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29
Nearest-neighbor searching and metric space dimensions
- In Nearest-Neighbor Methods for Learning and Vision: Theory and Practice
, 2006
"... Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distan ..."
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Cited by 107 (0 self)
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Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distance function as a “black box”. The structure is able to speed up nearest neighbor searching in a variety of settings, for example: points in low-dimensional or structured Euclidean space, strings under Hamming and edit distance, and bit vector data from an OCR application. The data structures are observed to need linear space, with a modest constant factor. The preprocessing time needed per site is observed to match the query time. The data structure can be viewed as an application of a “kd-tree ” approach in the metric space setting, using Voronoi regions of a subset in place of axis-aligned boxes. 1
Complexity of the Delaunay triangulation of points on surfaces: the smooth case
- In Annual Symposium on Computational Geometry
, 2003
"... It is well known that the complexity of the Delaunay trian-gulation of N points in 3, i.e. the number of its faces, can be (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms rst construct the Delau-n ..."
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Cited by 54 (15 self)
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It is well known that the complexity of the Delaunay trian-gulation of N points in 3, i.e. the number of its faces, can be (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms rst construct the Delau-nay triangulation of a set of points measured on a surface. In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth sur-faces of 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N). Categories and Subject Descriptors F.2.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity|Geometrical problems and com-
Deformable spanners and applications
- In Proc. of the 20th ACM Symposium on Computational Geometry (SoCG’04
, 2004
"... For a set S of points in R d,ans-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d) edges, where ε is a spe ..."
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Cited by 49 (6 self)
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For a set S of points in R d,ans-spanner is a graph on S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε d) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decomposition, and approximate k-centers. 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 three-dimensional Delaunay triangulation of a finite set ..."
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Cited by 37 (10 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 three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time 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 worst-case, we show in this paper that it is only linear when the points are distributed on a fixed number of well-sampled 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
Collision detection for deforming necklaces
, 2004
"... 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, ropes, and other linear objects in the physical world. We exploit this linearity to develop geometr ..."
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Cited by 34 (9 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, ropes, and other linear objects in the physical world. We exploit this linearity to develop geometric structures associated with necklaces that are useful for collision detection 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 which can be used for collision and self-collision detection of deforming and moving necklaces. As our theoretical and experimental results show, such a hierarchy is easy to compute and, more importantly, is also easy to maintain when the necklace deforms. Using this hierarchy, we achieve a collision detection upper bound of ¦¨§�©�������©� � in two dimensions and ¦¨§�©����������� � in �-dimensions, ���� �. 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 used to
Instance-optimal geometric algorithms
"... ... in 2-d and 3-d, and off-line point location in 2-d. We prove the existence of an algorithm A for computing 2-d or 3-d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of ..."
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Cited by 17 (2 self)
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... in 2-d and 3-d, and off-line point location in 2-d. We prove the existence of an algorithm A for computing 2-d or 3-d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of A on input 〈s1,..., sn〉 is at most a constant factor times the maximum running time of A ′ on 〈s1,..., sn〉, where the maximum is taken over all permutations 〈s1,..., sn 〉 of S. In fact, we can establish a stronger property: for every S and A ′ , the maximum running time of A is at most a constant factor times the average running time of A ′ over all permutations of S. We call algorithms satisfying these properties instance-optimal in the order-oblivious and random-order setting. Such instance-optimal algorithms simultaneously subsume output-sensitive algorithms and distribution-dependent average-case algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input is given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instance-specific lower bound, we deviate from traditional Ben–Or-style proofs and adopt an interesting adversary argument. For 2-d convex hulls, we prove that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap (1995) already attains this lower bound. For 3-d convex hulls, we propose a new algorithm. To demonstrate the potential of the concept, we further obtain instance-optimal results for a few other standard problems in computational geometry, such as maxima in 2-d and 3-d, orthogonal line segment intersection in 2-d, finding bichromatic L∞-close pairs in 2-d, off-line orthogonal range searching in 2-d, off-line dominance reporting in 2-d and 3-d, off-line halfspace range reporting 1.
State of the Union (of Geometric Objects)
- CONTEMPORARY MATHEMATICS
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play ..."
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Cited by 11 (7 self)
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play a central role in the analysis of many geometric algorithms, and the techniques used to attain these bounds are interesting in their own right.
Well-separated pair decomposition for the unit-disk graph metric and its applications
- SIAM Journal on Computing
, 2003
"... Abstract. We extend the classic notion of well-separated pair decomposition [10] to the unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c ≥ 1, there ex ..."
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Cited by 10 (2 self)
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Abstract. We extend the classic notion of well-separated pair decomposition [10] to the unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c ≥ 1, there exists a c-wellseparated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where k ≥ 3, there exists a c-wellseparated pair decomposition with O(n 2−2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric. Keywords Well separated pair decomposition, Unit-disk graph, Approximation algorithm
Particle Swarm Optimization with Spatially Meaningful Neighbors
- in Proc. of IEEE Swarm Intelligence Symposium
, 2008
"... Particle swarm optimization with spatially meaningful neighbours ..."
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Cited by 7 (0 self)
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Particle swarm optimization with spatially meaningful neighbours
Kinetic convex hulls and delaunay triangulations in the black-box model
- In Proc. 27th Annu. Sympos. Comput. Geom
, 2011
"... Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the ob-jects are known in advance. This assumption severely limits the applicability ..."
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Cited by 6 (1 self)
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Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the ob-jects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the tradi-tional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum dis-placement of any point in one time step. We study the maintenance of the convex hull and the De-launay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ∈ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ∆k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(k∆k log 2 n) amortized time. For the Delaunay triangu-lation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2∆2k) at each time step.
