Approximate Distance Oracles with Improved Preprocessing Time

TL;DR

This paper introduces improved algorithms for constructing approximate distance oracles in graphs, significantly reducing preprocessing time while maintaining accuracy, especially for larger graphs and specific parameters.

Contribution

It presents new approximate distance oracles with faster preprocessing times that surpass previous bounds for a wide range of parameters, under certain conjectures.

Findings

Breaks quadratic preprocessing time bound for k ≥ 6

Improves Thorup and Zwick's time bound for dense graphs

Achieves near-optimal performance under girth conjecture

Abstract

Given an undirected graph $G$ with $m$ edges, $n$ vertices, and non-negative edge weights, and given an integer $k\geq 1$, we show that for some universal constant $c$, a $(2k-1)$-approximate distance oracle for $G$ of size $O(kn^{1 + 1/k})$ can be constructed in $O(\sqrt km + kn^{1 + c/\sqrt k})$ time and can answer queries in $O(k)$ time. We also give an oracle which is faster for smaller $k$. Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all $k\geq 6$ and improve the $O(kmn^{1/k})$ time bound of Thorup and Zwick except for very sparse graphs and small $k$. When $m = \Omega(n^{1 + c/\sqrt k})$ and $k = O(1)$, our oracle is optimal w.r.t.\ both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erd\H{o}s.