Approximate Single-Source Fault Tolerant Shortest Path

Abstract

Let G=(V,E) be an n -vertices m -edges directed graph with edge weights in the range [1, W ] for some parameter W , and sϵ V be a designated source. In this article, we address several variants of the problem of maintaining the (1+ε)-approximate shortest path from s to each v ϵ 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 Õ(ε -1 } n log W ) edges such that for any x,vϵ V , the graph H \ x contains a path whose length is a (1+ε)-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 (ε -1 n log W ) edges. Demetrescu, Thorup, Chowdhury, and Ramachandran (SICOMP 2008) showed that the size of a fault tolerant exact shortest path subgraph in weighted directed/undirected graphs is Ω ( m ). Parter and Peleg (ESA 2013) 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 ( n 1.5 ). 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 ϕ(ε -1 n log W ) size oracle that for any vϵ V reports a (1+ε)-approximate distance of v from s on a failure of any xϵ V in O(log log 1+ε ( nW )) time. We show that the size of the oracle is optimal (up to logarithmic factors) by proving a lower bound of Ω (ε -1 n log W log -1 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 ϕ(ε -1 log W ) bits. We present also a labeling scheme that assigns each vertex a label of ϕ(ε -1 log W ) bits. For any two vertices x,vϵ V , the labeling scheme outputs a (1+ε)-approximation of the distance from s to v in G \ x using only the labels of x and v .