Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

Ophir Setter, Micha Sharir, Dan Halperin
Tel-Aviv University
In Transactions on Computational Science IX, Vol. 6290 (2010), pp. 1-27, arXiv:0906.2760v1 [cs.CG]


   title={Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space},

   author={Halperin, D. and Setter, O. and Sharir, M.},




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We present a general framework for computing Voronoi diagrams of different classes of sites under various distance functions in $R^3$. Most diagrams mentioned in the paper are in the plane. However, the framework is sufficiently general to support diagrams embedded on a family of two-dimensional parametric surfaces in three-dimensions. The computation of the diagrams is carried out through the construction of envelopes of surfaces in 3-space provided by Cgal (the Computational Geometry Algorithm Library). The construction of the envelopes follows a divide-and-conquer approach. A straightforward application of the divide-and-conquer approach for Voronoi diagrams yields algorithms that are inefficient in the worst case. We prove that through randomization, the expected running time becomes near-optimal in the worst case. We also show how to apply the new framework and other existing tools from Cgal to compute minimum-width annuli of sets of disks, which requires the computation of two Voronoi diagrams of different types, and of the overlay of the two diagrams. We do not assume general position. Namely, we handle degenerate input, and produce exact results.
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