On optimal fill-preserving orderings of sparse matrices for parallel Cholesky factorizations

In this paper, we consider the problem of finding fill-preserving ordering of a sparse symmetric and positive definite matrix such that the reordered matrix is suitable for parallel factorization. We extended the unit-cost fill-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 satisfies two mandatory properties, we can deploy a greedy-based algorithm to find the corresponding optimal ordering. The complexity of our algorithm is O(q log q+κ), where q denotes the number of maximal cliques, and κ the sum of all maximal clique sizes in the filled 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)