Light euclidean spanners with steiner points

Hung Le, Shay Solomon

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Abstract

The FOCS’19 paper of Le and Solomon [59], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1 + )-spanner in Rd is Õ(−d) for any d = O(1) and any = Ω(n− d−1 1 ) (where Õ hides polylogarithmic factors of 1 ), and also shows the existence of point sets in Rd for which any (1 + )-spanner must have lightness Ω(−d).1 Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in R2 with lightness O(1 log ∆), where ∆ is the spread of the point set.2 In the regime of ∆ 21/, this provides an improvement over the lightness bound of [59]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications often controls the precision, and it sometimes needs to be much smaller than O(1/ log n). Moreover, for spread polynomially bounded in 1/, this upper bound provides a quadratic improvement over the non-Steiner bound of [59], We then demonstrate that such a light spanner can be constructed in O(n) time for polynomially bounded spread, where O hides a factor of poly(1 ). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ((d+1)/2 + 2 log ∆) for any 3 ≤ d = O(1) and any = Ω(n− d−1). 1

Original languageEnglish
Title of host publication28th Annual European Symposium on Algorithms, ESA 2020
EditorsFabrizio Grandoni, Grzegorz Herman, Peter Sanders
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771627
DOIs
StatePublished - 1 Aug 2020
Event28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, Italy
Duration: 7 Sep 20209 Sep 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume173
ISSN (Print)1868-8969

Conference

Conference28th Annual European Symposium on Algorithms, ESA 2020
Country/TerritoryItaly
CityVirtual, Pisa
Period7/09/209/09/20

Keywords

  • Euclidean spanners
  • Light spanners
  • Steiner spanners

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