Finding minimum height elimination trees for interval graphs in polynomial time
 Bengt Aspvall,
 Pinar Heggernes
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The elimination tree plays an important role in many aspects of sparse matrix factorization. The height of the elimination tree presents a rough, but usually effective, measure of the time needed to perform parallel elimination. Finding orderings that produce low elimination is therefore important. As the problem of finding minimum height elimination tree orderings is NPhard, it is interesting to concentrate on limited classes of graphs and find minimum height elimination trees for these efficiently. In this paper, we use clique trees to find an efficient algorithm for interval graphs which make an important subclass of chordal graphs. We first illustrate this method through an algorithm that finds minimum height elimination for chordal graphs. This algorithm, although of exponential time complexity, is conceptionally simple and leads to a polynomialtime algorithm for finding minimum height elimination trees for interval graphs.
 Bernstein, P. A., Goodman, N. (1981) Power of natural semijoins. SIAM J. Comput. 10: pp. 751771
 J. R. S. Blair and B. W. Peyton,An introduction to chordal graphs and clique trees, in George, Gilbert and Liu [11], pp. 1–30.
 Blair, J. R. S., Peyton, B. W. (1991) On finding minimumdiameter clique trees. Oak Ridge National Laboratory, Tennessee
 Booth, K. S., Lueker, G. S. (1976) Testing for the consecutive ones property, interval graphs, and graph planarity using PQtree algorithms. J. Comput. System Sci. 13: pp. 335379
 J. R. Bunch and D. J. Rose, eds.,Sparse Matrix Computations, Academic Press, 1976.
 E. Cuthill and J. McKee,Reducing the bandwidth of sparse symmetric matrices, in Proceedings of the 24th National Conference of the ACM, (1969), pp. 157–172.
 Deogun, J. S., Kloks, T., Kratsch, D., Müller, H. (1994) On vertex ranking for permutation and other graphs. Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science 775: pp. 747758
 Dirac, G. A. (1961) On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25: pp. 7176
 Fulkerson, D. R., Gross, O. A. (1965) Incidence matrices and interval graphs. Pacific J. Math. 15: pp. 835855
 Gavril, F. (1974) The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Combin. Theory Ser. B 16: pp. 4756
 J. A. George, J. R. Gilbert, and J. W. H. Liu, eds.,Graph Theory and Sparse Matrix Computations, IMA Volumes in Mathematics and Its Applications, Springer Verlag, 1993.
 Gilmore, P. C., Hoffman, A. J. (1964) A characterization of comparability graphs and of interval graphs. Canadian Journal of Mathematics 16: pp. 539548
 M. C. Golumbic,Algorithmic Graph Theory and Perfect Graphs, Academic Press, 1980.
 F. Harary, ed.,Proof Techniques in Graph Theory, Academic Press, 1969.
 Heggernes, P. (1992) Minimizing fillin size and elimination tree height in parallel Cholesky factorization. University of Bergen, Norway
 Ho, C. W., Lee, R. C. T. (1989) Counting clique trees and computing perfect elimination schemes in parallel. Information Processing Letter 31: pp. 6168
 Liu, J. W. H. (1990) The role of elimination trees in sparse factorization. SIAM J. Matrix Anal. Appl. 11: pp. 134172
 Manne, F. (1991) Reducing the height of an elimination tree through local reorderings. University of Bergen, Norway
 Manne, F. (1992) An algorithm for computing a minimum height elimination tree for a tree. University of Bergen, Norway
 A. Pothen,The complexity of optimal elimination trees, Tech. Report CS8813, Pennsylvania State University, 1988.
 R. E. Read, ed.,Graph Theory and Computing, Academic Press, 1972.
 F. S. Roberts,Indifference graphs, in Harary [14], pp. 139–146.
 D. J. Rose,A graphtheoretic study of the numerical solution of sparse positive definite systems of linear equation, in Read [21], pp. 183–217.
 R. E. Tarjan,Graph theory and Gaussian elimination, in Bunch and Rose [5], pp. 3–22.
 Tarjan, R. E., Yannakakis, M. (1984) Simple lineartime algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectivity reduce acyclic hypergraphs. SIAM J. Computing 8: pp. 566579
 Yannakakis, M. (1981) Computing the minimum fillin is NPcomplete. SIAM J. Alg. Disc. Meth. 2: pp. 7779
 Title
 Finding minimum height elimination trees for interval graphs in polynomial time
 Journal

BIT Numerical Mathematics
Volume 34, Issue 4 , pp 484509
 Cover Date
 19941201
 DOI
 10.1007/BF01934264
 Print ISSN
 00063835
 Online ISSN
 15729125
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 05C50
 65F50
 65Y05
 Sparse matrix computation
 elimination trees
 parallel computing
 interval graphs
 Industry Sectors
 Authors

 Bengt Aspvall ^{(1)}
 Pinar Heggernes ^{(1)}
 Author Affiliations

 1. Department of Informatics, University of Bergen, N5020, Norway