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.ISAAC.2019.62
URN: urn:nbn:de:0030-drops-115582
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11558/
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Barequet, Gill ; Papadopoulou, Evanthia ; Suderland, Martin

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

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Abstract

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S^(d-1). We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k,n-k} n^(d-1)), which is tight for n-k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d-1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n^2-n three-dimensional cells, when n >= 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(n^(d-1) alpha(n)) time, while if d=3, the time drops to worst-case optimal O(n^2).

BibTeX - Entry

@InProceedings{barequet_et_al:LIPIcs:2019:11558,
  author =	{Gill Barequet and Evanthia Papadopoulou and Martin Suderland},
  title =	{{Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{62:1--62:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Pinyan Lu and Guochuan Zhang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2019/11558},
  URN =		{urn:nbn:de:0030-drops-115582},
  doi =		{10.4230/LIPIcs.ISAAC.2019.62},
  annote =	{Keywords: Voronoi diagram, lines, line segments, higher-order, order-k, unbounded, hypersphere arrangement, great hyperspheres}
}

Keywords: Voronoi diagram, lines, line segments, higher-order, order-k, unbounded, hypersphere arrangement, great hyperspheres
Collection: 30th International Symposium on Algorithms and Computation (ISAAC 2019)
Issue Date: 2019
Date of publication: 28.11.2019

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