On Optimal Fill-Preserving Orderings of Sparse Matrices for Parallel Cholesky Factorizations

In this paper, we consider the problem of nding llpreserving ordering of a sparse symmetric and positive de nite matrix such that the reordered matrix is suitable for parallel factorization. We extended the unitcost ll-preserving ordering into a generalized class that can adopt various aspects in parallel factorization, such as computation, communication and algorithmic diversity. Based on the elimination tree model, we show that as long as the node cost function for factoring a column/row satis es two mandatory properties, we can deploy a greedy-based algorithm to nd the corresponding optimal ordering. The complexity of our algorithm is O(q log q+ ), where q denote the number of maximal cliques, and the sum of all maximal clique sizes in the lled graph. Our experiments reveal that on the average, our minimum completion cost ordering (MinCP) would reduce up to 17% the cost to factor than minimum height ordering (Jess-Kees).