Minimal Orderings Revisited

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.