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.ISAAC.2021.30
URN: urn:nbn:de:0030-drops-154633
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Aurenhammer, Franz ; Papadopoulou, Evanthia ; Suderland, Martin

Piecewise-Linear Farthest-Site Voronoi Diagrams

LIPIcs-ISAAC-2021-30.pdf (0.8 MB)


Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewise-linear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it.
We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex distance polytope.

BibTeX - Entry

  author =	{Aurenhammer, Franz and Papadopoulou, Evanthia and Suderland, Martin},
  title =	{{Piecewise-Linear Farthest-Site Voronoi Diagrams}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{30:1--30:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-154633},
  doi =		{10.4230/LIPIcs.ISAAC.2021.30},
  annote =	{Keywords: Voronoi diagram, farthest-site, polyhedral distance, polyhedral sites, general dimensions}

Keywords: Voronoi diagram, farthest-site, polyhedral distance, polyhedral sites, general dimensions
Collection: 32nd International Symposium on Algorithms and Computation (ISAAC 2021)
Issue Date: 2021
Date of publication: 30.11.2021

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