Linear-time approximation schemes for clustering problems in any dimensions

We present a general approach for designing approximation algorithms for a fundamental class of geometric clustering problems in arbitrary dimensions. More specifically, our approach leads to simple randomized algorithms for the <i>k</i>-means, <i>k</i>-median and discrete <i>k</i>-means problems that yield (1+ϵ) approximations with probability ≥ 1/2 and running times of <i>O</i>(2<sup>(<i>k</i>/ϵ)<sup><i>O</i>(1)</sup></sup> <i>dn</i>). These are the first algorithms for these problems whose running times are linear in the size of the input (<i>nd</i> for <i>n</i> points in <i>d</i> dimensions) assuming <i>k</i> and ϵ are fixed. Our method is general enough to be applicable to clustering problems satisfying certain simple properties and is likely to have further applications.

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