A Fast Algorithm for Reordering Sparse Matrices for Parallel Factorization

Abstract

Jess and Kees [IEEE Trans. Comput., C-31 (1982), pp. 231–239] introduced a method for ordering a sparse symmetric matrix A for efficient parallel factorization. The parallel ordering is computed in two steps. First, the matrix A is ordered by some fill-reducing ordering. Second, a parallel ordering of A is computed from the filled graph that results from symbolically factoring A using the initial fill-reducing ordering. Among all orderings whose fill lies in the filled graph, this parallel ordering achieves the minimum number of parallel steps in the factorization of A. Jess and Kees did not specify the implementation details of an algorithm for either step of this scheme. Liu and Mirzaian [SIAM J. Discrete Math., 2 (1989), pp. 100–107] designed an algorithm implementing the second step, but it has time and space requirements higher than the cost of computing common fill-reducing orderings. A new fast algorithm that implements the parallel ordering step by exploiting the clique tree representation of a chordal graph is presented. The cost of the parallel ordering step is reduced well below that of the fill-reducing step. This algorithm has time and space complexity linear in the number of compressed subscripts for L, i.e., the sum of the sizes of the maximal cliques of the filled graph. Running times nearly identical to Liu’s heuristic composite rotations algorithm, which approximates the minimum number of parallel steps, are demonstrated empirically.