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 $α>0$, a Delaunay edge $pq$ is called $α$-stable, if the (equal) angles at which $p$ and $q$ see the corresponding Voronoi edge $e_{pq}$ are at least $α$. A subgraph $G$ of $\mathrm{DT}(P)$ is called {\it $(cα, α)$-stable Delaunay graph} ($\mathrm{SDG}$ in short), for some constant $c \ge 1$, if every edge in $G$ is $α$-stable and every $cα$-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 $α$, 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 $6α$-stable edge in $\mathrm{DT}(P)$ is $α$-stable in the Delaunay triangulation under the distance function induced by a regular $k$-gon for $k \ge 2π/α$, and vice-versa. Exploiting this relationship and the analysis in~\cite{polydel}, we present a linear-size kinetic data structure (KDS) for maintaining an $(8α,α)$-$\mathrm{SDG}$ 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 $P$.