Optimal Euclidean Tree Covers

Hsien Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, Cuong Than

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

A (1 + ε)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1 + ε)-stretch path in one of the trees. The celebrated Dumbbell Theorem [Arya et al. STOC’95] states that any set of n points in d-dimensional Euclidean space admits a (1 + ε)-stretch tree cover with Od−d · log(1/ε)) trees, where the Od notation suppresses terms that depend solely on the dimension d. The running time of their construction is Od(n log n · log(1εd/ε) + n · ε−2d). Since the same point may occur in multiple levels of the tree, the maximum degree of a point in the tree cover may be as large as Ω(log Φ), where Φ is the aspect ratio of the input point set. In this work we present a (1 + ε)-stretch tree cover with Od−d+1 · log(1/ε)) trees, which is optimal (up to the log(1/ε) factor). Moreover, the maximum degree of points in any tree is an absolute constant for any d. As a direct corollary, we obtain an optimal routing scheme in low-dimensional Euclidean spaces. We also present a (1 + ε)-stretch Steiner tree cover (that may use Steiner points) with Od(−d+1)/2 · log(1/ε)) trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive Od(n log n) term; this improves over the running time underlying the Dumbbell Theorem.

Original languageEnglish
Title of host publication40th International Symposium on Computational Geometry, SoCG 2024
EditorsWolfgang Mulzer, Jeff M. Phillips
ISBN (Electronic)9783959773164
DOIs
StatePublished - Jun 2024
Event40th International Symposium on Computational Geometry, SoCG 2024 - Athens, Greece
Duration: 11 Jun 202414 Jun 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume293

Conference

Conference40th International Symposium on Computational Geometry, SoCG 2024
Country/TerritoryGreece
CityAthens
Period11/06/2414/06/24

Keywords

  • bounded-degree
  • net-tree
  • quadtree
  • routing
  • spanner
  • Steiner point
  • Tree cover

All Science Journal Classification (ASJC) codes

  • Software

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