Compact and Fast Sensitivity Oracles for Single-Source Distances

TL;DR

This paper introduces two efficient single-source sensitivity oracles for approximate shortest path distances in weighted graphs, achieving a trade-off between size, accuracy, and query time, with proven lower bounds for related structures.

Contribution

The paper presents novel compact sensitivity oracles with optimal size and stretch, including a 2-approximate oracle in constant time and a near-optimal $(1+psilon)$-approximate oracle with logarithmic query time.

Findings

A size-O(n) oracle reports 2-approximate distances in O(1) time.

A size-O(n/epsilon log(1/epsilon)) oracle reports (1+epsilon)-approximate distances in O(log n/epsilon log(1/epsilon)) time.

Lower bounds established for additively-stretched sensitivity oracles.

Abstract

Let $s$ denote a distinguished source vertex of a non-negatively real weighted and undirected graph $G$ with $n$ vertices and $m$ edges. In this paper we present two efficient \emph{single-source approximate-distance sensitivity oracles}, namely \emph{compact} data structures which are able to \emph{quickly} report an approximate (by a multiplicative stretch factor) distance from $s$ to any node of $G$ following the failure of any edge in $G$. More precisely, we first present a sensitivity oracle of size $O(n)$ which is able to report 2-approximate distances from the source in $O(1)$ time. Then, we further develop our construction by building, for any $0<\epsilon<1$, another sensitivity oracle having size $O\left(n\cdot \frac{1}{\epsilon} \log \frac{1}{\epsilon}\right)$, and which is able to report a $(1+\epsilon)$-approximate distance from $s$ to any vertex of $G$ in $O\left(\log n\cdot \frac{1}{\epsilon} \log \frac{1}{\epsilon}\right)$ time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source \emph{additively-stretched} sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.