On kinetic Delaunay triangulations: A near-quadratic bound for unit speed motions

Natan Rubin

doi:10.1145/2746228

On kinetic Delaunay triangulations: A near-quadratic bound for unit speed motions

Natan Rubin

Department of Computer Science

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

6 Scopus citations

Abstract

Let P be a collection of n points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of O(n^2+ε), for any ε > 0, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.

Original language	English
Article number	25
Journal	Journal of the ACM
Volume	62
Issue number	3
DOIs	https://doi.org/10.1145/2746228
State	Published - 1 Jun 2015

Keywords

Algorithms
Combinatorial complexity
Computational geometry
Computing methodologies → computer graphics
Delaunay
Design
Discrete changes
Geometric arrangements
Kinetic data structures
Mathematics of computing → combinatorics; combinatorial algorithms
Moving points
Theory
Theory of computation → computational geometry
Triangulation
Voronoi diagram

ASJC Scopus subject areas

Software
Control and Systems Engineering
Information Systems
Hardware and Architecture
Artificial Intelligence

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10.1145/2746228

Cite this

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abstract = "Let P be a collection of n points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of O(n2+ε), for any ε > 0, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.",

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N2 - Let P be a collection of n points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of O(n2+ε), for any ε > 0, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.

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KW - Combinatorial complexity

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KW - Computing methodologies → computer graphics

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KW - Design

KW - Discrete changes

KW - Geometric arrangements

KW - Kinetic data structures

KW - Mathematics of computing → combinatorics; combinatorial algorithms

KW - Moving points

KW - Theory

KW - Theory of computation → computational geometry

KW - Triangulation

KW - Voronoi diagram

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On kinetic Delaunay triangulations: A near-quadratic bound for unit speed motions

Abstract

Keywords

ASJC Scopus subject areas

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