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Discrete & Computational Geometry

, Volume 54, Issue 4, pp 871–904 | Cite as

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

  • Pankaj K. Agarwal
  • Haim Kaplan
  • Natan RubinEmail author
  • Micha Sharir
Article

Abstract

Let P be a set of n points and Q a convex k-gon in \({\mathbb {R}}^2\). We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of P, under the convex distance function defined by Q, as the points of P move along prespecified continuous trajectories. Assuming that each point of P moves along an algebraic trajectory of bounded degree, we establish an upper bound of \(O(k^4n\lambda _r(n))\) on the number of topological changes experienced by the diagrams throughout the motion; here \(\lambda _r(n)\) is the maximum length of an (nr)-Davenport–Schinzel sequence, and r is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework.

Keywords

Delaunay triangulation Voronoi diagram Moving points Discrete changes Convex distance function Kinetic data structure 

Notes

Acknowledgments

Work by P.A. and M.S. was supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation. Work by P.A. was also supported by NSF under Grants CCF-09-40671, CCF-10-12254, and CCF-11-61359, by an ARO contract W911NF-13-P-0018, and by an ERDC contract W9132V-11-C-0003. Work by H.K. has been supported by Grant 822/10 from the Israel Science Foundation, Grant 1161/2011 from the German-Israeli Science Foundation, and by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11). Work by N.R. was partially supported by Grants 975/06 and 338/09 from the Israel Science Fund, by Minerva Fellowship Program of the Max Planck Society, by the Fondation Sciences Mathématiques de Paris (FSMP), and by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098). Work by M. S. has also been supported by NSF Grant CCF-08-30272, by Grants 338/09 and 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Haim Kaplan
    • 2
  • Natan Rubin
    • 3
    Email author
  • Micha Sharir
    • 2
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Department of Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael

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