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Piecewise-Linear Farthest-Site Voronoi Diagrams

Authors Franz Aurenhammer , Evanthia Papadopoulou , Martin Suderland

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Author Details

Franz Aurenhammer
  • Institute for Theoretical Computer Science, TU Graz, Austria
Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
Martin Suderland
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland

Funding

The first author was supported by Projects I 1836-N15 and I 5270-N, Austria Science Fund (FWF). The last two authors were supported in part by Projects 200021E_154387 and 200021E_201356, Swiss National Science Foundation (SNF).

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Franz Aurenhammer, Evanthia Papadopoulou, and Martin Suderland. Piecewise-Linear Farthest-Site Voronoi Diagrams. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 30:1-30:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.30

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Voronoi diagram
  • farthest-site
  • polyhedral distance
  • polyhedral sites
  • general dimensions

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