Finding Optimal Ordering of Sparse Matrices for Column-Oriented Parallel Cholesky Factorization

Abstract

In this paper, we consider the problem of finding fill-preserving sparse matrix orderings for parallel factorization. That is, given a large sparse symmetric and positive definite matrix A that has been ordered by some fill-reducing ordering, we want to determine a reordering that is appropriate in terms of preserving the sparsity and minimizing the cost to perform the Cholesky factorization in parallel. Past researches on this problem all are based on the elimination tree model, in which each node represents the task for factoring a column, and thus, can be seen as a coarse-grained task dependence model. To exploit more parallelism, Joseph Liu proposed a medium-grained task model, called the column task graph, and showed that it is amenable to the shared-memory supercomputers. Based on the column task graph, we devise a greedy reordering algorithm, and show that our algorithm can find the optimal ordering among the class of all fill-preserving orderings of the given sparse matrix A.