# Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions

1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if $ß$ is a set of $n$ points in general position in $\Ee maximum complexity of its Voronoi diagram under the$\Linf$metric, and also under a simplicial distance function, are both shown to be$\Theta(n^{\lceil d/2 \rceil})$. The upper bound for the case of the$\Linf$metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of$n$axis-parallel hypercubes in$\E complexity isof $\Theta(n^{\ceil{d/2}})$, for $d \ge 1$, and it improves to $\Theta(n^{\floor{d/2}})$, for $d \ge 2$, if all the hypercubes have the same size. Under the $L_1$ metric, the maximum complexity of the Voronoi diagram of a set of $n$ points in general position in $\Eown to be d$\Theta(n^2)$. We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in degenerate configurations. Finally, on-line algorithms are proposed for computing the Voronoi diagram of$n$points in$\E under a simplicial or $\Linf$ distance function. Their randomized complexities are $O(n \log n + n ^{\ceil{d/2}})$ for simplicial diagrams and $O(n ^{\ceil{d/2}} \log ^{d-1} n)$ for $\Linf$-diagrams.
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Submitted on : Wednesday, May 24, 2006 - 2:24:18 PM
Last modification on : Saturday, January 27, 2018 - 1:31:03 AM
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### Identifiers

• HAL Id : inria-00074058, version 1

### Citation

Jean-Daniel Boissonnat, Micha Sharir, Boaz Tagansky, Mariette Yvinec. Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions. RR-2629, INRIA. 1995. ⟨inria-00074058⟩

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