Stable Delaunay Graphs

TL;DR

This paper introduces a notion of stability for edges in Euclidean Delaunay triangulations, explores their properties under various distance functions, and develops a kinetic data structure for maintaining stable Delaunay graphs as points move.

Contribution

It defines stable Delaunay edges, relates their stability across different convex distance functions, and presents a linear-size kinetic data structure for dynamic maintenance.

Findings

Stable edges in Euclidean Delaunay triangulations remain stable under nearby convex distance functions.

A linear-size kinetic data structure efficiently maintains stable Delaunay graphs during point motion.

Properties of Euclidean Delaunay triangulations are preserved in stable Delaunay graphs.

Abstract

Let $P$ be a set of $n$ points in $\mathrm{R}^2$, and let $\mathrm{DT}(P)$ denote its Euclidean Delaunay triangulation. We introduce the notion of an edge of $\mathrm{DT}(P)$ being {\it stable}. Defined in terms of a parameter $\alpha>0$, a Delaunay edge $pq$ is called $\alpha$-stable, if the (equal) angles at which $p$ and $q$ see the corresponding Voronoi edge $e_{pq}$ are at least $\alpha$. A subgraph $G$ of $\mathrm{DT}(P)$ is called {\it $(c\alpha, \alpha)$-stable Delaunay graph} ($\mathrm{SDG}$ in short), for some constant $c \ge 1$, if every edge in $G$ is $\alpha$-stable and every $c\alpha$-stable of $\mathrm{DT}(P)$ is in $G$. We show that if an edge is stable in the Euclidean Delaunay triangulation of $P$, then it is also a stable edge, though for a different value of $\alpha$, in the Delaunay triangulation of $P$ under any convex distance function that is sufficiently close to the Euclidean norm, and vice-versa. In particular, a $6\alpha$-stable edge in $\mathrm{DT}(P)$ is $\alpha$-stable in the Delaunay triangulation under the distance function induced by a regular $k$-gon for $k \ge 2\pi/\alpha$, and vice-versa. Exploiting this relationship and the analysis in~\cite{polydel}, we present a linear-size kinetic data structure (KDS) for maintaining an $(8\alpha,\alpha)$-$\mathrm{SDG}$ as the points of $P$ move. If the points move along algebraic trajectories of bounded degree, the KDS processes nearly quadratic events during the motion, each of which can processed in $O(\log n)$ time. Finally, we show that a number of useful properties of $\mathrm{DT}(P)$ are retained by $\mathrm{SDG}$ of $P$.