Parallel Sparse PLUQ Factorization modulo p

Abstract

In this paper, we present the results of our experiments to compute the rank of several large sparse matrices from Dumas's Sparse Integer Matrix Collection, by computing sparse PLUQ factorizations. Our approach consists in identifying as many pivots as possible before performing any arithmetic operation, based solely on the location of non-zero entries in the input matrix. These "structural" pivots are then all eliminated in parallel, in a single pass. We describe several heuristic structural pivot selection algorithms (the problem is NP-hard). These algorithms allows us to compute the ranks of several large sparse matrices in a few minutes, versus many days using Wiedemann's algorithm. Lastly, we describe a multi-thread implementation using OpenMP achieving 70% parallel efficiency on 24 cores on the largest benchmark.