@inproceedings{c5900d56525d4352b420386ae1c6904e,

title = "On euclidean steiner (1 + ϵ)-spanners",

abstract = "Lightness and sparsity are two natural parameters for Euclidean (1 + ϵ)-spanners. Classical results show that, when the dimension d ∈ N and ϵ > 0 are constant, every set S of n points in d-space admits an (1+ϵ)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ϵ > 0 for constant d ∈ N have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1 + ϵ)-spanner. They gave upper bounds of {\~O}(ϵ-(d+1)/2) for the minimum lightness in dimensions d ≥ 3, and {\~O}(ϵ-(d-1))/2) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1 + ϵ)-spanners of lightness O(ϵ-1 log Δ) in the plane, where Δ ∈ Ω(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1 + ϵ)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ϵ-d/2) for the lightness and Ω(ϵ-(d-1)/2) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2. We use the geometric insight from our lower bound analysis to construct Steiner (1 + ϵ)-spanners of lightness O(ϵ-1 log n) for n points in Euclidean plane.",

keywords = "(1 + ϵ)-spanner, Geometric spanner, Lightness, Minimum weight, Sparsity",

author = "Sujoy Bhore and T{\'o}th, {Csaba D.}",

note = "Funding Information: Funding Sujoy Bhore: Research on this paper was supported by the Fonds de la Recherche Scientifique-FNRS under Grant no MISU F 6001. Csaba D. T{\'o}th: Research on this paper was partially supported by the NSF award DMS-1800734. Publisher Copyright: {\textcopyright} Sujoy Bhore and Csaba D. T{\'o}th; licensed under Creative Commons License CC-BY 4.0.; 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021 ; Conference date: 16-03-2021 Through 19-03-2021",

year = "2021",

month = mar,

day = "1",

doi = "10.4230/LIPIcs.STACS.2021.13",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "Markus Blaser and Benjamin Monmege",

booktitle = "38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021",

address = "Germany",

}