Fault tolerant reachability for directed graphs
Abstract Let G=(V, E) be an n-vertices m-edges directed graph. Let s ∈ 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 ⊆ E \ T of minimum size such that on a failure of any vertex w ∈
V, the set of vertices reachable from s in T ∪ E \ {w\} is the same as the set of vertices
reachable from s in G \ {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 …
source vertex, and let T be an arbitrary reachability tree rooted at s. We address the problem
of finding a set of edges E ⊆ E \ T of minimum size such that on a failure of any vertex w ∈
V, the set of vertices reachable from s in T ∪ E \ {w\} is the same as the set of vertices
reachable from s in G \ {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 …
Abstract
Let be an n-vertices m-edges directed graph. Let 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 of minimum size such that on a failure of any vertex , the set of vertices reachable from s in is the same as the set of vertices reachable from s in . We obtain the following results:
- The optimal set for any arbitrary reachability tree T has at most edges.
- There exists an -time algorithm that computes the optimal set 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 by using semidominators with respect to DFS trees from the celebrated work of Lengauer and Tarjan [13]. Such a set 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 time algorithm for computing such semidominators. As a byproduct, we get an alternative algorithm for computing dominators in time.
Springer
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