Very Sparse Additive Spanners and Emulators

TL;DR

This paper advances the understanding of sparse graph structures that approximate shortest path distances, introducing new bounds for distance preservers and additive spanners with improved error guarantees.

Contribution

It provides new bounds on the size of distance preservers based on demand pairs and extends the path-buying framework to larger clusters for better additive spanners.

Findings

Distance preserver size bound of $O(n^{2/3}p^{2/3} + np^{1/3})$ edges.

Existence of sparse spanners with $O(n)$ edges and $+O(n^{3/7 + ext{epsilon}})$ additive error.

A lower bound on the limitations of the consistency property in distance preservers.

Abstract

We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers, which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work on distance preservers has exploited only a simple structural property of shortest paths, called consistency, stating that one can break shortest path ties such that no two paths intersect, split apart, and then intersect again later. We prove that consistency alone is not enough to understand distance preservers, by showing both a lower bound on the power of consistency and a new general upper bound that polynomially surpasses it. Specifically, our new upper bound is that any $p$ demand pairs in an $n$-node undirected unweighted graph have a distance preserver on $O(n^{2/3}p^{2/3} + np^{1/3})$ edges. We leave a conjecture that the right bound is $O(n^{2/3}p^{2/3} + n)$ or better. The second part of this paper leverages these distance preservers in a new construction of additive spanners, which are subgraphs that preserve all pairwise distances up to an additive error function. We give improved error bounds for spanners with relatively few edges; for example, we prove that all graphs have spanners on $O(n)$ edges with $+O(n^{3/7 + \varepsilon})$ error. Our construction can be viewed as an extension of the popular path-buying framework to clusters of larger radii.