License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.19
URN: urn:nbn:de:0030-drops-178696
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17869/
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Biniaz, Ahmad

Improved Bounds for Covering Paths and Trees in the Plane

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LIPIcs-SoCG-2023-19.pdf (0.9 MB)

Abstract

A covering path for a planar point set is a path drawn in the plane with straight-line edges such that every point lies at a vertex or on an edge of the path. A covering tree is defined analogously. Let π(n) be the minimum number such that every set of n points in the plane can be covered by a noncrossing path with at most π(n) edges. Let τ(n) be the analogous number for noncrossing covering trees. Dumitrescu, Gerbner, Keszegh, and Tóth (Discrete & Computational Geometry, 2014) established the following inequalities: 5n/9 - O(1) < π(n) < (1-1/601080391)n, and 9n/17 - O(1) < τ(n) ⩽ ⌊5n/6⌋. We report the following improved upper bounds: π(n) ⩽ (1-1/22)n, and τ(n) ⩽ ⌈4n/5⌉.
In the same context we study rainbow polygons. For a set of colored points in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color in its interior or on its boundary. Let ρ(k) be the minimum number such that every k-colored point set in the plane admits a perfect rainbow polygon of size ρ(k). Flores-Peñaloza, Kano, Martínez-Sandoval, Orden, Tejel, Tóth, Urrutia, and Vogtenhuber (Discrete Mathematics, 2021) proved that 20k/19 - O(1) < ρ(k) < 10k/7 + O(1). We report the improved upper bound of ρ(k) < 7k/5 + O(1).
To obtain the improved bounds we present simple O(nlog n)-time algorithms that achieve paths, trees, and polygons with our desired number of edges.

BibTeX - Entry

@InProceedings{biniaz:LIPIcs.SoCG.2023.19,
  author =	{Biniaz, Ahmad},
  title =	{{Improved Bounds for Covering Paths and Trees in the Plane}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{19:1--19:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17869},
  URN =		{urn:nbn:de:0030-drops-178696},
  doi =		{10.4230/LIPIcs.SoCG.2023.19},
  annote =	{Keywords: planar point sets, covering paths, covering trees, rainbow polygons}
}

Keywords: planar point sets, covering paths, covering trees, rainbow polygons
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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