Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings
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.
KeywordsSIAM Journal Gaussian Elimination Input Graph Chordal Graph Performance Guarantee
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- A. Agrawal, “Network Design and Network Cut Dualities: Approximation Algorithms and Applications,” Ph.D. thesis, Technical Report CS-91–60, Brown University (1991).Google Scholar
- H. L. Bodlaender, J. R. Gilbert, H. Hafsteinsson and T. Kloks, “Approximating treewidth, pathwidth, and minimum elimination tree height,” Technical Report CSL-90–01, Xerox Corporation, Palo Alto Research Center (1990).Google Scholar
- E. Cuthill, and J. McKee, “Reducing the bandwidth of sparse symmetric matrices,” Proceedings of the 24th National Conference of the ACM (1969), pp. 157–172.Google Scholar
- I. Duff, N. Gould, M. Lescrenier, and J. K. Reid, “The multifrontal method in a parallel environment,” in Advances in Numerical Computation, M. Cox and S. Hammarling, eds., Oxford University Press (1990).Google Scholar
- I. Duff, R. Grimes, and J. G. Lewis, “Users’ guide for the Harwell-Boeing sparse matrix collection,” Manuscript (1988).Google Scholar
- I. Duff, and J. K. Reid, Direct Methods for Sparse Matrices, Oxford University Press (1986).Google Scholar
- K. A. Gallivan et al. Parallel Algorithms for Matrix Computations, SIAM (1990).Google Scholar
- M. R. Garey and D. S. Johnson, Computers and Intractability: A guide to the theory of NP-completeness, W. H. Freeman, San Francisco (1979).Google Scholar
- George, J. A., “Computer implementation of a finite element method,” Tech. Report STANCS-208, Stanford University (1971).Google Scholar
- George, J. A., “Block elimination of finite element system of equations,” in Sparse Matrices and Their Applications, D. J. Rose and R. A. Willoughby, eds., Plenum Press (1972).Google Scholar
- George, J. A., and J. W. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall Inc. (1981).Google Scholar
- George, J. A., J. W. Liu, and E. G. Ng, “User’s guide for SPARSPAK: Waterloo sparse linear equations package,” Tech. Rep. CS78–30 (revised), Dept. of Computer Science, Univ. of Waterloo, Waterloo, Ontario, Canada (1980).Google Scholar
- J. R. Gilbert, “Some Nested Dissection Order is Nearly Optimal,” Information Processing Letters 26 (1987/88), pp. 325–328.Google Scholar
- J. R. Gilbert, personal communication (1989).Google Scholar
- J. R. Gilbert and H. Hafsteinsson, “Approximating treewidth, minimum front size, and minimum elimination tree height,” manuscript, 1989.Google Scholar
- J. R. Gilbert, and R. Schreiber, “Hightly parallel sparse Cholesky factorization,” Tech. Report CSL-90–7, Xerox Palo Alto Research Center, 1990.Google Scholar
- U. Kjwrulff, “Triangulation of graphs — Algorithms giving small total state space,” R 9009, Institute for Electronic Systems, Department of Mathematics and Computer Science, University of Aalborg (1990).Google Scholar
- P. N. Klein, “A parallel randomized approximation scheme for shortest paths,” Technical Report CS-91–56, Brown University (1991).Google Scholar
- P. N. Klein, A. Agrawal, R. Ravi and S. Rao, “Approximation through multicommodity flow,” Proceedings of the 31st Annual IEEE Conference on Foundations of Computer Science, (1990), pp. 726–737.Google Scholar
- P. N. Klein, and S. Kang, “Approximating concurrent flow with uniform demands and capacities: an implementation,” Technical Report CS-91–58, Brown University (1991).Google Scholar
- P. Klein, C. Stein and E. Tardos, “Leighton-Rao might be practical: faster approximation algorlthme for concurrent flow with uniform capacities,” Proceedings of the 22nd ACM Symposium on Theory of Computing (1990), pp. 310–321.Google Scholar
- F. T. Leighton and S. Rao, “An approximate max-flow min-cut theorem for uniform multicommodity flow problems with application to approximation algorithms,” Proceedings of the 29th Annual IEEE Conference on Foundations of Computer Science (1988), pp. 422–431.Google Scholar
- F. T. Leighton, F. Makedon and S. Tragoudas, personal communication, 1990Google Scholar
- C. Leiserson, and J. Lewis, “Orderings for parallel sparse symmetric factorization,” in Parallel Processing for Scientific Computing,G. Rodrigue, ed., Philadelphia, PA, 1987, SIAM, pp. 27–32.Google Scholar
- F. Makedon, and S. Tragoudas, “Approximating the minimum net expansion: near optimal solutions to circuit partitioning problems,” Manuscript (1991).Google Scholar
- A. Pothen, “The complexity of optimal elimination trees,” Tech. Report CS-88–16, Department of Computer Science, The Pennsylvania State University, University Park, PA, 1988.Google Scholar
- D. J. Rose, “A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations,” in Graph Theory and Computing, R. C. Read, ed., Academic Press (1972), pp. 183–217.Google Scholar