Distributed Distance Sensitivity Oracles

TL;DR

This paper introduces distributed algorithms for distance sensitivity oracles in directed graphs, providing tradeoffs between preprocessing and query efficiency, along with lower bounds and results for related shortest path problems.

Contribution

It presents the first distributed DSO algorithms with different tradeoffs, along with unconditional lower bounds in the CONGEST model and bounds for related shortest path problems.

Findings

Distributed DSO algorithms with optimized query response rounds.

Preprocessing-focused DSO algorithms with reduced query complexity.

Unconditional lower bounds for DSO and related shortest path problems.

Abstract

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 in $G$ from vertex $s$ to vertex $t$ avoiding edge $e$, for any $s,t \in V, e \in E$. DSO enables optimal re-routing under a link failure, and can serve as a key component for fault tolerance in a distributed setting. However, no non-trivial results for DSO are known in the distributed CONGEST model. We present DSO algorithms with different tradeoffs between preprocessing and query cost: one that optimizes query response rounds, and another that prioritizes preprocessing rounds. We complement these algorithms with unconditional CONGEST lower bounds for DSO. Our DSO lower bounds build on a lower bound we present for the $k$-source shortest paths problem ($k$-SSP), which may be of independent interest. Additionally, we present almost-optimal upper and lower bounds for the related all pairs second simple shortest path (2-APSiSP) problem.