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Abstract

Given a set P of n points on the real line and a (potentially infinite) family of functions, we investigate the problem of finding a small (weighted) subset \({\mathcal{S}} \subseteq P\), such that for any \(f \in {\mathcal{F}}\), we have that f(P) is a (1±ε)-approximation to \(f({\mathcal{S}})\). Here, f(Q) = ∑  q ∈ Q w(q) f(q) denotes the weighted discrete integral of f over the point set Q, where w(q) is the weight assigned to the point q.

We study this problem, and provide tight bounds on the size \({\mathcal{S}}\) for several families of functions. As an application, we present some coreset constructions for clustering.

Keywords

Real Line Discrete Algorithm Full Version Partition Scheme Discrete Integration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sariel Har-Peled
    • 1
  1. 1.Department of Computer ScienceUniversity of IllinoisUrbanaUSA

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