Maintaining Exact Distances under Multiple Edge Failures

TL;DR

This paper introduces a novel compact distance oracle capable of providing exact shortest path distances in undirected weighted graphs even after multiple arbitrary edge failures, balancing space and query efficiency.

Contribution

It presents the first exact distance oracle that tolerates multiple failures with a compact space complexity and efficient query time, advancing beyond previous approximate solutions.

Findings

Space complexity is O(d n^4)

Query time is d^{O(d)}

Preprocessing time is likely n^{Ω(d)}

Abstract

We present the first compact distance oracle that tolerates multiple failures and maintains exact distances. Given an undirected weighted graph $G = (V, E)$ and an arbitrarily large constant $d$, we construct an oracle that given vertices $u, v \in V$ and a set of $d$ edge failures $D$, outputs the exact distance between $u$ and $v$ in $G - D$ (that is, $G$ with edges in $D$ removed). Our oracle has space complexity $O(d n^4)$ and query time $d^{O(d)}$. Previously, there were compact approximate distance oracles under multiple failures [Chechik, Cohen, Fiat, and Kaplan, SODA'17; Duan, Gu, and Ren, SODA'21], but the best exact distance oracles under $d$ failures require essentially $\Omega(n^d)$ space [Duan and Pettie, SODA'09]. Our distance oracle seems to require $n^{\Omega(d)}$ time to preprocess; we leave it as an open question to improve this preprocessing time.