Dense point sets have sparse Delaunay triangulations

TL;DR

This paper proves that the Delaunay triangulation complexity for a set of points in three-dimensional space is bounded by a cubic function of the spread, with tight bounds for dense point sets and generalizations to other systems.

Contribution

It establishes tight bounds on the complexity of Delaunay triangulations based on point set spread and generalizes these bounds to various geometric configurations.

Findings

Delaunay triangulation complexity is O(D^3) for point sets in R^3 with spread D.

For dense point sets, the triangulation complexity is linear.

Constructed examples show lower bounds of Omega(nD) for certain configurations.

Abstract

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3). This bound is tight in the worst case for all D = O(sqrt{n}). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.