Piecewise-Linear Farthest-Site Voronoi Diagrams
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.
Voronoi diagram
farthest-site
polyhedral distance
polyhedral sites
general dimensions
Theory of computation~Computational geometry
30:1-30:11
Regular Paper
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).
Franz
Aurenhammer
Franz Aurenhammer
Institute for Theoretical Computer Science, TU Graz, Austria
https://orcid.org/0000-0003-4257-4021
Evanthia
Papadopoulou
Evanthia Papadopoulou
Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
https://orcid.org/0000-0003-0144-7384
Martin
Suderland
Martin Suderland
Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland
https://orcid.org/0000-0002-6604-6381
10.4230/LIPIcs.ISAAC.2021.30
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Franz Aurenhammer, Evanthia Papadopoulou, and Martin Suderland
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