Speeding up dynamic transitive closure for bounded degree graphs

In this paper we present an algorithm for solving two problems in dynamically maintaining the transitive closure of a digraph: In the first problem a sequence of edge insertions is performed on an initially empty graph, interspersed withp transitive closure queries of the form: is there a path froma tob in the graph. Our algorithm solves this problem in timeO (dm^*+p), whered is the maximum outdegree of the resulting graphG andm^* is the number of edges in the transitive closure ofG. In the second problem, a sequence of edge deletions is performed on anacyclic graph, interspersed withp transitive closure queries. Once again we solve this problem in timeO (dm^*+p), whered is the maximum outdegree of the initial graphG andm^* is the number of edges in the transitive closure ofG. For bounded degree graphs, this improves upon previous results. Our algorithms also work when insertions and deletions to the graph are intermixed. Finally, we show how to implement the operation findpath (x, y) which retrieves some path fromx toy in time proportional to the length of the path.