Abstract
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ℝ^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on ε > 0 and d ∈ ℕ of the minimum lightness of a (1+ε)spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1+ε)spanners of lightness O(ε^{1}logΔ) in the plane, where Δ ≥ Ω(√n) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness Õ(ε^{(d+1)/2}) in dimensions d ≥ 3. Recently, Bhore and Tóth (2020) established a lower bound of Ω(ε^{d/2}) for the lightness of Steiner (1+ε)spanners in ℝ^d, for d ≥ 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d ≥ 2.
In this work, we show that for every finite set of points in the plane and every ε > 0, there exists a Euclidean Steiner (1+ε)spanner of lightness O(ε^{1}); this matches the lower bound for d = 2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.
BibTeX  Entry
@InProceedings{bhore_et_al:LIPIcs.SoCG.2021.15,
author = {Bhore, Sujoy and T\'{o}th, Csaba D.},
title = {{Light Euclidean Steiner Spanners in the Plane}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {15:115:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771849},
ISSN = {18688969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13814},
URN = {urn:nbn:de:0030drops138145},
doi = {10.4230/LIPIcs.SoCG.2021.15},
annote = {Keywords: Geometric spanner, lightness, minimum weight}
}
Keywords: 

Geometric spanner, lightness, minimum weight 
Collection: 

37th International Symposium on Computational Geometry (SoCG 2021) 
Issue Date: 

2021 
Date of publication: 

02.06.2021 