TY - GEN

T1 - Constructing two-dimensional Voronoi diagrams via divide-and-conquer of envelopes in space

AU - Setter, Ophir

AU - Sharir, Micha

AU - Halperin, Dan

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Envelopes

KW - Minimum-width annulus

KW - Randomization

KW - Voronoi diagrams

UR - http://www.scopus.com/inward/record.url?scp=77951461522&partnerID=8YFLogxK

U2 - 10.1109/ISVD.2009.20

DO - 10.1109/ISVD.2009.20

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AN - SCOPUS:77951461522

SN - 9780769537818

T3 - 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

SP - 43

EP - 52

BT - 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

T2 - 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

Y2 - 23 June 2009 through 26 June 2009

ER -