A fast algorithm for reordering sparse matrices for parallel factorization
Jess and Kees 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 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 (1987) designed an algorithm implementing the second step, but it has time and space requirements higher than the cost of computing common fill-reducing orderings. We present here a new fast algorithm that implements the parallel ordering step by exploiting the clique tree representation of a chordal graph. We succeed in reducing the cost of the parallel ordering step well below that of the fill-reducing step. Our algorithm has time and space complexity linear in the number of compressed subscripts of L, i.e., the sum of the sizes of the maximal cliques of the filled graph. Empirically we demonstrate running times nearly identical to Liu's heuristic Composite Rotations algorithm that approximates the minimum number of parallel steps. 21 refs., 3 figs., 4 tabs.
- Research Organization:
- Oak Ridge National Lab., TN (USA)
- DOE Contract Number:
- AC05-84OR21400
- OSTI ID:
- 6487761
- Report Number(s):
- ORNL/TM-11040; ON: DE89007873
- Resource Relation:
- Other Information: Portions of this document are illegible in microfiche products
- Country of Publication:
- United States
- Language:
- English
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