Linear-Time Approximation Schemes for Clustering Problems

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.