Parallel Ordering Using Edge Contraction

Computing a ll-reducing ordering of a sparse matrix is a central problem in the solution of sparse linear systems using direct methods. In recent years, there has been signiicant research in developing a sparse direct solver suitable for message-passing multiprocessors. However, computing the ordering step in parallel remains a challenge and there are very few methods available. This paper describes a new scheme called parallel contracted ordering which is a combination of a new parallel nested dissection heuristic and any serial ordering method. The new nested dissection heuristic called Shrink-Split ND (SSND) is based on parallel graph contraction. For a system with N unknowns, the complexity of SSND is O ? N P logP using P processors in a hypercube; the overall complexity is O ? N P log N when the serial ordering method chosen is graph exploration based nested dissection. We provide extensive empirical results on the quality of the ordering. We also report on the parallel performance of a preliminary implementation on three diierent message passing multiprocessors.