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Practical Ways to Accelerate Delaunay Triangulations

Pedro de Castro 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : This thesis proposes several new practical ways to speed-up some of the most important operations in a Delaunay triangulation. We propose two approaches to compute a Delaunay triangulation for points on or close to a sphere. The first approach computes the Delaunay triangulation of points placed exactly on the sphere. The second approach directly computes the convex hull of the input set, and gives some guarantees on the output. Both approaches are based on the regular triangulation on the sphere. The second approach outperforms previous solutions. Updating a Delaunay triangulation when its vertices move is a bottleneck in several domains of application. Rebuilding the whole triangulation from scratch is surprisingly a viable option compared to relocating the vertices. However, when all points move with a small magnitude, or when only a fraction of the vertices moves, rebuilding is no longer the best option. We propose a filtering scheme based upon the concept of vertex tolerances. We conducted several experiments to showcase the behavior of the algorithm for a variety of data sets. The experiments showed that the algorithm is particularly relevant for convergent schemes such as the Lloyd iterations. In two dimensions, the algorithm presented performs up to an order of magnitude faster than rebuilding for Lloyd iterations. In three dimensions, although rebuilding the whole triangulation at each time stamp when all vertices move can be as fast as our algorithm, our solution is fully dynamic and outperforms previous dynamic solutions. This result makes it possible to go further on the number of iterations so as to produce higher quality meshes. Point location in spatial subdivision is one of the most studied problems in computational geometry. In the case of triangulations of Rd, we revisit the problem to exploit a possible coherence between the query points. We analyze, implement, and evaluate a distribution-sensitive point location algorithm based on the classical Jump & Walk, called Keep, Jump, & Walk. For a batch of query points, the main idea is to use previous queries to improve the retrieval of the current one. Regarding point location in a Delaunay triangulation, we show how the Delaunay hierarchy can be used to answer, under some hypotheses, a query q with a O(log ](pq)) randomized expected complexity, where p is a previously located query and ](s) indicates the number of simplices crossed by the line segment s. We combine the good distribution-sensitive behavior of Keep, Jump, & Walk, and the good complexity of the Delaunay hierarchy, into a novel point location algorithm called Keep, Jump, & Climb. To the best of our knowledge, Keep, Jump, & Climb is the first practical distribution-sensitive algorithm that works both in theory and in practice for Delaunay triangulations--in our experiments, it is faster than the Delaunay hierarchy regardless of the spatial coherence of queries, and significantly faster when queries have reasonable spatial coherence.
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Contributor : Pedro Machado Manhaes de Castro <>
Submitted on : Wednesday, November 3, 2010 - 5:02:38 PM
Last modification on : Tuesday, December 3, 2019 - 4:26:49 PM
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Pedro de Castro. Practical Ways to Accelerate Delaunay Triangulations. Software Engineering [cs.SE]. Université Nice Sophia Antipolis, 2010. English. ⟨tel-00531765⟩

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