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DOI: 10.4230/LIPIcs.SoCG.2016.21
URN: urn:nbn:de:0030-drops-59135
URL: http://drops.dagstuhl.de/opus/volltexte/2016/5913/
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Bohler, Cecilia ; Klein, Rolf ; Liu, Chih-Hung

An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

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LIPIcs-SoCG-2016-21.pdf (0.7 MB)


Abstract

Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before.

BibTeX - Entry

@InProceedings{bohler_et_al:LIPIcs:2016:5913,
  author =	{Cecilia Bohler and Rolf Klein and Chih-Hung Liu},
  title =	{{An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5913},
  URN =		{urn:nbn:de:0030-drops-59135},
  doi =		{10.4230/LIPIcs.SoCG.2016.21},
  annote =	{Keywords: Order-k Voronoi Diagrams, Abstract Voronoi Diagrams, Randomized Geometric Algorithms}
}

Keywords: Order-k Voronoi Diagrams, Abstract Voronoi Diagrams, Randomized Geometric Algorithms
Seminar: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 09.06.2016


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