Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space

Ophir Setter*, Micha Sharir, Dan Halperin

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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-andconquer 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 minimumwidth 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.

Original languageEnglish
Title of host publication6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009
Pages43-52
Number of pages10
DOIs
StatePublished - 2009
Event6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 - Copenhagen, Denmark
Duration: 23 Jun 200926 Jun 2009

Publication series

Name6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

Conference

Conference6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009
Country/TerritoryDenmark
CityCopenhagen
Period23/06/0926/06/09

Keywords

  • Envelopes
  • Minimum-width annulus
  • Randomization
  • Voronoi diagrams

Fingerprint

Dive into the research topics of 'Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space'. Together they form a unique fingerprint.

Cite this