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 most often 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 minimum degree 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 efficient block computations. Experimental results show the performance improvement of this hybridization, both in terms of fill-in reduction and increa..