Improved Approximate Distance Oracles: Bypassing the Thorup-Zwick Bound in Dense Graphs

TL;DR

This paper introduces new approximate distance oracles for dense graphs that achieve near-optimal stretch bounds with subquadratic space, advancing the state of the art in graph distance approximation.

Contribution

It presents the first distance oracles with multiplicative stretch close to 1 plus small additive stretch for dense graphs, surpassing previous bounds.

Findings

Achieved a multiplicative stretch of 1+ε with small additive stretch in dense graphs.

Constructed a family of oracles with stretch 2k-1+ε using sublinear space.

Improved the understanding of distance oracle capabilities in dense graph settings.

Abstract

Despite extensive research on distance oracles, there are still large gaps between the best constructions for spanners and distance oracles. Notably, there exist sparse spanners with a multiplicative stretch of $1+\varepsilon$ plus some additive stretch. A fundamental open problem is whether such a bound is achievable for distance oracles as well. Specifically, can we construct a distance oracle with multiplicative stretch better than 2, along with some additive stretch, while maintaining subquadratic space complexity? This question remains a crucial area of investigation, and finding a positive answer would be a significant step forward for distance oracles. Indeed, such oracles have been constructed for sparse graphs. However, in the more general case of dense graphs, it is currently unknown whether such oracles exist. In this paper, we contribute to the field by presenting the first distance oracles that achieve a multiplicative stretch of $1+\varepsilon$ along with a small additive stretch while maintaining subquadratic space complexity. Our results represent an advancement particularly for constructing efficient distance oracles for dense graphs. In addition, we present a whole family of oracles that, for any positive integer $k$, achieve a multiplicative stretch of $2k-1+\varepsilon$ using $o(n^{1+1/k})$ space.