2010 International Symposium on Voronoi Diagrams in Science and Engineering (2009)

Copenhagen, Denmark

June 23, 2009 to June 26, 2009

ISBN: 978-0-7695-3781-8

pp: 43-52

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/ISVD.2009.20

ABSTRACT

We present a general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions. 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 two 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.Additional material is available at:http://acg.cs.tau.ac.il/projects/internal-projects/vd-via-dc-of-envelopes/project-page

INDEX TERMS

envelopes, Voronoi diagrams, randomization, minimum-width annulus

CITATION

Micha Sharir,
Ophir Setter,
Dan Halperin,
"Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space",

*2010 International Symposium on Voronoi Diagrams in Science and Engineering*, vol. 00, no. , pp. 43-52, 2009, doi:10.1109/ISVD.2009.20