On the Number of Higher Order Delaunay Triangulations

TL;DR

This paper investigates the bounds and expected counts of higher order Delaunay triangulations, revealing their combinatorial complexity and how they vary with point distribution and order.

Contribution

It provides new bounds on the number of higher order Delaunay triangulations and analyzes their expected counts for random point sets.

Findings

Maximum number of first order Delaunay triangulations is 2^{n-3}.

Expected number of first order Delaunay triangulations is at least 2^{ρ_1 n(1+o(1))}.

Expected number of higher order Delaunay triangulations increases with order.

Abstract

Higher order Delaunay triangulations are a generalization of the Delaunay triangulation which provides a class of well-shaped triangulations, over which extra criteria can be optimized. A triangulation is order-$k$ Delaunay if the circumcircle of each triangle of the triangulation contains at most $k$ points. In this paper we study lower and upper bounds on the number of higher order Delaunay triangulations, as well as their expected number for randomly distributed points. We show that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delaunay triangulations, the maximum number is $2^{n-3}$. Next we show that uniformly distributed points have an expected number of at least $2^{\rho_1 n(1+o(1))}$ first order Delaunay triangulations, where $\rho_1$ is an analytically defined constant ($\rho_1 \approx 0.525785$), and for $k > 1$, the expected number of order-$k$ Delaunay triangulations (which are not order-$i$ for any $i < k$) is at least $2^{\rho_k n(1+o(1))}$, where $\rho_k$ can be calculated numerically.