Approximate Single-Source Fault Tolerant Shortest Path

Let G = (V, E) be an n-vertices m-edges directed graph with edge weights in the range [1, W] and L = log(W). Let s ∈ V be a designated source. In this paper we address several variants of the problem of maintaining the (1 + ∈)-approximate shortest path from s to each v ∈ V \ {s} in the presence of a failure of an edge or a vertex. From the graph theory perspective we show that G has a subgraph H with O(nL/∈) edges such that for any x, v ∈ V, the graph H \ x contains a path whose length is a (1 + ∈)-approximation of the length of the shortest path from s to v in G \ x. We show that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/∈) edges. Demetrescu, Thorup, Chowdhury and Ramachandran [12] showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is Ω(m). Parter and Peleg [18] showed that even in the restricted case of unweighted undirected graphs the size of any subgraph for the exact shortest path is at least Ω(n1.5). Therefore, a (1 + ∈)-approximation is the best one can hope for. We consider also the data structure problem and show that there exists an O(nL/∈) size oracle that for any v ∈ V reports a (1 + ∈)-approximate distance of v from s on a failure of any x ∈ V in O(log log1+∈(n W)) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of Ω(nL/∈ log n). Finally, we present two distributed algorithms. We present a single source routing scheme that can route on a (1 + ∈)-approximation of the shortest path from a fixed source s to any destination t in the presence of a fault. Each vertex has a label and a routing table of O(L/∈) bits. We present also a labeling scheme that assigns each vertex a label of O(L/∈) bits. For any two vertices x, v ∈ V the labeling scheme outputs a (1 + ∈)-approximation of the distance from s to v in G \ x using only the labels of x and v.