2002/07/15 by Daniel A. Spielman, Dan A. Spielman, Shang-hua Teng +6
Agricultural and Biological Sciences · Computer Science · Environmental Science · #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #F.2.2 #FOS: Computer and information sciences #Remote Sensing and LiDAR Applications #Soil erosion and sediment transport #cs.CG
paper · pdf · doi:10.48550/arxiv.cs/0207063
12 pages (short version); 2 figures; see also http://www.cs.duke.edu/~ungor/abstracts/parallelDelRef.html
arxiv created 2002/07/15 · openalex publication_date 2002/07/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of ``independent'' points to insert into the domain, and then update the Delaunay triangulation. We show that such a set of independent points can be constructed efficiently in parallel and that the number of iterations needed is O(log2(L/s)), where L is the diameter of the domain, and s is the smallest edge in the output mesh. In addition, we show that the insertion of each independent set of points can be realized sequentially by Ruppert's method in two dimensions and Shewchuk's in three dimensions. Therefore, our parallel Delaunay refinement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasi-uniform meshes, such as those produced by Chew's method, we show that the number of iterations can be reduced to O(log(L/s)). To the best of our knowledge, these are the first provably polylog(L/s) parallel time Delaunay meshing algorithms that generate well-shaped meshes of size optimal to within a constant.