A Fast Reordering Algorithm for Parallel Sparse Triangular Solution

A space-efficient partitioned representation of the inverse of a unit lower triangular matrix L may be used for efficiently solving sparse triangular systems on massively parallel computers. The number of steps required in the parallel triangular solution is equal to the number of subsets of elementary triangular matrices in the partitioned representation of the inverse. Alvarado and Schreiber have recently described two partitioning algorithms that compute the minimum number of subsets in the partition over all permutations of L which preserve the lower triangular structure of the matrix. Their algorithms require space linear and time nonlinear in the number of nonzeros in L. This paper describes a partitioning algorithm that requires only $\mathcal{O}(n)$ time and space for computing an optimal partition, when L is restricted to be a Cholesky factor. (Here n is the order of L.) The savings result from the observation that instead of working with the structure of L, it is sufficient to work with its tran...

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