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17
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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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 15 (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
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 10 (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. 1
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 8 (4 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.
Simple and Fast Nearest Neighbor Search
 In: 2010 Proceedings of the Twelfth Workshop on Algorithm Engineering and Experiments
"... 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. 1 ..."
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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. 1
Markov Incremental Constructions
, 2008
"... A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be“locally”random. Much att ..."
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Cited by 2 (2 self)
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A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be“locally”random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology. We generalize Mulmuley’s theory of Θseries and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving ClarksonShor type bounds for this new model; (iii) applying the results to classical geometric problems. We hope that this work will pioneer a new approach to averagecase analysis in computational geometry.
LinearTime Delaunay Triangulations Simplified ∗
"... Recently it was shown that — under reasonable assumptions — Voronoi diagrams and Delaunay triangulations of planar point sets can be computed in time o(n log n), beating the classical comparisonbased lower bound. A number of increasingly faster randomized algorithms have been proposed, most recently ..."
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Recently it was shown that — under reasonable assumptions — Voronoi diagrams and Delaunay triangulations of planar point sets can be computed in time o(n log n), beating the classical comparisonbased lower bound. A number of increasingly faster randomized algorithms have been proposed, most recently a lineartime algorithm based on a randomized incremental construction that uses a combination of nearest neighbor graphs and the history structure to speed up point location. We present a simpler variant of this approach relying only on nearest neighbor graphs. The algorithm and its analysis generalize to higher dimensions, with an expected performance that is proportional to the structural change of the randomized incremental construction. As a byproduct, we analyze an interesting class of insertion orders for randomized incremental constructions. 1
VALIDATION OF PLANAR PARTITIONS USING CONSTRAINED TRIANGULATIONS
"... Planar partitions—full tessellations of the plane into nonoverlapping polygons—are frequently used in GIS to model concepts such as land cover, cadastral parcels or administrative boundaries. Since in practice planar partitions are often stored as a set of individual objects (polygons) to which att ..."
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Planar partitions—full tessellations of the plane into nonoverlapping polygons—are frequently used in GIS to model concepts such as land cover, cadastral parcels or administrative boundaries. Since in practice planar partitions are often stored as a set of individual objects (polygons) to which attributes are attached (e.g. stored with a shapefile), and since different errors/mistakes can be introduced during their construction, manipulation or exchange, several inconsistencies will often arise in practice. The inconsistencies are for instance overlapping polygons, gaps and unconnected polygons. We present in this paper a novel algorithm to validate such planar partitions. It uses a constrained triangulation as a support for the validation, and permits us to avoid different problems that arise with existing solutions based on the construction of a planar graph. We describe in the paper the details of our algorithm, our implementation, how inconsistencies can be detected, and the experiments we have made with realworld data (the CORINE2000 dataset).
LowEntropy Computational Geometry
, 2010
"... The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional informa ..."
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The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional information can affect the design and analysis of geometric algorithms. First, we consider hereditary convex hulls: given a threedimensional convex polytope and a twocoloring of its vertices, we can find the individual monochromatic polytopes in linear expected time. This can be generalized in many ways, eg, to more than two colors, and to the offlineproblem where we wish to preprocess a polytope so that any large enough subpolytope can be found quickly. Our techniques can also be used to give a simple analysis of the selfimproving algorithm for planar Delaunay triangulations by Clarkson and Seshadhri [58]. Next, we assume that the point coordinates have a bounded number of bits, and that we can do standard bit manipulations in constant time. Then Delaunay triangulations can be found in expected time O(n √ log log n). Our result is based on a new connection between quadtrees and Delaunay triangulations, which also lets us generalize a recent result by Löffler and Snoeyink about Delaunay triangulations for imprecise points [110]. Finally, we consider randomized incremental constructions when the input permutation is generated by a boundeddegree Markov chain, and show that the resulting running time is almost optimal for chains with a constant eigenvalue gap.
Delaunay Triangulations in Linear Time? (Part I) ∗
, 812
"... We present a new and simple randomized algorithm for constructing the Delaunay triangulation using nearestneighbor graphs for point location. It runs in linear expected time for points in the plane with polynomially bounded spread, i.e., if the ratio between the largest and smallest pointwise dista ..."
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We present a new and simple randomized algorithm for constructing the Delaunay triangulation using nearestneighbor graphs for point location. It runs in linear expected time for points in the plane with polynomially bounded spread, i.e., if the ratio between the largest and smallest pointwise distance is polynomially bounded. This also holds for point sets with bounded spread in higher dimensions as long as the expected complexity of the Delaunay triangulation of a sample of the points is linear in the sample size. Chan and Patracu [6, 7] presented o(N log N) randomized algorithms for constructing Voronoi Diagrams of points in the plane (from which the Delaunay triangulation can be computed in linear time and viceversa) under suitable models of computation. We improve on these results by presenting an O(N) randomized algorithm for the Delaunay triangulation in the plane in a di erent model. The algorithm is not restricted to two dimensions and it runs in linear expected time as long as the expected complexity of the Delaunay triangulation of a random sample of the input points is linear in the sample size. An example of linear complexity Delaunay triangulation are suitably sampled (d − 1)dimensional polyhedra in