A Fast Algorithm for Well-Spaced Points and Approximate Delaunay Graphs

SOCG: ACM Symposium on Computational Geometry
2013

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.

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.

@inproceedings{miller13fast, Title = {A Fast Algorithm for Well-Spaced Points and Approximate Delaunay Graphs}, Author = {Gary L. Miller and Donald R. Sheehy and Ameya Velingker}, Booktitle = {Proceedings of the 29th annual Symposium on Computational Geometry}, Pages = {289--298}, Year = {2013}}