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 2 V be a designated source. In this paper we address several variants of the problem of maintaining the (1 +ϵ)-approximate shortest path from s to each v 2 V n fsg 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 Õ (nL) edges such that for any x; v 2 V, the graph H n x contains a path whose length is a (1 +ϵ)-approximation of the length of the shortest path from s to v in G n x. We show that the size of the subgraph H is optimal (up to logarithmic factors) by proving a lower bound of (nL/ϵ) 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 (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 (n1:5). Therefore, a (1 +ϵ)-approximation is the best one can hope for. We consider also the data structure problem and show that there exists an O(nL/ϵ) size oracle that for any v 2 V reports a (1 +ϵ)-approximate distance of v from s on a failure of any x 2 V in O(log log1+ϵ (nW)) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of (nL/log n). Finally, we present two distributed algorithms. We present a single source routing scheme that can route on a (1 +ϵ)-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/ϵ) bits. We present also a labeling scheme that assigns each vertex a label of eO (L/ϵ) bits. For any two vertices x; v 2 V the labeling scheme outputs a (1 +ϵ)-approximation of the distance from s to v in Gnx using only the labels of x and v.