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33
A Replacement for Voronoi Diagrams of Near Linear Size
 In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci
, 2001
"... For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an εapproximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d ≥ 2. This contrasts with the standard Voronoi diagram that has ..."
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Cited by 87 (6 self)
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For a set P of n points in R^d, we define a new type of space decomposition. The new diagram provides an εapproximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d ≥ 2. This contrasts with the standard Voronoi diagram that has complexity Ω(n^⌈d/2⌉) in the worst case.
SmoothSurface Reconstruction in Near Linear Time
, 2001
"... A surface reconstruction algorithm takes as input a set of sample points from an unknown closed and smooth surface in 3d space, and produces a piecewise linear approximation of the surface that contains the sample points. Variants of this problem have received considerable attention in computer ..."
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Cited by 42 (6 self)
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A surface reconstruction algorithm takes as input a set of sample points from an unknown closed and smooth surface in 3d space, and produces a piecewise linear approximation of the surface that contains the sample points. Variants of this problem have received considerable attention in computer vision and computer graphics and more recently in computational geometry. In the latter area, three different algorithms (Amenta and Bern `98, and refined in Amenta, Choi, Dey and Leekha `00; Amenta, Choi and Kolluri `00; Boissonnat and Cazals `00) have been proposed. These algorithms have a correctness guarantee: if the sample is sufficiently dense then the output is a good approximation to the original surface. They have unfortunately a worstcase running time that is quadratic in the size of the input. This is so because they are based on the construction of 3d Voronoi diagrams or Delaunay tetrahedrizations, which can have quadratic size. Even worse, according to recent work (Erickson `01), there are surfaces for which this is the case even when the sample set is "locally uniform" on the surface. In this paper, we describe a new algorithm that also has a correctness guarantee but whose worstcase running time is almost linear. In fact, O(n log n) where n is the input size. As in some of the previous algorithms, the piecewise linear approximation produced by the new algorithm is a subset of the 3d Delaunay tetrahedrization; however, this is obtained by computing only the relevant parts of the 3d Delaunay structure. The algorithm first estimates for each sample point the surface normal and a parameter that is then used to "decimate" the set of samples. The resulting subset of sample points is locally uniform and so a reconstruction based on it can be compu...
On ConflictFree Coloring of Points and Simple Regions in the Plane
"... In this paper, we study coloring problems related to frequency assignment problems in cellular networks. In abstract setting, the problems are of the following two types: CFcoloring of regions: Given a finite family of n regions of some fixed type (such as discs, pseudodiscs, axisparallel rec ..."
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Cited by 37 (8 self)
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In this paper, we study coloring problems related to frequency assignment problems in cellular networks. In abstract setting, the problems are of the following two types: CFcoloring of regions: Given a finite family of n regions of some fixed type (such as discs, pseudodiscs, axisparallel rectangles, etc.), what is the minimum integer k, such that one can assign a color to each region of using a total of at most k colors, such that the resulting coloring has the following property: For each point p b#S b there is at least one region b # S that contains p in its interior, whose color is unique among all regions in that contain p in their interior (in this case we say that p is being `served' by that color). We refer to such a coloring as a conflictfree coloring of (CFcoloring in short).
The Delaunay hierarchy
 Internat. J. Found. Comput. Sci
"... 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 s ..."
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Cited by 36 (6 self)
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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.
Collision detection for deforming necklaces
 IN SYMP. ON COMPUTATIONAL GEOMETRY
, 2002
"... In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, rope, and other ‘linear ’ objects in the physical world. In this paper, we exploit this linearity ..."
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Cited by 34 (11 self)
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In this paper, we propose to study deformable necklaces — flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macromolecules, muscles, rope, and other ‘linear ’ objects in the physical world. In this paper, we exploit this linearity to develop geometric structures associated with necklaces that are useful in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres built on a necklace. Such a hierarchy is easy to compute and is suitable for maintenance when the necklace deforms, as our theoretical and experimental results show. This hierarchy can be used for collision and selfcollision detection. In particular, we achieve an upper bound of O(nlog n) in two dimensions and O(n 2−2/d) in ddimensions, d ≥ 3, for collision checking. To our knowledge, this is the first subquadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be also used to detect selfcollisions of a necklace in certain ways complementary to the sphere hierarchy.
Provably Good Surface Sampling and Approximation
, 2003
"... We present an algorithm for meshing surfaces that is a simple adaptation of a greedy "farthest point" technique proposed by Chew. Given a surface S, it progressively adds points on S and updates the 3dimensional Delaunay triangulation of the points. The method is very simple and works in 3dspace ..."
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Cited by 34 (0 self)
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We present an algorithm for meshing surfaces that is a simple adaptation of a greedy "farthest point" technique proposed by Chew. Given a surface S, it progressively adds points on S and updates the 3dimensional Delaunay triangulation of the points. The method is very simple and works in 3dspace without requiring to parameterize the surface. Taking advantage of recent results on the restricted Delaunay triangulation, we prove that the algorithm can generate good samples on S as well as triangulated surfaces that approximate S. More precisely, we show that the restricted Delaunay triangulation Del # S of the points has the same topology type as S, that the Hausdorff distance between Del # S and S can be made arbitrarily small, and that we can bound the aspect ratio of the facets of Del # S . The algorithm has been implemented and we report on experimental results that provide evidence that it is very effective in practice. We present results on implicit surfaces, on CSG models and on polyhedra. Although most of our theoretical results are given for smooth closed surfaces, the method is quite robust in handling smooth surfaces with boundaries, and even nonsmooth surfaces.
Sparse Voronoi Refinement
 In Proceedings of the 15th International Meshing Roundtable
, 2006
"... a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in outputsensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, ..."
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Cited by 32 (21 self)
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a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in outputsensitive time O(nlog(L/s) + m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, this matches the optimal time bound of Θ(n log n + m). Our new technique uses interleaving: we maintain a sparse mesh as we mix the recovery of input features with the addition of Steiner vertices for quality improvement. 1
A linear bound on the complexity of the Delaunay triangulations of points on polyhedral surfaces
 Proc. 7th Annu. ACM Sympos. Solid Modeling Appl
"... Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the threedimensional Delaunay triangulation of a finite set ..."
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Cited by 31 (7 self)
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Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the threedimensional Delaunay triangulation of a finite set of points scattered over a surface. Their runningtime therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in may be quadratic in the worstcase, we show in this paper that it is only linear when the points are distributed on a fixed number of wellsampled facets of (e.g. the facets of a polyhedron). Our bound is deterministic and the constants are explicitly given. Categories and Subject Descriptors I.3.5 [Computing Methodologies]: Computational Geometry and
Shape Dimension and Approximation from Samples
, 2003
"... There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to ..."
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Cited by 31 (6 self)
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There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to. Based on this dimension detection, we also present an algorithm to approximate shapes of arbitrary dimension from their samples. Our empirical results with data sets in three dimensions support our theory.
Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Cited by 30 (2 self)
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating