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28
Faster CoreSet Constructions and Data Stream Algorithms in Fixed Dimensions
 Comput. Geom. Theory Appl
, 2003
"... We speed up previous (1 + ")factor approximation algorithms for a number of geometric optimization problems in xed dimensions: diameter, width, minimumradius enclosing cylinder, minimumwidth annulus, minimumvolume bounding box, minimumwidth cylindrical shell, etc. ..."
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Cited by 82 (6 self)
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We speed up previous (1 + ")factor approximation algorithms for a number of geometric optimization problems in xed dimensions: diameter, width, minimumradius enclosing cylinder, minimumwidth annulus, minimumvolume bounding box, minimumwidth cylindrical shell, etc.
Practical Methods for Shape Fitting and Kinetic Data Structures using Core Sets
 In Proc. 20th Annu. ACM Sympos. Comput. Geom
, 2004
"... The notion of εkernel was introduced by Agarwal et al. [5] to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the expanded slab (1 + ε)W contains P. They illustrat ..."
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Cited by 30 (8 self)
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The notion of εkernel was introduced by Agarwal et al. [5] to set up a unified framework for computing various extent measures of a point set P approximately. Roughly speaking, a subset Q ⊆ P is an εkernel of P if for every slab W containing Q, the expanded slab (1 + ε)W contains P. They illustrated the significance of εkernel by showing that it yields approximation algorithms for a wide range of geometric optimization problems. We present a simpler and more practical algorithm for computing the εkernel of a set P of points in R d. We demonstrate the practicality of our algorithm by showing its empirical performance on various inputs. We then describe an incremental algorithm for fitting various shapes and use the ideas of our algorithm for computing εkernels to analyze the performance of this algorithm. We illustrate the versatility and practicality of this technique by implementing approximation algorithms for minimum enclosing cylinder, minimumvolume bounding box, and minimumwidth annulus. Finally, we show that εkernels can be effectively used to expedite the algorithms for maintaining extents of moving points. 1
Streaming geometric optimization using graphics hardware
 In Proc. 11th European Sympos. Algorithms, Lect. Notes Comput. Sci
, 2003
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A Practical Approximation Algorithm for the LMS Line Estimator
, 1997
"... The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Robust estimators are particularly important because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is it ..."
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Cited by 19 (4 self)
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The problem of fitting a straight line to a finite collection of points in the plane is an important problem in statistical estimation. Robust estimators are particularly important because of their lack of sensitivity to outlying data points. The basic measure of the robustness of an estimator is its breakdown point, that is, the fraction (up to 50%) of outlying data points that can corrupt the estimator. Rousseeuw's least medianofsquares (LMS) regression (line) estimator [11] is among the best known 50% breakdown point estimators. The best exact algorithms known for this problem run in O(n 2 ) time, where n is the number of data points. Because of this high running time, many practitioners prefer to use a simple O(n log n) Monte Carlo algorithm, which is quite efficient but provides no guarantees of accuracy (even probabilistic) unless the data set satisfies certain assumptions. In this paper, we present two algorithms in an attempt to close the gap between theory and practice. ...
Computing the Diameter of a Point Set
"... Given a finite set of points P in R^d, the diameter of P is defined as the maximum distance between two points of P. We propose a very simple algorithm to compute the diameter of a finite set of points. Although the algorithm is not worstcase optimal, it appears to be extremely fast for a large var ..."
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Cited by 14 (0 self)
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Given a finite set of points P in R^d, the diameter of P is defined as the maximum distance between two points of P. We propose a very simple algorithm to compute the diameter of a finite set of points. Although the algorithm is not worstcase optimal, it appears to be extremely fast for a large variety of point distributions.
Antipole Tree indexing to support range search and Knearest neighbor search in metric spaces
 IEEE/TKDE
, 2005
"... Range and knearest neighbor searching are core problems in pattern recognition. Given a database S of objects in a metric space M and a query object q in M, in a range searching problem the target is to find the objects of S within some threshold distance to q, whereas in a knearest neighbor searc ..."
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Cited by 11 (0 self)
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Range and knearest neighbor searching are core problems in pattern recognition. Given a database S of objects in a metric space M and a query object q in M, in a range searching problem the target is to find the objects of S within some threshold distance to q, whereas in a knearest neighbor searching problem, the k elements of S closest to q must be produced. These problems can obviously be solved with a linear number of distance calculations, by comparing the query object against every object in the database. However, the goal is to solve such problems much faster. We combine and extend ideas from the MTree, the MultiVantage Point structure, and the FQTree to create a new structure in the “bisector tree ” class, called the Antipole Tree. Bisection is based on the proximity to an “Antipole ” pair of elements generated by a suitable linear randomized tournament. The final winners a, b of such a tournament are far enough apart to approximate the diameter of the splitting set. If dist(a, b) is larger than the chosen cluster diameter threshold, then the cluster is split. The proposed data structure is an indexing scheme suitable for (exact and approximate) best match searching on generic metric spaces. The Antipole Tree compares very well with existing structures such as List of Clusters, MTrees and others, and in many cases it achieves better results.
Optimal Location of Transportation Devices
, 2007
"... We consider algorithms for finding the optimal location of a simple transportation device, that we call a moving walkway, consisting of a pair of points in the plane between which the travel speed is high. More specifically, one can travel from one endpoint of the walkway to the other at speed v> ..."
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Cited by 3 (1 self)
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We consider algorithms for finding the optimal location of a simple transportation device, that we call a moving walkway, consisting of a pair of points in the plane between which the travel speed is high. More specifically, one can travel from one endpoint of the walkway to the other at speed v> 1, but can only travel at unit speed between any other pair of points. The travel time between any two points in the plane is the minimum between the actual geometric distance, and the time needed to go from one point to the other using the walkway. A location for a walkway is said to be optimal with respect to a given finite set of points if it minimizes the maximum travel time between any two points of the set. We give a simple lineartime algorithm for finding an optimal location in the case where the points are on a line. We also give an Ω(n log n) lower bound for the problem of computing the travel time diameter of a set of n points on a line with respect to a given walkway. Then we describe an O(n log n) algorithm for locating a walkway with the additional restriction that the walkway must be horizontal. This algorithm is based on a recent generic method for solving quasiconvex programs with implicitly defined constraints. It is used in a (1+ε)approximation algorithm for optimal location of a walkway of arbitrary orientation.
Shapebased transfer functions for volume visualization
 In Proceedings of the IEEE Pacific Visualization Symposium
, 2010
"... We present a novel classification technique for volume visualization that takes the shape of volumetric features into account. The presented technique enables the user to distinguish features based on their 3D shape and to assign individual optical properties to these. Based on a rough presegmentat ..."
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We present a novel classification technique for volume visualization that takes the shape of volumetric features into account. The presented technique enables the user to distinguish features based on their 3D shape and to assign individual optical properties to these. Based on a rough presegmentation that can be done by windowing, we exploit the curveskeleton of each volumetric structure in order to derive a shape descriptor similar to those used in current shape recognition algorithms. The shape descriptor distinguishes three main shape classes: longitudinal, surfacelike, and blobby shapes. In contrast to previous approaches, the classification is not performed on a pervoxel level but assigns a uniform shape descriptor to each feature and therefore allows a more intuitive user interface for the assignment of optical properties. By using the proposed technique, it becomes for instance possible to distinguish blobby heart structures filled with contrast agents from potentially occluding vessels and rib bones. After introducing the basic concepts, we show how the presented technique performs on real world data, and we discuss current limitations.
A tight lower bound for computing the diameter of a 3D convex polytope
, 2007
"... The diameter of a set P of n points in R d is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is s ..."
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Cited by 2 (0 self)
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The diameter of a set P of n points in R d is the maximum Euclidean distance between any two points in P. If P is the vertex set of a 3dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(n log n) time in the algebraic computation tree model. It shows that the O(n log n) time algorithm of Ramos for computing the diameter of a point set in R 3 is optimal for computing the diameter of a 3polytope. We also give a linear time reduction from Hopcroft’s problem of finding an incidence between points and lines in R 2 to the diameter problem for a point set in R 7.