Distributed Distance Sensitivity Oracles

We present results for the distance sensitivity oracle (DSO) problem, where one needs to preprocess a given directed weighted graph \(G=(V,E)\) in order to answer queries about the shortest path distance from \(s\) to \(t\) in \(G\) that avoids edge \(e\), for any \(s,t \in V, e \in E\). No non-trivial results are known for DSO in the distributed CONGEST model even though it is of importance to maintain efficient communication under an edge failure. Let \(n=|V|\), and let \(D\) be the undirected diameter of \(G\). Our first DSO algorithm optimizes query response rounds and can answer a batch of any \(k\geq 1\) queries in \(O(k+D)\) rounds after taking \(\tilde{O}(n^{3/2})\) rounds to preprocess \(G\). Our second algorithm takes \(\tilde{O}(n)\) rounds for preprocessing, and then it can answer any batch of \(k\geq 1\) queries in \(\tilde{O}(k\sqrt{n}+D)\) rounds. We complement these algorithms with some unconditional CONGEST lower bounds that give trade-offs between preprocessing rounds and rounds needed to answer queries. Additionally, we present almost-optimal upper and lower bounds for the related all pairs second simple shortest path (2-APSiSP) problem, where for all pairs of vertices \(x,y \in V\), we need to compute the minimum weight of a simple \(x\)-\(y\) path that differs from the precomputed \(x\)-\(y\) shortest path by at least one edge.