Approximate Single Source Dual Fault Tolerant Distance Oracle

2026-07-03

AI Summary

This work addresses the problem of approximate single-source shortest path queries in weighted undirected graphs under up to two edge failures. It presents the first fault-tolerant distance oracle whose space complexity breaks the Ω(n²) barrier that has long been considered inherent in multi-failure settings. By integrating graph sparsification, hierarchical data structures, and failure-sensitive distance estimation techniques, the proposed oracle achieves a space usage of Õ(n√n) while supporting query time of Õ(1) and returning (1+O(ε))-approximate shortest path distances. This represents a significant improvement over existing approaches in terms of space efficiency, without compromising query speed or approximation quality.

Abstract

We are given an undirected weighted graph $G$ with $n$ vertices and $m$ edges, edge weights in $[1, W]$, and a designated source vertex $s$. We design a single source dual fault tolerant distance oracle for $G$. Given a destination vertex $t$ and a set $F$ of at most two faulty edges, the oracle returns a $(1 + O(ε))$-approximation of the weight of the shortest path from the source $s$ to $t$ avoiding $F$. Our oracle uses $\tilde{O}(n\sqrt{n})$ space and has $\tilde{O}(1)$ query time. Prior to our result, single source single fault tolerant oracles were known to return a $(1+ε)$ approximation of the weight of the shortest path using $\tilde{O}(n)$ space and $O(1)$ query time. However, extending these approaches to multiple faults remained an open problem. Indeed, all $(1+ε)$-approximate distance oracles that handle multiple faults require $Ω(n^2)$ space. We break this bound by presenting the first dual fault tolerant distance oracle with $o(n^2)$ space.

Problem

• fault tolerant
• distance oracle
• shortest path
• graph algorithms
• approximation

Innovation

• dual fault tolerant
• distance oracle
• approximate shortest paths
• subquadratic space
• single source