we present the first polynomial-time ordering algorithm that guarantees approximately optimal fill. Our algorithm is a variant of the well-known nested dissection algorithm. Our ordering performs particularly well when the number of elements in each row (and hence each column) of the coefficient matrix is small. Fortunately, many problems in practice, especially those arising from finite-element methods, have such a property due to the physical constraints of the problems being modeled.

Our ordering heuristic guarantees not only low fill, but also approximately optimal operation count, and approximately optimal elimination height. Elimination orderings with small height and low fill are of much interest when performing factorization on parallel machines. No previous ordering heuristic guaranteed even small elimination height.

We will describe our ordering algorithm and prove its performance bounds. We shall also present some experimental results comparing the quality of the orderings produced by our heuristic to those produced by two other well-known heuristics.

Some of the work reported in this paper first appeared in an extended abstract in the Proceedings of the 31st Annual IEEE Conference on the Foundations of Computer Science, 1990 [33].

Brown University, Providence, RI 02912. Research supported by NSF grant CCR-9012357 and an NSF PYI award, together with PYI matching funds from Thinking Machines Corporation and Xerox Corporation. Additional support provided by ONR and DARPA contract N00014-83-K-0146 and ARPA Order No. 6320, Amendment 1.

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Author information

Authors and Affiliations

  1. Digital Equipment Corp., Massively Parallel Systems Group, 146 Main Street, Maynard, MA, 01574, USA

    Ajit Agrawal

  2. Brown University, Providence, RI, 02912, USA

    R. Ravi

Authors
  1. Ajit Agrawal
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  2. Philip Klein
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  3. R. Ravi
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Editors and Affiliations

  1. University of Waterloo, Needles Hall, Waterloo, Ontario, N2L 3G1, Canada

    Alan George

  2. Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, CA, 94304-1314, USA

    John R. Gilbert

  3. Department of Computer Science, York University, North York, Ontario, M3J 1P3, Canada

    Joseph W. H. Liu

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© 1993 Springer-Verlag New York, Inc.

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Agrawal, A., Klein, P., Ravi, R. (1993). Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds) Graph Theory and Sparse Matrix Computation. The IMA Volumes in Mathematics and its Applications, vol 56. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8369-7_2

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  • DOI: https://doi.org/10.1007/978-1-4613-8369-7_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8371-0

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