### Abstract

A set X of points in ℛ^{d} is (k, b)-clusterable if X can be partitioned into k subsets (clusters) so that the diameter (alternatively, the radius) of each cluster is at most b. We present algorithms that, by sampling from a set X, distinguish between the case that X is (k, b)-clusterable and the case that X is ε-far from being (k, b′)-clusterable for any given 0 < ε ≤ 1 and for b′ ≥ b. By ε-far from being (k, b′)-clusterable we mean that more than ε · |X| points should be removed from X so that it becomes (k, b′)-clusterable. We give algorithms for a variety of cost measures that use a sample of size independent of |X| and polynomial in k and 1/ε. Our algorithms can also be used to find approximately good clusterings. Namely, these are clustering of all but an ε-fraction of the points in X that have optimal (or close to optimal) cost. The benefit of our algorithms is that they construct an implicit representation of such clusterings in time independent of |X|. That is, without actually having to partition all points in X, the implicit representation can be used to answer queries concerning the cluster to which any given point belongs.

Original language | English (US) |
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Pages (from-to) | 393-417 |

Number of pages | 25 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 16 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2003 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*SIAM Journal on Discrete Mathematics*,

*16*(3), 393-417. https://doi.org/10.1137/S0895480102410973