Results 1  10
of
19
Nearestneighbor searching and metric space dimensions
 In NearestNeighbor 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 87 (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 lowdimensional 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 “kdtree ” approach in the metric space setting, using Voronoi regions of a subset in place of axisaligned boxes. 1
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,ansspanner 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 35 (5 self)
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For a set S of points in R d,ansspanner 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), wellseparated pair decomposition, and approximate kcenters. 1
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 35 (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.
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 30 (6 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
Wellseparated pair decomposition for the unitdisk graph metric and its applications
 SIAM Journal on Computing
, 2003
"... Abstract. We extend the classic notion of wellseparated pair decomposition [10] to the unitdisk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unitdisk graph metric of n points in the plane and for any constant c ≥ 1, there ex ..."
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Cited by 8 (1 self)
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Abstract. We extend the classic notion of wellseparated pair decomposition [10] to the unitdisk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unitdisk graph metric of n points in the plane and for any constant c ≥ 1, there exists a cwellseparated 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 unitball graph metric in k dimensions where k ≥ 3, there exists a cwellseparated pair decomposition with O(n 2−2/k) pairs, and the bound is tight in the worst case. We present the application of the wellseparated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unitdisk graph metric. Keywords Well separated pair decomposition, Unitdisk graph, Approximation algorithm
Instanceoptimal geometric algorithms
"... ... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d 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 7 (1 self)
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... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d 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 instanceoptimal in the orderoblivious and randomorder setting. Such instanceoptimal algorithms simultaneously subsume outputsensitive algorithms and distributiondependent averagecase 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 instancespecific lower bound, we deviate from traditional Ben–Orstyle proofs and adopt an interesting adversary argument. For 2d 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 3d convex hulls, we propose a new algorithm. To demonstrate the potential of the concept, we further obtain instanceoptimal results for a few other standard problems in computational geometry, such as maxima in 2d and 3d, orthogonal line segment intersection in 2d, finding bichromatic L∞close pairs in 2d, offline orthogonal range searching in 2d, offline dominance reporting in 2d and 3d, offline halfspace range reporting 1.
Combinatorial and experimental methods for approximate point pattern matching
 Algorithmica
, 2003
"... Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary ..."
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Cited by 6 (0 self)
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Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods.
EmptyEllipse Graphs
, 2008
"... We define and study a geometric graph over points in the plane that captures the local behavior of Delaunay triangulations of points on smooth surfaces in IR 3. Two points in a planar point set P are neighbors in the emptyellipse graph if they lie on an axisaligned ellipse with no point of P in it ..."
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Cited by 4 (1 self)
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We define and study a geometric graph over points in the plane that captures the local behavior of Delaunay triangulations of points on smooth surfaces in IR 3. Two points in a planar point set P are neighbors in the emptyellipse graph if they lie on an axisaligned ellipse with no point of P in its interior. The emptyellipse graph can be a clique in the worst case, but it is usually much less dense. Specifically, the emptyellipse graph of n points has complexity Θ(∆n) in the worst case, where ∆ is the ratio between the largest and smallest pairwise distances. For points generated uniformly at random in a rectangle, the emptyellipse graph has expected complexity Θ(n log n). As an application of our proof techniques, we show that the Delaunay triangulation of n random points on a circular cylinder has expected complexity Θ(n log n).
State of the Union (of Geometric Objects): A Review
, 2007
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometr ..."
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Cited by 3 (1 self)
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These problems play a central role in the design and analysis of many geometric algorithms arising in robotics, molecular modeling, solid modeling, and shape matching, and the techniques used for their solutions are interesting in their own right.
New Bounds on the Size of Optimal Meshes
"... The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents ..."
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Cited by 3 (1 self)
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The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing φ, then the number of vertices in an optimal mesh will be O(φ d n), where d is the input dimension. We give a new analysis of this integral showing that the output size is only Θ(n + nlogφ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n).