TY - JOUR

T1 - Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

AU - Agarwal, Pankaj K.

AU - Kaplan, Haim

AU - Rubin, Natan

AU - Sharir, Micha

N1 - Funding Information:
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.
Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2015/9/8

Y1 - 2015/9/8

N2 - Let P be a set of n points and Q a convex k-gon in R2. 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(k4nλr(n)) on the number of topological changes experienced by the diagrams throughout the motion; here λr(n) is the maximum length of an (n, r)-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.

AB - Let P be a set of n points and Q a convex k-gon in R2. 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(k4nλr(n)) on the number of topological changes experienced by the diagrams throughout the motion; here λr(n) is the maximum length of an (n, r)-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.

KW - Convex distance function

KW - Delaunay triangulation

KW - Discrete changes

KW - Kinetic data structure

KW - Moving points

KW - Voronoi diagram

UR - http://www.scopus.com/inward/record.url?scp=84945436333&partnerID=8YFLogxK

U2 - 10.1007/s00454-015-9729-3

DO - 10.1007/s00454-015-9729-3

M3 - Article

AN - SCOPUS:84945436333

SN - 0179-5376

VL - 54

SP - 871

EP - 904

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -