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10
Approximate clustering via coresets
 In Proc. 34th Annu. ACM Sympos. Theory Comput
, 2002
"... In this paper, we show that for several clustering problems one can extract a small set of points, so that using those coresets enable us to perform approximate clustering efficiently. The surprising property of those coresets is that their size is independent of the dimension. Using those, we pre ..."
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Cited by 111 (15 self)
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In this paper, we show that for several clustering problems one can extract a small set of points, so that using those coresets enable us to perform approximate clustering efficiently. The surprising property of those coresets is that their size is independent of the dimension. Using those, we present a ¡ 1 ¢ ε £approximation algorithms for the kcenter clustering and kmedian clustering problems in Euclidean space. The running time of the new algorithms has linear or near linear dependency on the number of points and the dimension, and exponential dependency on 1 ¤ ε and k. As such, our results are a substantial improvement over what was previously known. We also present some other clustering results including ¡ 1 ¢ ε £approximate 1cylinder clustering, and kcenter clustering with outliers. 1
Approximation Algorithms for kLine Center
, 2002
"... Given a set P of n points in Rd and an integer k> = 1, let w * denote the minimumvalue so that P can be covered by k cylinders of width at most w*. We describe analgorithm that, given P and an "> 0, computes k cylinders of width at most (1 + ")w*that cover P. The running time of the algori ..."
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Cited by 33 (5 self)
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Given a set P of n points in Rd and an integer k> = 1, let w * denote the minimumvalue so that P can be covered by k cylinders of width at most w*. We describe analgorithm that, given P and an "> 0, computes k cylinders of width at most (1 + ")w*that cover P. The running time of the algorithm is O(n log n), with the constant ofproportionality depending on k, d, and ". The running times of the fastest algorithmsthat compute w * exactly are of the order of nO(dk). An approximation algorithm withnearlinear dependence on n for k> 1 was only known for the planar 2line centerproblem, i.e., the case k = 2, d = 2.We believe that the techniques used in showing this result are quite useful in themselves. We first show that there exists a small "certificate " Q ` P, whose size doesnot depend on n, such that for any kcylinders that cover Q, an enlargement of thesecylinders by a factor of (1 + ") covers P. We only establish the existence of a small certificate and our proof does not give us an efficient way of constructing one. We then observe that a wellknown scheme based on sampling and iterated reweighting gives usan efficient algorithm for solving the problem. Only the existence of a small certificate is used to establish the correctness of the algorithm. This technique is quite generaland can be used in other contexts as well.
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
Clustering Motion
 In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci
, 2003
"... Given a set of moving points in IR , we show how to cluster them in advance, using a small number of clusters, so that at any time this static clustering is competitive with the optimal kcenter clustering at that time. The advantage of this approach is that it avoids updating the clustering a ..."
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Cited by 29 (5 self)
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Given a set of moving points in IR , we show how to cluster them in advance, using a small number of clusters, so that at any time this static clustering is competitive with the optimal kcenter clustering at that time. The advantage of this approach is that it avoids updating the clustering as time passes. We also show how to maintain this static clustering eciently under insertions and deletions.
Shape Fitting with Outliers
 SIAM J. Comput
, 2003
"... we present an algorithm that "approximates the extent between the top and bottom k levels of the arrangement of H in time O(n+(k=") ), where c is a constant depending on d. The algorithm relies on computing a subset of H of size O(k=" ), in near linear time, such that the klevel of the a ..."
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Cited by 28 (11 self)
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we present an algorithm that "approximates the extent between the top and bottom k levels of the arrangement of H in time O(n+(k=") ), where c is a constant depending on d. The algorithm relies on computing a subset of H of size O(k=" ), in near linear time, such that the klevel of the arrangement of the subset approximates that of the original arrangement. Using this algorithm, we propose ecient approximation algorithms for shape tting with outliers for various shapes. This is the rst algorithms to handle outliers eciently for the shape tting problems considered.
HighDimensional Shape Fitting in Linear Time
 Discrete Comput. Geom
, 2002
"... The radius of a kdimensional at F with respect to P , denoted by RD(F ; P ), is de ned to be max p2P dist(F ; p), where dist(F ; p) denotes the Euclidean distance between p and its projection onto F . The kat radius of P , which we denote by R k (P ), is the minimum, over all kdimensional at ..."
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Cited by 17 (6 self)
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The radius of a kdimensional at F with respect to P , denoted by RD(F ; P ), is de ned to be max p2P dist(F ; p), where dist(F ; p) denotes the Euclidean distance between p and its projection onto F . The kat radius of P , which we denote by R k (P ), is the minimum, over all kdimensional ats F , of RD(F ; P ). We consider the problem of computing R k (P ) for a given set of points P .
An Improved Algorithm for Approximating the Radii of Point Sets
 In RANDOMAPPROX
, 2003
"... We consider the problem of computing the outerradii of point sets. In this problem, we are given integers n; d; k where k d, and a set P of n points in R ats F , of max p2P d(p; F ), where d(p; F ) is the Euclidean distance between the point p and at F . Computing the radii of point sets is a ..."
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Cited by 7 (2 self)
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We consider the problem of computing the outerradii of point sets. In this problem, we are given integers n; d; k where k d, and a set P of n points in R ats F , of max p2P d(p; F ), where d(p; F ) is the Euclidean distance between the point p and at F . Computing the radii of point sets is a fundamental problem in computational convexity with signi cantly many applications. The problem admits a polynomial time algorithm when the dimension d is constant [9]. Here we are interested in the general case when the dimension d is not xed and can be as large as n, where the problem becomes NPhard even for k = 1.
Approximating a Voronoi Cell
"... called sites, we consider the problem of approximating the Voronoi cell of a site p by a convex polyhedron with a small number of facets or, equivalently, of finding a small set of approximate Voronoi neighbors of p. More precisely, we define an approximate Voronoi neighborhood of p, denoted AVN ..."
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Cited by 6 (0 self)
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called sites, we consider the problem of approximating the Voronoi cell of a site p by a convex polyhedron with a small number of facets or, equivalently, of finding a small set of approximate Voronoi neighbors of p. More precisely, we define an approximate Voronoi neighborhood of p, denoted AVN (p; S), to be a subset of S satisfying the following property: p is an approximate nearest neighbor for any point q inside the convex polyhedron defined by the bisectors between p and the sites in AVN (p; S).
Approximating the radii of point sets in high dimensions
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Abstract Let P be a set of n points in Rd. For any 1 < = k < = d, the outer kradius of P, denotedby Rk(P), is the minimum, over all (d k)dimensional flats F, of maxp2P d(p, F),where d(p, F) is the Euclidean distance between the point p and flat F. We considerthe scenario when the dimension d is n ..."
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Cited by 4 (2 self)
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Abstract Let P be a set of n points in Rd. For any 1 < = k < = d, the outer kradius of P, denotedby Rk(P), is the minimum, over all (d k)dimensional flats F, of maxp2P d(p, F),where d(p, F) is the Euclidean distance between the point p and flat F. We considerthe scenario when the dimension d is not fixed and can be as large as n. Computing thevarious radii of point sets is a fundamental problem in computational convexity with many applications (See [18]).The main result of this paper is a randomized polynomial time algorithm that approximates Rk(P) to within a factor of O(plog n * log d) for any 1 < = k < = d. Thisalgorithm is obtained using techniques from semidefinite programming and dimension reduction. Previously, good approximation algorithms were known only for the case k = 1 and for the case when k = d c for any constant c; there are polynomial time algorithms that approximate Rk(P) to within a factor of (1+"), for any "> 0, when dkis any fixed constant [23, 7]. On the other hand, some results from the mathematical programming community on approximating certain kinds of quadratic programs [27, 26]imply an O(plog n) approximation for R1(P), the width of the point set P.We also prove an inapproximability result for computing Rk(P), which easily yieldsthe conclusion that our approximation algorithm performs quite well for a large range
On the number of cylindrical shells
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2003
"... Given a set P of n points in three dimensions, a cylindrical shell (or zone cylinder) is formed by two circular cylinders with the same axis such that all points of P are between the two cylinders. We prove that the number of cylindrical shells enclosing P passing through combinatorially different s ..."
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Cited by 3 (1 self)
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Given a set P of n points in three dimensions, a cylindrical shell (or zone cylinder) is formed by two circular cylinders with the same axis such that all points of P are between the two cylinders. We prove that the number of cylindrical shells enclosing P passing through combinatorially different subsets of P has size Ω(n 3) and O(n 4) (the previously known bound was O(n 5)). As a consequence, the minimum enclosing shell can be found in O(n 4) time.