Abstract not found

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation.

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.

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the level below. Point location is done by walking in a triangulation to determine the nearest neighbor of the query at that level, then the walk restarts from that neighbor at the level below. Using a small subset (3%) to sample a level allows a small memory occupation; the walk and the use of the nearest neighbor to change levels quickly locate the query.

In this paper we are concerned with developing a practical parallel algorithm for Delaunay triangulation that works well on general distributions, particularly those that arise in Scientific Computation. Although there have been many theoretical 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 3D convex hull of points on a sphere or paraboloid. A variant of the Edelsbrunner and Shi 3D convex hull is used, but for the special case when the point set lies on either a sphere or a paraboloid. Our variant greatly reduces the constant costs from the 3D convex hull algorithm and seems to be a more promising for a practical implementation than other parallel approaches. We have run experiments on the algorithm using a variety of distributions that are motivated by various problems that use Delau...

Abstract not found

Many problems in computer graphics deal with complex 3D scenes where visibility, proximity, collision detection queries have to be answered. Due to the complexity of these queries and the one of the models they are applied to, data structures most often based on hierarchical decompositions have been proposed to solve them. As a result of these involved algorithms/data structures, most of the analysis have been carried out in the worst case and fail to report good average case performances in a vast majority of cases. The goal of this work is therefore to investigate geometric probability tools to characterize average case properties of standard scenes such as architectural scenes, natural models, etc under some standard visibility and proximity requests. In the first part we recall some fundamentals of integral geometry and discuss the classical assumption of measures invariant under the group of motions in the context of non uniform models. In the second one we present simple generali...

Data structures based on uniform subdivisions of the space ---also known as bucketing--- have the nice properties that they can be walked through very easily and can provide neighborhood relations at low cost. For data sets which are uniformly scattered in 2D or 3D space, this makes the implementation of algorithms such as ray tracing, nearest neighbors computation or Delaunay triangulation almost trivial. But should the processed data set admit dense clusters, the spatial partitioning does not result in data partitioning so that the performances are collapsing.

This paper describes the derivation of an empirically efficient parallel two-dimensional Delaunay triangulation program from a theoretically efficient CREW PRAM algorithm. Compared to previous work, the resulting implementation is not limited to datasets with a uniform distribution of points, achieves significantly better speedups over good serial code, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for a loosely-coupled cluster of workstations, a distributed-memory multicomputer, and a shared-memory multiprocessor. The Machiavelli toolkit used to transform the nested data parallelism inherent in the divide-and-conquer algorithm into achievable task and data parallelism is also described and compared to previous techniques.

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