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Approximating extent measure of points
 Journal of ACM
"... We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our tec ..."
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Cited by 96 (28 self)
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We present a general technique for approximating various descriptors of the extent of a set of points in�when the dimension�is an arbitrary fixed constant. For a given extent measure�and a parameter��, it computes in time a subset�of size, with the property that. The specific applications of our technique include�approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms. 1
Geometric approximation via coresets
 Combinatorial and Computational Geometry, MSRI
, 2005
"... Abstract. The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem o ..."
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Cited by 60 (7 self)
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Abstract. The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem on Q using a relatively inefficient algorithm. The solution for Q is then translated to an approximate solution to the original point set P. This paper describes the ways in which this paradigm has been successfully applied to various optimization and extent measure problems. 1.
Projective Clustering in High Dimensions using CoreSets
, 2002
"... Let P be a set of n points in IRd, and for any integer 0 < = k < = d 1, let RDk(P) denote the minimum over all kflats F of maxp2P dist(p, F). We present an algorithm that computes, for any 0 < " < 1, a kflat that is within a distance of (1 + ")RDk(P) from each point of P. The running ti ..."
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Cited by 32 (9 self)
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Let P be a set of n points in IRd, and for any integer 0 < = k < = d 1, let RDk(P) denote the minimum over all kflats F of maxp2P dist(p, F). We present an algorithm that computes, for any 0 < " < 1, a kflat that is within a distance of (1 + ")RDk(P) from each point of P. The running time of the algorithm is dnO(k/" 5 log(1/")). The crucial step in obtaining this algorithm is a structural result that says that there is a nearoptimal flat that lies in an affine subspace spanned by a small subset of points in P. The size of this "coreset" depends on k and ε but is independent of the dimension. This
Maintaining Approximate Extent Measures of Moving Points
 In Proc. 12th ACMSIAM Sympos. Discrete Algorithms
, 2001
"... We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smal ..."
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Cited by 31 (4 self)
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We present approximation algorithms for maintaining various descriptors of the extent of moving points in R d . We first describe a data structure for maintaining the smallest orthogonal rectangle containing the point set. We then use this data structure to maintain the approximate diameter, smallest enclosing disk, width, and smallest area or perimeter bounding rectangle of a set of moving points in R 2 so that the number of events is only a constant. This contrasts with\Omega\Gamma n 2 ) events that data structures for the maintenance of those exact properties have to handle. 1 Introduction With the rapid advances in positioning systems, e.g., GPS, adhoc networks, and wireless communication, it is becoming increasingly feasible to track and record the changing position of continuously moving objects. These developments have raised a wide range of challenging geometric problems involving moving objects, including efficient data structures for answering proximity queries, fo...
Computing Diameter in the Streaming and SlidingWindow Models
 Algorithmica
, 2002
"... We investigate the diameter problem in the streaming and slidingwindow models. We show that, for a stream of n points or a sliding window of size n, any exact algorithm for diameter requires Ω(n) bits of space. We present a simple ɛapproximation 1algorithm for computing the diameter in the streami ..."
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Cited by 28 (3 self)
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We investigate the diameter problem in the streaming and slidingwindow models. We show that, for a stream of n points or a sliding window of size n, any exact algorithm for diameter requires Ω(n) bits of space. We present a simple ɛapproximation 1algorithm for computing the diameter in the streaming model. Our main result is an ɛapproximation algorithm that maintains the diameter in two dimensions in the sliding windows model using O ( 1 ɛ3/2 log 3 n(log R + log log n + log 1 ɛ)) bits of space, where R is the maximum, over all windows, of the ratio of the diameter to the minimum nonzero distance between any two points in the window. 1 introduction In recent years, massive data sets have become increasingly important in a wide range of applications. In many applications, the input can be viewed as a data stream [12, 7] that the
On efficient representation and computation of reachable sets for hybrid systems
 In HSCC’2003, LNCS 2289
, 2003
"... Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approxim ..."
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Cited by 28 (6 self)
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Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the overapproximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.
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 27 (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
MinimumVolume Enclosing Ellipsoids and Core Sets
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 2005
"... We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly improved ..."
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Cited by 25 (4 self)
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We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly improved complexity bound of O(nd (0, 1). As a byproduct, our algorithm returns a core set with the property that the minimum volume enclosing ellipsoid of provides a good approximation to that of S.
Penetration Depth of Two Convex Polytopes in 3D
 Nordic J. Computing
, 2000
"... with m and n facets, respectively. The penetration depth of A and B, denoted as (A; B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes (A; B) in O(m + m ) expected time, for any constant " > 0. I ..."
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Cited by 24 (2 self)
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with m and n facets, respectively. The penetration depth of A and B, denoted as (A; B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes (A; B) in O(m + m ) expected time, for any constant " > 0. It also computes a vector t such that ktk = (A; B) and int(A + t) \ B = ;. We show that if the Minkowski sum B ( A) has K facets, then the expected running time of our algorithm is O K , for any " > 0.
Sublinear geometric algorithms
 In Proc. of the 35th Annual ACM Symp. on Theory of Computing
, 2003
"... Abstract. We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in twodimensional triangulations and Voronoi diagrams, and ray shooting in convex ..."
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Cited by 19 (2 self)
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Abstract. We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in twodimensional triangulations and Voronoi diagrams, and ray shooting in convex polyhedra, all of which run in expected time O ( √ n), where n is the size of the input. We also provide sublinear solutions for the approximate evaluation of the volume of a convex polytope and the length of the shortest path between two points on the boundary. Key words. sublinear algorithms, approximate shortest paths, polyhedral intersection