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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)

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)

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)

- 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)

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)

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)

We consider the problem of embedding general metrics into trees. We give the first non-trivial approximation algorithm for minimizing the multiplicative distortion. Our algorithm produces an embedding with distortion (<i>c</i> log <i>n</i>)<sup><i>O</i>(√log Δ)</sup>, where <i>c</i> is the optimal distortion, and Δ is the spread of the… (More)

- Mihai Badoiu
- SODA
- 2003

In this paper, we present a polynomial-time approximation algorithm for computing an embedding of an arbitrary metric into a two-dimensional space. The algorithm finds an embedding whose additive distortion is at most <i>c</i>ε*, where ε* is the smallest additive distortion possible and <i>c</i> is an absolute constant. To our knowledge, this is… (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)