Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Pankaj K. Agarwal; Haim Kaplan; Natan Rubin; Micha Sharir

doi:10.1007/s00454-015-9729-3

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Pankaj K. Agarwal, Haim Kaplan, Natan Rubin^*, Micha Sharir

^*Corresponding author for this work

School of Computer Science and AI

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

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(k⁴nλ_r(n)) on the number of topological changes experienced by the diagrams throughout the motion; here λ_r(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.

Original language	English
Pages (from-to)	871-904
Number of pages	34
Journal	Discrete and Computational Geometry
Volume	54
Issue number	4
DOIs	https://doi.org/10.1007/s00454-015-9729-3
State	Published - 8 Sep 2015

Funding

Funders	Funder number
Agence Nationale de la Recherche	CCF-08-30272, 892/13, ANR-10-LABX-0098
Army Research Office	W911NF-13-P-0018
Engineer Research and Development Center	822/10, W9132V-11-C-0003
German-Israeli Science Foundation
Israel Science Foundation	1161/2011
Israel Science Fund
National Science Foundation	CCF-10-12254, CCF-09-40671, CCF-11-61359
Tel Aviv University
United States-Israel Binational Science Foundation	2012/229
Max-Planck-Gesellschaft
Israeli Centers for Research Excellence	338/09, 4/11, 975/06
Fondation Sciences Mathématiques de Paris

Keywords

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

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10.1007/s00454-015-9729-3

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title = "Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions",

abstract = "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.",

keywords = "Convex distance function, Delaunay triangulation, Discrete changes, Kinetic data structure, Moving points, Voronoi diagram",

author = "Agarwal, \{Pankaj K.\} and Haim Kaplan and Natan Rubin and Micha Sharir",

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

AU - Agarwal, Pankaj K.

AU - Kaplan, Haim

AU - Rubin, Natan

AU - Sharir, Micha

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

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SN - 0179-5376

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JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

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