A Single-Pass Algorithm for Transitive Closure

One of the most attractive methods for computing the transitive closure of a binary relation is to apply depth-rst search to the corresponding directed graph, recognize the strongly connected components, and, at the same time, compute the successor lists for all nodes. Recognizing the strongly connected components is important because the straightforward successor list computation computes correctly only the successors of the component roots, and these successor lists have to be distributed to all members of the strong components. This scheme has some obvious ineeciences: e.g., if the graph consists of a single cycle of length n, then during the search n sets of size O(n) will be computed, and, nally, the set containing all nodes will be distributed to all nodes in the cycle. A much better way would be to compute directly only the successor list of the root (the node where the search for the cycle starts). For other nodes in the cycle, the successor list is obtained directly via the root node value stored in every node. In the present paper we generalize this idea for all directed graphs: whenever the search of a node exits and we know that the node is not a component root, we will directly add the successor set of the node to the current candidate of the component root.