Fast parallel algorithms that compute transitive closure of a fuzzy relation

The notion of a transitive closure of a fuzzy relation is very useful for clustering in pattern recognition, for fuzzy databases, etc. The original algorithm proposed by L. Zadeh (1971) requires the computation time O(n(sup 4)), where n is the number of elements in the relation. In 1974, J. C. Dunn proposed a O(n(sup 2)) algorithm. Since we must compute n(n-1)/2 different values s(a, b) (a not equal to b) that represent the fuzzy relation, and we need at least one computational step to compute each of these values, we cannot compute all of them in less than O(n(sup 2)) steps. So, Dunn's algorithm is in this sense optimal. For small n, it is ok. However, for big n (e.g., for big databases), it is still a lot, so it would be desirable to decrease the computation time (this problem was formulated by J. Bezdek). Since this decrease cannot be done on a sequential computer, the only way to do it is to use a computer with several processors working in parallel. We show that on a parallel computer, transitive closure can be computed in time O((log(sub 2)(n))2).