Voronoi diagrams in higher dimensions under certain polyhedral distance functions

Jean Daniel Boissonnat, Micha Sharir, Boaz Tagansky, Mariette Yvinec

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in IRd, the complexity of its Voronoi diagram under the metric, and also under a simplicial distance function, are both shown to be O(n[d/2]). The upper bound for the case of the L metric follows from a new upper bound, also proved in this paper, on the complexity of the union of n axis-parallel hypercubes in IRd. This complexity is O(n[d/2]), for d ≥ 1, and it improves to Θ(n[d/2]), for d ≥ 2, if all the hypercubes have the same size. Under the L1 metric, the complexity of the Voronoi diagram of a set of n points in general position in IR3 is shown to be Θ(n2). We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in degenerate configurations.

Original languageEnglish
Title of host publicationProceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995
PublisherAssociation for Computing Machinery
Pages79-88
Number of pages10
ISBN (Electronic)0897917243
DOIs
StatePublished - 1 Sep 1995
Event11th Annual Symposium on Computational Geometry, SCG 1995 - Vancouver, Canada
Duration: 5 Jun 19957 Jun 1995

Publication series

NameProceedings of the Annual Symposium on Computational Geometry
VolumePart F129372

Conference

Conference11th Annual Symposium on Computational Geometry, SCG 1995
Country/TerritoryCanada
CityVancouver
Period5/06/957/06/95

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