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

TL;DR

This paper introduces an efficient algorithm for generating well-spaced point sets and approximate Delaunay graphs in any dimension, optimizing for output size and computational complexity.

Contribution

The authors develop a new algorithm that efficiently produces well-spaced points and approximate Delaunay graphs with guaranteed size bounds and improved expected running time.

Findings

Expected running time is $O(2^{O(d)}(n ext{log} n + m))$.

Produces a hierarchical well-spaced superset with size $2^{O(d)}n$ in $2^{O(d)}n ext{log} n$ time.

Maintains approximate Voronoi diagram without full storage, enabling efficient neighbor structure computation.

Abstract

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.