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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 <i>core-sets</i> 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… (More)

Given a set of points <i>P</i> ⊂ <i>R</i><sup><i>d</i></sup> and value ∊ > 0, an ∊-core-set <i>S</i> ⊂ <i>P</i> has the property that the smallest ball containing <i>S</i> is an ∊-approximation of the smallest ball containing <i>P</i>. This paper shows that any point-set has an ∊-core-set of size [2/∊]. We also give… (More)

Given a set of points P ⊂ R d and value > 0, an-core-set S ⊂ P has the property that the smallest ball containing S is within of the smallest ball containing P. This paper shows that any point set has an-core-set of size 1//, and this bound is tight in the worst case. A faster algorithm given here finds an-core-set of size at most 2//. These results imply… (More)

A low-distortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Low-distortion embeddings have recently found numerous applications in computer science.Most of the known embedding results are "absolute",that is, of the form: any metric <i>Y</i> from a given… (More)

We present a dynamic comparison-based search structure that supports insertions, deletions, and searches within the unified bound. The unified bound specifies that it is quick to access an element that is near a recently accessed element. More precisely, if w(y) distinct elements have been accessed since the last access to element y, and d(x, y) denotes the… (More)

- Mihai Badoiu, Kedar Dhamdhere, Anupam Gupta, Yuri Rabinovich, Harald Räcke, R. Ravi +1 other
- SODA
- 2005

We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the two-dimensional plane. Among other results, we give an <i>O</i>(√<i>n</i>)-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative… (More)

A frequently arising problem in computational geometry is when a physical structure, such as an ad-hoc wireless sensor network or a protein backbone, can measure local information about its geometry (e.g., distances, angles, and/or orientations), and the goal is to reconstruct the global geometry from this partial information. More precisely, we are given a… (More)

In this paper we present a randomized constant factor approximation algorithm for the problem of computing the optimal cost of the metric Minimum Facility Location problem, in the case of uniform costs and uniform demands, and in which every point can open a facility. By exploiting the fact that we are approximating the optimal cost without computing an… (More)

We introduce a new notion of embedding, called <i>minimum-relaxation ordinal embedding</i>, parallel to the standard notion of minimum-distortion (metric) embedding. In an ordinal embedding, it is the relative order between pairs of distances, and not the distances themselves, that must be preserved as much as possible. The (multiplicative) relaxation of an… (More)

This paper studies how to optimally embed a general metric , represented by a graph, into a target space while preserving the relative magnitudes of most distances. More precisely, in an ordinal embedding , we must preserve the relative order between pairs of distances (which pairs are larger or smaller), and not necessarily the values of the distances… (More)