Parallel 3D Delaunay Triangulation P. Cignoni † , C. Montani ‡ , R. Perego † , R. Scopigno † † CNUCE – Consiglio Nazionale delle Ricerche , Via S. Maria 36, 56126 Pisa, ITALY ‡ I.E.I. – Consiglio Nazionale delle Ricerche, Via S. Maria 46, 56126 Pisa, ITALY Abstract The paper deals with the parallelization of Delaunay triangulation algorithms, giving more emphasis to pratical issues and implementation than to theoretical complexity. Two parallel implementations are pre- sented. The first one is built on DeWall, an E d triangulator based on an original interpretation of the divide & conquer paradigm. The second is based on an incremental construction algorithm. The paral- lelization strategies are presented and evaluated. The target parallel machine is a distributed computing environment, composed of coarse grain processing nodes. Results of first implementations are reported and compared with the performance of the serial versions running on a Unix workstation. Keywords: Delaunay triangulation, divide & conquer, uniform grids, parallel processing, distributed computing. 1 Introduction Triangulation is a well known topic of computational geometry. It is routinely used in a broad range of applications, such as robotics, computer vision and image synthesis, as well as in mathematics and natural sciences. Delaunay Triangulation (DT) is a particular type of triangulation well known in Computational Geometry; many algorithms have been proposed for the DT of a set of sites in E 2 ,E 3 or E d [2]. Volume Rendering [16] is one of the latest applications of DT. A volume dataset consists of sampled points in E 3 space, with one or more scalar or vector values associated with each point. The spatial arrangement of pointsets can be either structured, with explicit or implicit topological relations between the sites, or unstructured. In the latter case, the triangulation of the set of points in E 3 is a prerequisite for execution of a class of surface reconstruction or direct rendering algorithms. The large number of sites Volume Rendering applications have to manage imposes strong efficiency constraints on the triangulator used. The same constraints are also common to other applications, such as, for example, digital terrain modeling [12] where digital maps with O(100K) triangular elements are common. High efficiency of a triangulator can be achieved by the use of optimization techniques, which can reduce the complexity of the basic algorithm, and by the exploitation of parallelism. This paper describes the parallelization of two Delaunay triangulation algorithms whose sequential imple- mentations were presented in detail in a previous paper [5]. The first algorithm is based on a particular reinterpretation of the divide & conquer paradigm, which makes it possible to specify both a generalized solution to DT in any dimension and a simple parallel implementation. The second is an optimized implementation of the incremental construction method. The paper is organized as follows. In Section 2 definitions and a classification of Delaunay triangulation algorithms are given, together with an overview of other works regarding parallelism and Delaunay triangulation. In Section 3 we describe the sequential divide & conquer algorithm and the optimization techniques used, followed in Section 4 by the description of the parallel divide & conquer solution. In Section 5 we present and evaluate a parallel incremental construction algorithm. Concluding remarks are in Section 6. 1