Approximate Distance Oracles with Improved Query Time

TL;DR

This paper introduces improved approximate distance oracles for graphs that significantly reduce query time and size while maintaining good approximation ratios, advancing the efficiency of distance queries in large graphs.

Contribution

It presents new constructions of approximate distance oracles with faster query times and better size-approximation tradeoffs, improving upon prior work by Thorup, Zwick, Mendel, and Naor.

Findings

Achieves $O(rac{1}{ ext{epsilon}})$ query time with $(2+ ext{epsilon})k$ approximation.

Constructs oracles with size $O(kn^{1+1/k})$ and improved query times.

Approaches the theoretical best tradeoff between size and stretch under a girth conjecture.

Abstract

Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 2$, we show that a $(2k-1)$-approximate distance oracle for $G$ of size $O(kn^{1 + 1/k})$ and with $O(\log k)$ query time can be constructed in $O(\min\{kmn^{1/k},\sqrt km + kn^{1 + c/\sqrt k}\})$ time for some constant $c$. This improves the $O(k)$ query time of Thorup and Zwick. Furthermore, for any $0 < \epsilon \leq 1$, we give an oracle of size $O(kn^{1 + 1/k})$ that answers $((2 + \epsilon)k)$-approximate distance queries in $O(1/\epsilon)$ time. At the cost of a $k$-factor in size, this improves the $128k$ approximation achieved by the constant query time oracle of Mendel and Naor and approaches the best possible tradeoff between size and stretch, implied by a widely believed girth conjecture of Erd\H{o}s. We can match the $O(n^{1 + 1/k})$ size bound of Mendel and Naor for any constant $\epsilon > 0$ and $k = O(\log n/\log\log n)$.