A Kinetic Triangulation Scheme for Moving Points in The Plane

TL;DR

This paper introduces a simple randomized kinetic triangulation scheme for moving points in the plane, maintaining the triangulation efficiently with improved bounds on discrete changes compared to previous methods.

Contribution

It presents a new randomized scheme for kinetic triangulation with better bounds and simpler implementation than prior solutions.

Findings

Expected number of discrete changes is O(n^2β_{s+2}(n)log^2n).

Scheme maintains all kinetic data structure requirements.

Improves bounds on triangulation updates over previous work.

Abstract

We present a simple randomized scheme for triangulating a set $P$ of $n$ points in the plane, and construct a kinetic data structure which maintains the triangulation as the points of $P$ move continuously along piecewise algebraic trajectories of constant description complexity. Our triangulation scheme experiences an expected number of $O(n^2\beta_{s+2}(n)\log^2n)$ discrete changes, and handles them in a manner that satisfies all the standard requirements from a kinetic data structure: compactness, efficiency, locality and responsiveness. Here $s$ is the maximum number of times where any specific triple of points of $P$ can become collinear, $\beta_{s+2}(q)=\lambda_{s+2}(q)/q$, and $\lambda_{s+2}(q)$ is the maximum length of Davenport-Schinzel sequences of order $s+2$ on $n$ symbols. Thus, compared to the previous solution of Agarwal et al.~\cite{AWY}, we achieve a (slightly) improved bound on the number of discrete changes in the triangulation. In addition, we believe that our scheme is simpler to implement and analyze.