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An Empirical Comparison of Techniques for Updating Delaunay Triangulations
, 2004
"... The computation of Delaunay triangulations from static point sets has been extensively studied in computational geometry. When the points move with known trajectories, kinetic data structures can be used to maintain the triangulation. However, there has been little work so far on how to maintain the ..."
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Cited by 31 (3 self)
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The computation of Delaunay triangulations from static point sets has been extensively studied in computational geometry. When the points move with known trajectories, kinetic data structures can be used to maintain the triangulation. However, there has been little work so far on how to maintain the triangulation when the points move without explicit motion plans, as in the case of a physical simulation. In this paper we examine how to update Delaunay triangulations after small displacements of the defining points, as might be provided by a physicsbased integrator. We have implemented a variety of update algorithms, many new, toward this purpose. We ran these algorithms on a corpus of data sets to provide running time comparisons and determined that updating Delaunay can be significantly faster than recomputing.
Lecture Notes on Delaunay Mesh Generation
, 1999
"... purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ..."
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Cited by 22 (0 self)
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purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the
I/OEfficient Construction of Constrained Delaunay Triangulations
"... In this paper, we designed and implemented an I/Oefficient algorithm for constructing constrained Delaunay triangulations. If the number of constraining segments is smaller than the memory size, our algorithm runs in expected O ( N B logM/B N B) I/Os for triangulating N points in the plane, where M ..."
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Cited by 19 (5 self)
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In this paper, we designed and implemented an I/Oefficient algorithm for constructing constrained Delaunay triangulations. If the number of constraining segments is smaller than the memory size, our algorithm runs in expected O ( N B logM/B N B) I/Os for triangulating N points in the plane, where M is the memory size and B is the disk block size. If there are more constraining segments, the theoretical bound does not hold, but in practice the performance of our algorithm degrades gracefully. Through an extensive set of experiments with both synthetic and real data, we show that our algorithm is significantly faster than existing implementations.
Delaunay Triangulations in O(sort(n)) Time and More
"... We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports ..."
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Cited by 15 (6 self)
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We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports the shuffleoperation in constant time; (ii) if we know the ordering of a planar point set in x and in ydirection, its DT can be found by a randomized algebraic computation tree of expected linear depth; (iii) given a universe U of points in the plane, we construct a data structure D for Delaunay queries: for any P ⊆ U, D can find the DT of P in time O(P  log log U); (iv) given a universe U of points in 3space in general convex position, there is a data structure D for convex hull queries: for any P ⊆ U, D can find the convex hull of P in time O(P (log log U) 2); (v) given a convex polytope in 3space with n vertices which are colored with χ> 2 colors, we can split it into the convex hulls of the individual color classes in time O(n(log log n) 2). The results (i)–(iii) generalize to higher dimensions. We need a wide range of techniques. Most prominently, we describe a reduction from DTs to nearestneighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling.
Incremental Construction of the Delaunay Triangulation and the Delaunay Graph in Medium Dimension
, 2009
"... We describe a new implementation of the wellknown incremental algorithm for constructing Delaunay triangulations in any dimension. Our implementation follows the exact computing paradigm and is fully robust. Extensive comparisons show that our implementation outperforms the best currently available ..."
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Cited by 14 (1 self)
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We describe a new implementation of the wellknown incremental algorithm for constructing Delaunay triangulations in any dimension. Our implementation follows the exact computing paradigm and is fully robust. Extensive comparisons show that our implementation outperforms the best currently available codes for exact convex hulls and Delaunay triangulations, compares very well to the fast nonexact Qhull implementation and can be used for quite big input sets in spaces of dimensions up to 6. To circumvent prohibitive memory usage, we also propose a modi cation of the algorithm that uses and stores only the Delaunay graph (the edges of the full triangulation). We show that a careful implementation of the modi ed algorithm performs only 6 to 8 times slower than the original algorithm while drastically reducing memory usage in dimension 4 or above.
MEIJERS M.: Topologically consistent 3D city models obtain by extrusion
 International Journal of Geographical Information Science, In Press, LEE J., ZLATANOVA S.: A
, 2008
"... One of the simplest methods to construct a 3D city model is to extrude building footprints to obtain “blockshaped ” polyhedra representing buildings. While the method is wellknown and easy to implement, if the 2D topological relationships between the footprints are not taken into account, the resu ..."
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Cited by 8 (1 self)
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One of the simplest methods to construct a 3D city model is to extrude building footprints to obtain “blockshaped ” polyhedra representing buildings. While the method is wellknown and easy to implement, if the 2D topological relationships between the footprints are not taken into account, the resulting 3D city models will not necessarily be topologically consistent (i.e. primitives shared by 3D buildings will be duplicated and/or intersect each others). As a result, the model will be of little use for most applications, besides visualisation that is. In this paper, we present a new extrusion procedure to construct topologically correct 3D city models. It is based on the use of a constrained triangulation, is conceptually simple, and offers great flexibility to create city models in different formats (e.g. CityGML or a surfacebased representation). We have implemented the procedure, tested it with realworld datasets, and validated it. 1
Simple and Fast Nearest Neighbor Search
 IN: 2010 PROCEEDINGS OF THE TWELFTH WORKSHOP ON ALGORITHM ENGINEERING AND EXPERIMENTS
, 2010
"... We present a simple randomized data structure for twodimensional point sets that allows fast nearest neighbor queries in many cases. An implementation outperforms several previous implementations for commonly used benchmarks. ..."
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Cited by 8 (1 self)
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We present a simple randomized data structure for twodimensional point sets that allows fast nearest neighbor queries in many cases. An implementation outperforms several previous implementations for commonly used benchmarks.
DELAUNAY TRIANGULATION OF IMPRECISE POINTS: PREPROCESS AND ACTUALLY GET A FAST QUERY TIME
 JOURNAL OF COMPUTATIONAL GEOMETRY
, 2011
"... We propose a new algorithm to preprocess a set of n disjoint unit disks in O(n log n) expected time, allowing to compute the Delaunay triangulation of a set of n points, one from each disk, in O(n) expected time. Our algorithm has the same asymptotic complexity as previous ones for this problem, bu ..."
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Cited by 8 (0 self)
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We propose a new algorithm to preprocess a set of n disjoint unit disks in O(n log n) expected time, allowing to compute the Delaunay triangulation of a set of n points, one from each disk, in O(n) expected time. Our algorithm has the same asymptotic complexity as previous ones for this problem, but our algorithm is much simpler and it runs faster in practice than a direct computation of the Delaunay triangulation.
Deterministic Galois: Ondemand, Portable and Parameterless
 In Proc. of ASPLOS
, 2014
"... Nondeterminism in program execution can make program development and debugging difficult. In this paper, we argue that solutions to this problem should be ondemand, portable and parameterless. Ondemand means that the programming model should permit the writing of nondeterministic programs since ..."
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Cited by 7 (0 self)
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Nondeterminism in program execution can make program development and debugging difficult. In this paper, we argue that solutions to this problem should be ondemand, portable and parameterless. Ondemand means that the programming model should permit the writing of nondeterministic programs since these programs often perform better than deterministic programs for the same problem. Portable means that the program should produce the same answer even if it is run on different machines. Parameterless means that if there are machinedependent scheduling parameters that must be tuned for good performance, they must not affect the output. Although many solutions for deterministic program execution have been proposed in the literature, they fall short along one or more of these dimensions. To remedy this, we propose a new approach, based on the Galois programming model, in which (i) the programming model permits the writing of nondeterministic programs and (ii) the runtime system executes these programs deterministically if needed. Evaluation of this approach on a collection of benchmarks from the PARSEC, PBBS, and Lonestar suites shows that it delivers deterministic execution with substantially less overhead than other systems in the literature.
Parallel geometric algorithms for multicore computers. Research Report 6749
, 2008
"... 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 ddimensional algorithms we describe are (a) s ..."
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Cited by 6 (1 self)
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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 ddimensional algorithms we describe are (a) spatial sorting of points, as is typically used for preprocessing before using incremental algorithms, (b) kdtree construction, (c) axisaligned box intersection computation, and finally (d) bulk insertion of points in Delaunay triangulations for mesh generation algorithms or simply computing Delaunay triangulations. We show experimental results for these algorithms in 3D, using our implementations based on the Computational Geometry Algorithms Library (CGAL1). 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.