Dense point sets have sparse Delaunay triangulations
or "...but not too nasty"

Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, 125-134, 2002.

To appear in Discrete & Computational Geometry.

The spread of a set of points is the ratio between the largest ans smallest pairwise distances. We show that the Delaunay triangulation of any set of n points in R3 with spread D has complexity O(D3). 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 any D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D.

Publications - Jeff Erickson ( 14 Sep 2004