Compact and Fast Sensitivity Oracles for Single-Source Distances

Davide Bilò; Luciano Gualà; Stefano Leucci; Guido Proietti

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arXiv · 1608.04769

Compact and Fast Sensitivity Oracles for Single-Source Distances

Davide Bilò·Luciano Gualà·Stefano Leucci·Guido Proietti

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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.

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Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti. 2016-08-16. Compact and Fast Sensitivity Oracles for Single-Source Distances. https://arxiv.org/abs/1608.04769

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