Design and Implementation of a Practical Parallel Delaunay Algorithm

Abstract

This paper describes the design and implementation of a practical parallel algorithm for Delaunay triangulation that works well on general distributions. Although there have been many theoretical parallel algorithms for the problem, and some implementations based on bucketing that work well for uniform distributions, there has been little work on implementations for general distributions. We use the well known reduction of 2D Delaunay triangulation to find the 3D convex hull of points on a paraboloid. Based on this reduction we developed a variant of the Edelsbrunner and Shi 3D convex hull algorithm, specialized for the case when the point set lies on a paraboloid. This simplification reduces the work required by the algorithm (number of operations) from O(n log 2 n) to O(n log n). The depth (parallel time) is O(log 3 n) on a CREW PRAM. The algorithm is simpler than previous O(n log n) work parallel algorithms leading to smaller constants.Initial experiments using a variety of distributions showed that our parallel algorithm was within a factor of 2 in work from the best sequential algorithm. Based on these promising results, the algorithm was implemented using C and an MPI-based toolkit. Compared with previous work, the resulting implementation achieves significantly better speedups over good sequential code, does not assume a uniform distribution of points, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for the IBM SP2, Cray T3D, SGI Power Challenge, and DEC AlphaCluster.

Incremental constructions con BRIO

“…[7]) but even the relatively simple early divide-and-conquer Delaunay triangulation algorithm of Clarkson [10] seems difficult to implement robustly. An exception is the practical parallel two-dimensional Delaunay triangulation algorithm of Blelloch, Miller, and Talmor [6]. Their approach, however, does not immediately apply to either three dimensions or to out-of-core computation.…”

Incremental constructions con BRIO

“…The happy buddha data set (2,643,633 points) is taken from the Stanford 3D scanning repository 6 . We use the raw scanner data as an example of typical input to a surface reconstruction computation.…”

A Parallel Dynamic-Mesh Lagrangian Method for Simulation of Flows With Dynamic Interfaces

“…To remedy this problem our parallel Delaunay algorithm uses a somewhat different approach [5]. We still use divide-and-conquer, but instead of doing most of the work when the recursive calls return we do most of the work at the divide step.…”

“…To maintain mesh quality and permit adaptivity, we generate in parallel an entirely new Delaunay triangulation of the dynamically-evolving grid points at each time step. The key to such an aggressive approach to meshing is our recent development of a parallel Delaunay triangulation algorithm that is theoretically optimal in work, requires polylogarithmic depth, and is very efficient in practice [5]. Given an arbitrary set of grid points, distributed across the processors, the algorithm returns a Delaunay triangulation of the points and at the same time partitions them for load balance and minimal communication across the processor boundaries.…”