Stable Delaunay Graphs

Abstract

Let P be a set of n points in ${\mathbb R}^2$, and let $\mathop {{\mathrm {DT}}}(P)$ denote its Euclidean Delaunay triangulation. We introduce the notion of the stability of edges of $\mathop {{\mathrm {DT}}}(P)$. Specifically, 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 $\mathop {{\mathrm {DT}}}(P)$ is called a $(c\alpha , \alpha )$ -stable Delaunay graph ($\mathop {{\mathrm {SDG}}}$ in short), for some absolute constant $c \ge 1$, if every edge in G is $\alpha$-stable and every $c\alpha$-stable edge of $\mathop {{\mathrm {DT}}}(P)$ is in G. Stability can also be defined, in a similar manner, for edges of Delaunay triangulations under general convex distance functions, induced by arbitrary compact convex sets Q. 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 $\mathop {{\mathrm {DT}}}(P)$ 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. This relationship, along with the analysis in the companion paper [3], yields a linear-size kinetic data structure (KDS) for maintaining an $(8\alpha ,\alpha )$-$\mathop {{\mathrm {SDG}}}$ as the points of P move. If the points move along algebraic trajectories of bounded degree, the KDS processes a nearly quadratic number of events during the motion, each of which can be processed in $O(\log n)$ time. We also show that several useful properties of $\mathop {{\mathrm {DT}}}(P)$ are retained by any SDG of P (although some other properties are not).