Computing the transitive closure of symmetric matrices

Abstract

Certain real life binary relations are symmetric (map point connections, “iff” mathematical statements, multiprocessor links). Such relations can be represented by symmetric 0–1 matrices. Algorithms which take advantage of the symmetry when acting on such matrices, are more efficient than algorithms that are “good for all cases” by assuming a generic (non-symmetric) matrix. No algorithm, to our knowledge, focusing on symmetric matrices has been designed up to date for the computation of the transitive closure. In this paper, four algorithms - G, Symmetric, 0-1-G, 1-Symmetric - are given for computing the transitive closure of a symmetric binary relation which is represented by a 0–1 matrix. Algorithms G and 0-1-G pose no restriction on the type of the input matrix, while algorithms Symmetric and 1-Symmetric require it to be symmetric. These four algorithms are compared to Warren's algorithm in terms of the number of page faults incurred. Experimental results indicate that the new algorithms (with the exception of algorithm G) are about 2 times faster than Warren's algorithm for sparse matrices, 10 to 100 times faster for dense matrices, and about 1.4 times faster for medium dense matrices.