Approximating Approximate Distance Oracles

TL;DR

This paper investigates how to optimize approximate distance oracles for specific metric spaces, providing approximation algorithms and bounds for well-known classes like Thorup-Zwick and Ptrau-Roditty.

Contribution

It introduces approximation algorithms for finding minimal size oracles within certain classes, with proven bounds and integrality gaps, and extends to outlier scenarios using SDP relaxations.

Findings

O(( log n))-approximation for Thorup-Zwick and Ptrau-Roditty oracles

Matching ( log n)) lower bounds for these problems

Approximation of oracles with outliers using SDP relaxations

Abstract

Given a finite metric space $(V,d)$, an approximate distance oracle is a data structure which, when queried on two points $u,v \in V$, returns an approximation to the the actual distance between $u$ and $v$ which is within some bounded stretch factor of the true distance. There has been significant work on the tradeoff between the important parameters of approximate distance oracles (and in particular between the size, stretch, and query time), but in this paper we take a different point of view, that of per-instance optimization. If we are given an particular input metric space and stretch bound, can we find the smallest possible approximate distance oracle for that particular input? Since this question is not even well-defined, we restrict our attention to well-known classes of approximate distance oracles, and study whether we can optimize over those classes. In particular, we give an $O(\log n)$-approximation to the problem of finding the smallest stretch $3$ Thorup-Zwick distance oracle, as well as the problem of finding the smallest P\v{a}tra\c{s}cu-Roditty distance oracle. We also prove a matching $\Omega(\log n)$ lower bound for both problems, and an $\Omega(n^{\frac{1}{k}-\frac{1}{2^{k-1}}})$ integrality gap for the more general stretch $(2k-1)$ Thorup-Zwick distance oracle. We also consider the problem of approximating the best TZ or PR approximate distance oracle \emph{with outliers}, and show that more advanced techniques (SDP relaxations in particular) allow us to optimize even in the presence of outliers.