In this paper, we show that for several clustering problems one can extract a small set of points, so that using those core-sets enable us to perform approximate clustering efficiently. The surprising property of those core-sets is that their size is independent of the dimension. Using those, we present a ¡ 1 ¢ ε £-approximation algorithms for the k-center clustering and k-median 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 1-cylinder clustering, and k-center clustering with outliers. 1

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 k-flats F of maxp2P dist(p, F). We present an algorithm that computes, for any 0 < &quot; < 1, a k-flat 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 near-optimal flat that lies in an affine subspace spanned by a small subset of points in P. The size of this "core-set" depends on k and ε but is independent of the dimension. This

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 &quot;> 0, computes k cylinders of width at most (1 + &quot;)w*that cover P. The running time of the algorithm is O(n log n), with the constant ofproportionality depending on k, d, and &quot;. The running times of the fastest algorithmsthat compute w * exactly are of the order of nO(dk). An approximation algorithm withnear-linear dependence on n for k> 1 was only known for the planar 2-line 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 &quot;certificate &quot; Q ` P, whose size doesnot depend on n, such that for any k-cylinders that cover Q, an enlargement of thesecylinders by a factor of (1 + &quot;) 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 well-known scheme based on sampling and iterated re-weighting 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.

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 k-level 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.

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 k-center 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.

The radius of a k-dimensional 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 k-at radius of P , which we denote by R k (P ), is the minimum, over all k-dimensional ats F , of RD(F ; P ). We consider the problem of computing R k (P ) for a given set of points P .

We consider the problem of computing the outer-radii 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 NP-hard even for k = 1.

Abstract Let P be a set of n points in Rd. For any 1 < = k < = d, the outer k-radius 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 al-gorithms that approximate Rk(P) to within a factor of (1+&quot;), for any &quot;> 0, when d-kis 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

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).

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.