Making an arbitrary filled graph minimal by removing fill edges
We consider the problem of removing fill edges from a filled graph G′ to get a minimal chordal supergraph M of the original graph G; thus G\(\subseteq\)M\(\subseteq\)G′. We show that a greedy strategy can be applied if fill edges are processed for removal in the reverse order of their introduction. For a filled graph with f fill edges and e original edges, we give a simple O(f(e + f)) algorithm which solves the problem and computes a corresponding minimal elimination ordering. We believe that in practice the runtime of our algorithm is usually better than the worst-case bound of O(f(e+f)).
Unable to display preview. Download preview PDF.
- 1.A.Agrawal, P.Klein, and R.Ravi. Cutting down on fill using nested dissection: provably good elimination orderings. In J. A. George, J. R. Gilbert, and J. W. H. Liu, editors, Sparse matrix computations: Graph theory issues and algorithms, pages 31–55. Springer Verlag, 1993. IMA Volumes in Mathematics and its Applications, Vol. 56.Google Scholar
- 2.C. Beeri, R. Fagin, D. Maier, and M. Yannakakis. On the desirability of acyclic database systems. J. Assoc. Comput. Mach., 30:479–513, 1983.Google Scholar
- 3.F. R. K. Chung and D. Mumford. Chordal completions of planar graphs. J. Comb. Theory, 31:96–106, 1994.Google Scholar
- 4.E. Dahlhaus and M. Karpinski. An efficient parallel algorithm for the minimal elimination ordering of an arbitrary graph. Proceedings FOCS, pages 454–459, 1989.Google Scholar
- 5.R. E. England, J. R. S. Blair, and M. G. Thomason. Independent computations in a probablistic knowledge-based system. Technical Report CS-90-128, Department of Computer Science, The University of Tennessee, Knoxville, Tennessee, 1991.Google Scholar
- 6.A. George and J.W-H. Liu. Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1981.Google Scholar
- 7.S. L. Lauritzen and D. J. Spiegelhalter. Local computations with probabilities on graphical structures and their applications to expert systems. J. Royal Statist. Soc., ser B, 50:157–224, 1988.Google Scholar
- 8.J. G. Lewis, B. W. Peyton, and A. Pothen. A fast algorithm for reordering sparse matrices for parallel factorization. SIAM J. Sci. Stat. Comput., 10:1156–1173, 1989.Google Scholar
- 9.J. W-H. Liu and A. Mirzaian. A linear reordering algorithm for parallel pivoting of chordal graphs. SIAM J. Disc. Math., 2:100–107, 1989.Google Scholar
- 10.T. Ohtsuki, L.K. Cheung, and T. Fujisawa. Minimal triangulation of a graph and optimal pivoting ordering in a sparse matrix. J. Math. Anal. Appl., 54:622–633, 1976.Google Scholar
- 11.B. W. Peyton. Some applications of clique trees to the solution of sparse linear systems. PhD thesis, Dept. of Mathematical Sciences, Clemson University, 1986.Google Scholar
- 12.D. J. Rose. A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In R. C. Read, editor, Graph Theory and Computing, pages 183–217. Academic Press, 1972.Google Scholar
- 15.M. Yannakakis. Computing the minimum fill-in is NP-complete. SIAM J. Alg. Disc. Meth., 2:77–79, 1981.Google Scholar