Summary: A Dynamic Algorithm for Well-Spaced Point Sets
Umut A. Acar
Figure 1: A well-spaced superset.
Solid points (·) are input. Empty
points () are Steiner vertices.
The Motivation and the Problem. A set of points is well-spaced if
the Voronoi cell of each point has a bounded aspect ratio, i.e., the ratio
of the distance to the farthest point in the Voronoi cell divided by the
nearest neighbor distance is small. Informally, a set of points is well-
spaced if its density varies smoothly. Well-spaced points sets relate
strongly to meshing and triangulation for scientific computing: with
minimal processing, they lead to quality meshes (e.g., no small angles)
in two and higher dimensions. The Voronoi diagram of a well-spaced
point set is also immediately useful for the Control Volume Method.
Given a finite set of points N in the d-dimensional unit hypercube
[0, 1]d, the static well-spaced superset problem is to find a small, well-
spaced set M N by inserting so called Steiner points. This problem