Improved Dynamic Reachability Algorithms for Directed Graphs

Abstract

We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the transitive closure of a directed graph, through an arbitrary sequence of edge deletions, in $O(mn)$ total expected time, essentially the time needed for computing the transitive closure of the initial graph. Such a result was previously known only for acyclic graphs. (ii) Two fully dynamic algorithms for answering reachability queries. The first is deterministic and has an amortized insert/delete time of $O(m\sqrt{n})$, and worst-case query time of $O(\sqrt{n})$. The second is randomized and has an amortized insert/delete time of $O(m^{0.58}n)$ and worst-case query time of $O(m^{0.43})$. This significantly improves the query times of algorithms with similar update times. (iii) A fully dynamic algorithm for maintaining the transitive closure of an acyclic graph. The algorithm is deterministic and has a worst-case insert time of $O(m)$, constant amortized delete time of $O(1)$, and a worst-case query time of $O(n/\log n)$. Our algorithms are obtained by combining several new ideas, one of which is a simple sampling idea used for detecting decompositions of strongly connected components, with techniques of Even and Shiloach [J. ACM, 28 (1981), pp. 1-4], Italiano [Inform. Process. Lett., 28 (1988), pp. 5-11], Henzinger and King [Proceedings of the $36$th Annual Symposium on Foundations of Computer Science, Milwaukee, WI, 1995, pp. 664-672], and Frigioni et al. [ACM J. Exp. Algorithmics, 6 (2001), (electronic)].