Approximate Shortest Paths Avoiding a Failed Vertex: Optimal Size Data Structures for Unweighted Graph

TL;DR

This paper introduces compact data structures for unweighted graphs that efficiently answer approximate shortest path queries avoiding a failed vertex, with sizes close to static solutions and optimal query times.

Contribution

It presents novel data structures for approximate shortest paths avoiding a failed vertex, achieving near-static size and optimal query performance.

Findings

Data structures for single-source and all-pairs queries

Optimal query time guarantees

Size nearly matches static counterparts

Abstract

Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path $\textbf{P}$ between any two vertices $u,v\in V$ is said to be $t$-approximate shortest path if its length is at most $t$ times the length of the shortest path between $u$ and $v$. We consider the problem of building a compact data structure for a given graph $G$ which is capable of answering the following query for any $u,v,z\in V$ and $t>1$: Report $t$-approximate shortest path between $u$ and $v$ when vertex $z$ fails We present data structures for the single source as well all-pairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size {\em nearly} match the size of their best static counterparts.