Parallel Output-Sensitive Algorithms for Combinatorial and Linear Algebra Problems

Abstract

This paper gives output-sensitive parallel algorithms whose performance depends on the output size and are significantly more efficient tan previous algorithms for problems with sufficiently small output size. Inputs are n×n matrices over a fixed ground field. Let P(n) and M(n) be the PRAM processor bounds for O(logn) time multiplication of two degree n polynomials, and n×n matrices, respectively. Let T(n) be the time bounds, using M(n) processors, for testing if an n×n matrix is nonsingular, and if so, computing its inverse. We compute the rankR of a matrix in randomized parallel time O(logn+T(R)logR) using nP(n)+M(R) processors (P(n)+RP(R) processors for constant displacement rank matrices, e.g., Toeplitz matrices). We find a maximum linearly independent subset (MLIS) of an n-set of n-dimensional vectors in time O(T(n)logn) using M(n) randomized processors and we also give output-sensitive algorithms for this problem. Applications include output-sensitive algorithms for finding: (i) a size R maximum matching in an n-vertex graph using time O(T(R)logn) and nP(n)/T(R)+RM(R) processors, and (ii) a maximum matching in an n-vertex bipartite graph, with vertex subsets of sizes n1⩽n2, using time O(T(n1)logn) and nP(n)/T(n1)+n1M(n1) processors.