Fault Tolerant Reachability for Directed Graphs

Let G = (V, E) be an n-vertices m-edges directed graph. Let s is an element of V be any designated source vertex, and let T be an arbitrary reachability tree rooted at s. We address the problem of finding a set of edges E subset of E\T of minimum size such that on a failure of any vertex w is an element of V, the set of vertices reachable from s in T boolean OR E\{w} is the same as the set of vertices reachable from s in G\{w}. We obtain the following results: The optimal set E for any arbitrary reachability tree T has at most n - 1 edges. There exists an O(mlog n)-time algorithm that computes the optimal set E for any given reachability tree T. For the restricted case when the reachability tree T is a Depth-First-Search (DFS) tree it is straightforward to bound the size of the optimal set E by n - 1 using semidominators with respect to DFS trees from the celebrated work of Lengauer and Tarjan [13]. Such a set E can be computed in O(m) time using the algorithm of Buchsbaum et. al [4]. To bound the size of the optimal set in the general case we define semidominators with respect to arbitrary trees. We also present a simple O(mlog n) time algorithm for computing such semidominators. As a byproduct, we get an alternative algorithm for computing dominators in O(mlog n) time.