Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

TL;DR

This paper studies how Voronoi diagrams and Delaunay triangulations change when points move under polygonal distance functions, providing bounds on topological changes and an efficient kinetic maintenance algorithm.

Contribution

It establishes bounds on topological changes of diagrams under polygonal distance functions and introduces a kinetic data structure for their efficient maintenance.

Findings

Bound of O(k^4 n λ_r(n)) on topological changes during motion.

Algorithm for maintaining diagrams using kinetic data structures.

Analysis assuming algebraic trajectories of bounded degree.

Abstract

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. 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 $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ 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.