License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2020.67
URN: urn:nbn:de:0030-drops-129331
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12933/
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Le, Hung ; Solomon, Shay

Light Euclidean Spanners with Steiner Points

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LIPIcs-ESA-2020-67.pdf (0.6 MB)


Abstract

The FOCS'19 paper of Le and Solomon [Hung Le and Shay Solomon, 2019], culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy (1+ε)-spanner in ℝ^d is Õ(ε^{-d}) for any d = O(1) and any ε = Ω(n^{-1/(d-1)}) (where Õ hides polylogarithmic factors of 1/ε), and also shows the existence of point sets in ℝ^d for which any (1+ε)-spanner must have lightness Ω(ε^{-d}). 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(ε^{-1} log Δ), where Δ is the spread of the point set. In the regime of Δ ≪ 2^(1/ε), this provides an improvement over the lightness bound of [Hung Le and Shay Solomon, 2019]; 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 ε often controls the precision, and it sometimes needs to be much smaller than O(1/log n). Moreover, for spread polynomially bounded in 1/ε, this upper bound provides a quadratic improvement over the non-Steiner bound of [Hung Le and Shay Solomon, 2019], We then demonstrate that such a light spanner can be constructed in O_ε(n) time for polynomially bounded spread, where O_ε hides a factor of poly(1/(ε)). Finally, we extend the construction to higher dimensions, proving a lightness upper bound of Õ(ε^{-(d+1)/2} + ε^{-2} log Δ) for any 3 ≤ d = O(1) and any ε = Ω(n^{-1/(d-1)}).

BibTeX - Entry

@InProceedings{le_et_al:LIPIcs:2020:12933,
  author =	{Hung Le and Shay Solomon},
  title =	{{Light Euclidean Spanners with Steiner Points}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{67:1--67:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12933},
  URN =		{urn:nbn:de:0030-drops-129331},
  doi =		{10.4230/LIPIcs.ESA.2020.67},
  annote =	{Keywords: Euclidean spanners, Steiner spanners, light spanners}
}

Keywords: Euclidean spanners, Steiner spanners, light spanners
Collection: 28th Annual European Symposium on Algorithms (ESA 2020)
Issue Date: 2020
Date of publication: 26.08.2020


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