Covering Metric Spaces by Few Trees

TL;DR

This paper introduces efficient algorithms for constructing tree covers and Ramsey tree covers in various metric spaces, highlighting differences in their capabilities and focusing on low-distortion, small-number-of-trees scenarios.

Contribution

It provides the first efficient algorithms for tree covers and Ramsey tree covers in general, planar, and doubling metrics, with a focus on low distortion and few trees.

Findings

Large separation between tree covers and Ramsey tree covers.

Algorithms achieve low distortion with a small number of trees.

Applicable to general, planar, and doubling metrics.

Abstract

A {\em tree cover} of a metric space $(X,d)$ is a collection of trees, so that every pair $x,y\in X$ has a low distortion path in one of the trees. If it has the stronger property that every point $x\in X$ has a single tree with low distortion paths to all other points, we call this a {\em Ramsey} tree cover. Tree covers and Ramsey tree covers have been studied by \cite{BLMN03,GKR04,CGMZ05,GHR06,MN07}, and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by \cite{ADMSS95}. In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.