Fault Tolerant Reachability for Directed Graphs

Abstract

Let $G=(V,E)$ be an n-vertices m-edges directed graph. Let $s\in V$ 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 $\mathcal{E}\subseteq E\backslash T$ of minimum size such that on a failure of any vertex $w\in V$, the set of vertices reachable from s in $T\cup \mathcal{E} \backslash \{w\}$ is the same as the set of vertices reachable from s in $G\backslash \{w\}$. We obtain the following results:

• The optimal set $\mathcal E$ for any arbitrary reachability tree T has at most $n-1$ edges.

• There exists an $O(m \log n)$-time algorithm that computes the optimal set $\mathcal{E}$ 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 $\mathcal{E}$ by $n-1$ using semidominators with respect to DFS trees from the celebrated work of Lengauer and Tarjan [13]. Such a set $\mathcal E$ 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 $O(m \log n)$ time algorithm for computing such semidominators. As a byproduct, we get an alternative algorithm for computing dominators in $O(m \log n)$ time.

This research was partially supported by Israel Science Foundation (ISF) and University Grants Commission (UGC) of India. The research of the second author was partially supported by Google India under the Google India PhD Fellowship Award.