Parallel geometric algorithms for multi-core computers

Abstract

Computers with multiple processor cores using shared memory are now ubiquitous. In this paper, we present several parallel geometric algorithms that specifically target this environment, with the goal of exploiting the additional computing power. The algorithms we describe are (a) 2-/3-dimensional spatial sorting of points, as is typically used for preprocessing before using incremental algorithms, (b) d-dimensional axis-aligned box intersection computation, and finally (c) 3D bulk insertion of points into Delaunay triangulations, which can be used for mesh generation algorithms, or simply for constructing 3D Delaunay triangulations. For the latter, we introduce as a foundational element the design of a container data structure that both provides concurrent addition and removal operations and is compact in memory. This makes it especially well-suited for storing large dynamic graphs such as Delaunay triangulations.We show experimental results for these algorithms, using our implementations based on the Computational Geometry Algorithms Library (CGAL). This work is a step towards what we hope will become a parallel mode for CGAL, where algorithms automatically use the available parallel resources without requiring significant user intervention.

Parallel d-D Delaunay Triangulations in Shared and Distributed Memory

“…Parallel incremental insertion algorithms are generally bootstrapped with a sequentially obtained initial triangulation of a subset of the input points. Subsequently, the rest of the points can be inserted in parallel by identifying the surrounding simplex for each point, removing it and re-triangulating the resulting cavity with the inserted point and the facets of the surrounding simplices [2,7,15]. The Delaunay property of the re-triangulated region is ensured by performing local flips [14,15].…”

“…To avoid simultaneous access to the same simplex during re-triangulation, locks need to be employed. Various locking strategies are studied in [2,15]. The algorithm of Batista et al is the basis for the parallel DT algorithm found in the CGAL library [12].…”

“…. , p n } and a recursion level r, if the number of points is below a certain Kohout et al [15] 3.7 (4 PEs) Batista et al [2] 7 (8 PEs) Lo [18] 10 (12 PEs)…”

“…Therefore, many algorithms to efficiently compute the DT have been proposed [23] and well implemented codes exist [12,20]. With ever increasing input sizes, research interest has shifted from sequential algorithms towards parallel ones [2,4,6,9,11,15], with shared memory parallelism for algorithms in two dimensions receiving most of the attention. Distributed memory algorithms however -as studied by Cignoni et al [9] and Lee et al [17] -are required to cope with triangulations exceeding the memory limitations of one machine.…”

Parallel algorithms for planar and spherical Delaunay construction with an application to centroidal Voronoi tessellations

“…Amato and Preparata (1993); Cignoni et al (1998); Zhou et al (2001); Batista et al (2010). Typically, such algorithms divide the point set up into several smaller subsets, each of which can then be triangulated independently from the others.…”