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

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

12 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

Keywords

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

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s00454-015-9729-3

Cite this

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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",

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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

UR - https://www.scopus.com/pages/publications/84945436333

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 -

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

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Keywords

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