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 - 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

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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 -