Approximate Distance Oracles Subject to Multiple Vertex Failures

TL;DR

This paper introduces new approximate distance oracles that efficiently handle multiple vertex failures in undirected graphs, providing fast query responses with small stretch factors and polynomial preprocessing time.

Contribution

It presents the first poly-logarithmic query time approximate distance oracles for any constant number of vertex failures in general undirected graphs, with improved size and stretch guarantees.

Findings

Two types of oracles with different size and stretch trade-offs

Query time is polynomial in logarithmic factors and number of failures

Preprocessing time is polynomial in space complexity

Abstract

Given an undirected graph $G=(V,E)$ of $n$ vertices and $m$ edges with weights in $[1,W]$, we construct vertex sensitive distance oracles (VSDO), which are data structures that preprocess the graph, and answer the following kind of queries: Given a source vertex $u$, a target vertex $v$, and a batch of $d$ failed vertices $D$, output (an approximation of) the distance between $u$ and $v$ in $G-D$ (that is, the graph $G$ with vertices in $D$ removed). An oracle has stretch $\alpha$ if it always holds that $\delta_{G-D}(u,v)\le\tilde{\delta}(u,v)\le\alpha\cdot\delta_{G-D}(u,v)$, where $\delta_{G-D}(u,v)$ is the actual distance between $u$ and $v$ in $G-D$, and $\tilde{\delta}(u,v)$ is the distance reported by the oracle. In this paper we construct efficient VSDOs for any number $d$ of failures. For any constant $c\geq 1$, we propose two oracles: $\bullet$ The first oracle has size $n^{2+1/c}(\log n/\epsilon)^{O(d)}\cdot \log W$, answers a query in ${\rm poly}(\log n,d^c,\log\log W,\epsilon^{-1})$ time, and has stretch $1+\epsilon$, for any constant $\epsilon>0$. $\bullet$ The second oracle has size $n^{2+1/c}{\rm poly}(\log (nW),d)$, answers a query in ${\rm poly}(\log n,d^c,\log\log W)$ time, and has stretch ${\rm poly}(\log n,d)$. Both of these oracles can be preprocessed in time polynomial in their space complexity. These results are the first approximate distance oracles of poly-logarithmic query time for any constant number of vertex failures in general undirected graphs. Previously there are $(1+\epsilon)$-approximate $d$-edge sensitive distance oracles [Chechik et al. 2017] answering distance queries when $d$ edges fail, which have size $O(n^2(\log n/\epsilon)^d\cdot d\log W)$ and query time ${\rm poly}(\log n, d, \log\log W)$.