Improving Parallel Ordering of Sparse Matrices using Genetic Algorithms

In the direct solution of sparse symmetric and positive definite lin-ear systems, finding an ordering of the matrix to minimize the height 1 of the elimination tree (an indication of the number of parallel elimi-nation steps) is crucial for effectively computing the Cholesky factor in parallel. This problem is known to be NP-hard. Though many ef-fective heuristics have been proposed, the problems of how good these heuristics are near optimal and how to further reduce the height of the elimination tree remain unanswered. This paper is an effort for this investigation. We introduce a genetic algorithm tailored to this parallel ordering problem, which is characterized by two novel genetic operators, adaptive merge crossover and tree rotate mutation. Exper-iments showed that our approach is cost effective in the number of generations evolved to reach a better solution in reducing the height of the elimination tree