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93
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 119 (30 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
"... 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 usin ..."
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Cited by 84 (10 self)
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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.
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 44 (11 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.
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 ..."
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Cited by 39 (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 < &quot; < 1, a kflat that is within a distance of (1 + &quot;)RDk(P) from each point of P. The running time of the algorithm is dnO(k/&quot; 5 log(1/&quot;)). 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
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 imp ..."
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Cited by 38 (5 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.
On Khachiyan’s Algorithm for the Computation of Minimum Volume Enclosing Ellipsoids
, 2005
"... Given A: = {a 1,..., a m} ⊂ R d whose affine hull is R d, we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan’s barycentric ..."
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Cited by 36 (4 self)
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Given A: = {a 1,..., a m} ⊂ R d whose affine hull is R d, we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan’s barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using twosided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a new algorithm which computes an approximate rounding of the convex hull of A, and which can also be used to compute an approximation to the minimum volume enclosing ellipsoid of A. Our algorithm is a modification of the algorithm of Kumar and Yıldırım, which combines Khachiyan’s BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small “core set.” We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum volume
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 33 (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 32 (2 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
Realistic realtime rendering of landscapes using billboard clouds
 In Eurographics
"... We present techniques for realistic realtime rendering of complex landscapes that consist of many highly detailed plant models. The plants are approximated by dynamically changing sets of billboards. Realistic illumination is approximated using spherical harmonics. Since even the rendering of simpl ..."
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Cited by 31 (2 self)
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We present techniques for realistic realtime rendering of complex landscapes that consist of many highly detailed plant models. The plants are approximated by dynamically changing sets of billboards. Realistic illumination is approximated using spherical harmonics. Since even the rendering of simple billboard cloud plants is too time consuming, the landscape in the background is approximated with shell textures. The combination of these techniques allows us to render large scenes in realtime with varying illumination, which is interesting for computer games and interactive visualization in landscaping and architecture as well as modelling.