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] for some parameter W, and sμ 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 vμ 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} nlog W) edges such that for any x,vμ 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 (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 φ(ϵ -1 n log W) size oracle that for any vμ V reports a (1+ϵ)-approximate distance of v from s on a failure of any xμ 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 nlog 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 -1log W) bits. For any two vertices x,vμ 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.