On computing the transitive closure of a relation

Summary

An algorithm is presented for computing the transitive closure of an arbitrary relation which is based upon Tarjan's algorithm [7] for finding the strongly connected components of a directed graph. A new formulation, justifying a somewhat simplified statement of the latter, characterises weaker restrictions on the form of the graph traversal than Tarjan's depth first conditions and reveals aspects of the behaviour of this algorithm which have been obscure hitherto.

If V is the number of vertices in the directed graph representing the relation then the worst case behaviour, O(V ³) is inferior to existing algorithms [1, 2] which require O(V ³/log V) and $O(V^{log_2 7} log V)$ operations respectively. The best case performance, O(V ²) operations, is better. Viewed in this way, it is similar to other algorithms [5, 6, 8] but it combines the improved efficiency in the presence of strongly connected components which characterises the algorithms in [5, 6] with the advantages of Warshall's algorithm [8], namely, succinctness, a single traversal of the directed graph and ability to exploit the availability of Boolean vector operations.