Hybridizing Nested Dissection and Halo Approximate Minimum Degree for Efficient Sparse Matrix Ordering

Minimum Degree and Nested Dissection are the two most popular reordering schemes used to reduce ll-in and operation count when factoring and solving sparse matrices. Most of the state-of-the-art ordering packages hybridize these methods by performing incomplete Nested Dissection and ordering by Minimum Degree the subgraphs associated with the leaves of the separation tree, but to date only loose couplings have been achieved, resulting in poorer performance than could have been expected. This paper presents a tight coupling of the Nested Dissection and Halo Approximate Minimum Degree algorithms, which allows the MinimumDegree algorithm to use exact degrees on the boundaries of the subgraphs passed to it, and to yield back not only the ordering of the nodes of the subgraph, but also the amalgamated assembly subtrees, for e cient block computations. Experimental results show the performance improvement, both in terms of ll-in reduction and concurrency during numerical factorization.