A faster and simpler fully dynamic transitive closure

We obtain a new fully dynamic algorithm for maintaining the transitive closure of a directed graph. Our algorithm maintains the transitive closure matrix in a total running time of O(mn+(ins+del)·n²), where ins (del) is the number of insert (delete) operations performed. Here n is the number of vertices in the graph and m is the initial number of edges in the graph. Obviously, reachability queries can be answered in constant time. The space required by the algorithm is O(n²). Our algorithm can also support path queries. If v is reachable from u, the algorithm can produce a path from u to v in time proportional to the length of the path. The best previously known algorithm for the problem is due to Demetrescu and Italiano [3]. Their algorithm has total running time of O(n³ + (ins+del)·n²). The query time is also constant. We also present an algorithm for directed acyclic graphs (DAGs) with a total running time of O(mn + ins · n² + del). Our algorithms are obtained by combining some new ideas with techniques of Italiano [7], King [8], King and Thorup [10] and Frigioni et al. [4]. We also note that our algorithms are extremely simple and can be easily implemented.