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.


Real Line Discrete Algorithm Full Version Partition Scheme Discrete Integration 
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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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