@inproceedings{7a45b86a9c8045ee903543b5785b2031,
title = "Optimal Euclidean Tree Covers",
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{\textquoteright}95] states that any set of n points in d-dimensional Euclidean space admits a (1 + ε)-stretch tree cover with Od(ε−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(ε−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(ε(−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.",
keywords = "bounded-degree, net-tree, quadtree, routing, spanner, Steiner point, Tree cover",
author = "Chang, {Hsien Chih} and Jonathan Conroy and Hung Le and Lazar Milenkovi{\'c} and Shay Solomon and Cuong Than",
note = "Publisher Copyright: {\textcopyright} Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenkovi{\'c}, Shay Solomon, and Cuong Than.; 40th International Symposium on Computational Geometry, SoCG 2024 ; Conference date: 11-06-2024 Through 14-06-2024",
year = "2024",
month = jun,
doi = "https://doi.org/10.4230/LIPIcs.SoCG.2024.37",
language = "الإنجليزيّة",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
editor = "Wolfgang Mulzer and Phillips, {Jeff M.}",
booktitle = "40th International Symposium on Computational Geometry, SoCG 2024",
}