TY - GEN

T1 - Voronoi diagrams in higher dimensions under certain polyhedral distance functions

AU - Boissonnat, Jean Daniel

AU - Sharir, Micha

AU - Tagansky, Boaz

AU - Yvinec, Mariette

N1 - Publisher Copyright:
© 1995 ACM.

PY - 1995/9/1

Y1 - 1995/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84990660121&partnerID=8YFLogxK

U2 - 10.1145/220279.220288

DO - 10.1145/220279.220288

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AN - SCOPUS:84990660121

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 79

EP - 88

BT - Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995

PB - Association for Computing Machinery

T2 - 11th Annual Symposium on Computational Geometry, SCG 1995

Y2 - 5 June 1995 through 7 June 1995

ER -