Maximum Cardinality Search for Computing Minimal Triangulations
WG '02 Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science
Efficient Implementation of a Minimal Triangulation Algorithm
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Computing minimal triangulations in time O(nα log n) = o(n2.376)
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
A wide-range algorithm for minimal triangulation from an arbitrary ordering
Journal of Algorithms
Theoretical Computer Science
A wide-range algorithm for minimal triangulation from an arbitrary ordering
Journal of Algorithms
Treewidth computations I. Upper bounds
Information and Computation
Characterizing minimal interval completions towards better understanding of profile and pathwidth
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Simple and efficient modifications of elimination orderings
PARA'04 Proceedings of the 7th international conference on Applied Parallel Computing: state of the Art in Scientific Computing
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When minimum orderings proved too difficult to deal with, Rose, Tarjan, and Lueker instead studied minimal orderings and how to compute them [SIAM J. Comput., 5 (1976), pp. 266--283]. This paper introduces an algorithm that is capable of computing much better minimal orderings much more efficiently than the algorithm of Rose, Tarjan, and Lueker. The new insight is a way to use certain structures and concepts from modern sparse Cholesky solvers to reexpress one of the basic results of Rose, Tarjan, and Lueker. The new algorithm begins with any initial ordering and then refines it until a minimal ordering is obtained. It is simple to obtain high-quality low-cost minimal orderings by using fill-reducing heuristic orderings as initial orderings for the algorithm. We examine several such initial orderings in some detail. Our results here and previous work by others indicate that the improvements obtained over the initial heuristic orderings are relatively small because the initial orderings are minimal or nearly minimal. Nested dissection orderings provide some significant exceptions to this rule.