Covering Metric Spaces by Few Trees

Authors Yair Bartal, Nova Fandina, Ofer Neiman



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Author Details

Yair Bartal
  • Department of Computer Science, Hebrew University of Jerusalem, Israel
Nova Fandina
  • Department of Computer Science, Hebrew University of Jerusalem, Israel
Ofer Neiman
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Acknowledgements

We are grateful to Michael Elkin and Shay Solomon for fruitful discussions.

Cite AsGet BibTex

Yair Bartal, Nova Fandina, and Ofer Neiman. Covering Metric Spaces by Few Trees. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.20

Abstract

A 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 Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], 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 [S. Arya et al., 1995]. 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Sparsification and spanners
Keywords
  • tree cover
  • Ramsey tree cover
  • probabilistic hierarchical family

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