Efficient computation of transitive closures

Abstract We describe an efficient, time-space balanced algorithm for computation of the transitive max-min closure of a proximity relation, i.e. of a fuzzy relation that is reflexive and symmetric. The algorithm creates a binary tree representation of the transitive closure in O( m log 2 m ) time and O( m ) space, where m is the number of edges in the proximity graph. A central idea in algorithm is to order the edges after decreasing strength and then to process them in that order. For any pair of vertices, their similarity, i.e. their membership value in the transitive closure, is looked up in the tree in O(log 2 n ) time, where n is the number of vertices in the proximity graph. We compare the performance of the algorithm with the performances of two other presented algorithms that compute transitive closures of broader classes of fuzzy relations.