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Optimal Euclidean Tree Covers

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

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Author Details

Hsien-Chih Chang
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Jonathan Conroy
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Hung Le
  • Manning CICS, UMass Amherst, MA, USA
Lazar Milenković
  • Tel Aviv University, Israel
Shay Solomon
  • Tel Aviv University, Israel
Cuong Than
  • Manning CICS, UMass Amherst, MA, USA

Funding

Hung Le and Cuong Than are supported by the NSF CAREER Award No. CCF-2237288 and an NSF Grant No. CCF-2121952. Shay Solomon is funded by the European Union (ERC, DynOpt, 101043159). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Shay Solomon is also supported by the Israel Science Foundation(ISF) grant No.1991/1. Shay Solomon and Lazar Milenković are supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and the United States National Science Foundation(NSF).

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Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Optimal Euclidean Tree Covers. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.37

Abstract

A (1+e)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1+e)-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+e)-stretch tree cover with O_d(e^{-d} ⋅ log(1/e)) trees, where the O_d notation suppresses terms that depend solely on the dimension d. The running time of their construction is O_d(n log n ⋅ log(1/e)/e^d + n ⋅ e^{-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+e)-stretch tree cover with O_d(e^{-d+1} ⋅ log(1/e)) trees, which is optimal (up to the log(1/e) 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+e)-stretch Steiner tree cover (that may use Steiner points) with O_d(e^{(-d+1)/2} ⋅ log(1/e)) 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 O_d(n log n) term; this improves over the running time underlying the Dumbbell Theorem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Tree cover
  • spanner
  • Steiner point
  • routing
  • bounded-degree
  • quadtree
  • net-tree

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