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Transactions on Computational Science IX pp 1–27Cite as

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

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Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 6290))

Abstract

We present a general framework for computing Voronoi diagrams of different classes of sites under various distance functions in \({\mathbb R}^{2}\). 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.

Work on this paper has been supported in part by the Hermann Minkowski–Minerva Center for Geometry at Tel-Aviv University. Work by Ophir Setter and Dan Halperin has also been supported in part by the Israel Science Foundation (Grant no. 236/06), and by the German-Israeli Foundation (Grant no. 969/07). Work by Micha Sharir was also partially supported by NSF Grants CCF-05-14079 and CCF-08-30272, by Grant 2006/194 from the U.S.-Israeli Binational Science Foundation, by Grants 155/05 and 338/09 from the Israel Science Fund, Israeli Academy of Sciences, and by a grant from the French-Israeli AFIRST program.

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Authors and Affiliations

  1. School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel

    Ophir Setter, Micha Sharir & Dan Halperin

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  1. Ophir Setter
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  2. Micha Sharir
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  3. Dan Halperin
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  1. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  2. Exascala Ltd., Unit 9, 97 Rickman Drive, B15 2AL, Birmingham, UK

    C. J. Kenneth Tan

  3. Department of Informatics and Mathematical Modeling, Technical University of Denmark, Richard Petersens Plads, 2800, Lyngby, Denmark

    François Anton

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Setter, O., Sharir, M., Halperin, D. (2010). Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space. In: Gavrilova, M.L., Tan, C.J.K., Anton, F. (eds) Transactions on Computational Science IX. Lecture Notes in Computer Science, vol 6290. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16007-3_1

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  • DOI: https://doi.org/10.1007/978-3-642-16007-3_1

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