#### Date of Original Version

4-2-2013

#### Type

Article

#### Rights Management

© ACM, 2013. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published at http://doi.acm.org/10.1145/nnnnnn.nnnnnn

#### Abstract or Description

We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time O(2^{O(d)}(n log n + m)), where n is the input size, m is the output point set size, and d is the ambient dimension. The constants only depend on the desired element quality bounds.

To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on d is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size 2^{O(d)}m graph in 2^{O(d)}(n log n + m) expected time. If m is superlinear in n, then we can produce a hierarchically well-spaced superset of size 2^{O(d)}n in 2^{O(d)}n log n expected time.

#### DOI

10.1145/2462356.2462404

#### Published In

Proceedings of the Symposium on Computational Geometry (SoCG), 2013, 289-298.