Parallel Delaunay Refinement: Algorithms And Analyses

In this paper, we analyze the omplexity of natural parallelizations of Delaunay re nement methods for mesh generation. The parallelizations employ a simple strategy: at ea h iteration, they hoose a set of \independent" points to insert into the domain, and then update the Delaunay triangulation. We show that su h a set of independent points an be onstru ted eÆ iently 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 ea h independent set of points an be realized sequentially by Ruppert's method in two dimensions and Shew huk's in three dimensions. Therefore, our parallel Delaunay re nement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasi-uniform meshes, su h as those produ ed by Chew's method, we show that the number of iterations an be redu ed to O(log(L/s)). To the best of our knowledge, these are the rst provably polylog(L/s) parallel time Delaunay meshing algorithms that generate well-shaped meshes of size optimal to within a onstant.