Linear-time approximation schemes for clustering problems in any dimensions

Abstract

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 k -means, k -median and discrete k -means problems that yield (1+ε) approximations with probability ≥ 1/2 and running times of O (2 ( k /ε) O (1) dn ). These are the first algorithms for these problems whose running times are linear in the size of the input ( nd for n points in d dimensions) assuming k 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.