Abstract
Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done inO(n logn+k 3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k ⌈(d+1)/2⌉+1 n ⌊(d+1)/2⌋) and space complexityO(k ⌈(d+1)/2⌉ n ⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.
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Communicated by Franco P. Preparata.
This work has been supported in part by the ESPRIT Basic Research Action No. 3075 (ALCOM).
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Boissonnat, J.D., Devillers, O. & Teillaud, M. A semidynamic construction of higher-order voronoi diagrams and its randomized analysis. Algorithmica 9, 329–356 (1993). https://doi.org/10.1007/BF01228508
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DOI: https://doi.org/10.1007/BF01228508