Covering Metric Spaces by Few Trees
Yair Bartal, Nova Fandina, Ofer Neiman

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 is a collection of trees, so that every pair has a low distortion path in one of the trees. If it has the stronger property that every point 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…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Topology and Set Theory · Advanced Graph Theory Research
