Parallel sparse matrix ordering: quality improvement using genetic algorithms

Abstract

In the direct solution of sparse symmetric and positive definite linear systems, finding an ordering of the matrix to minimize the height of elimination tree (an indication of the number of parallel elimination steps) is crucial for effectively computing the Cholesky factor in parallel. This problem is known to be NP-hard. Though many effective heuristics have been proposed, the problems of how good these heuristics are near optimal and how to further reduce the height of elimination tree remain unanswered. This paper is an effort to this investigation. We introduce a genetic algorithm customized to this parallel ordering problem, which is characterized by two novel genetic operators, adaptive merge crossover and tree rotate mutation. Experiments showed that our approach is cost effective in the number of generations evolved to reach a better solution that having considerable improvement in reducing the height of elimination tree.