k-Means has Polynomial Smoothed Complexity
Abstract
The k-means method is one of the most widely used clustering algorithms, drawing its popularity from its speed in practice. Recently, however, it was shown to have exponential worst-case running time. In order to close the gap between practical performance and theoretical analysis, the k-means method has been studied in the model of smoothed analysis. But even the smoothed analyses so far are unsatisfactory as the bounds are still super-polynomial in the number n of data points. In this paper, we settle the smoothed running time of the k-means method. We show that the smoothed number of iterations is bounded by a polynomial in n and 1/\sigma, where \sigma is the standard deviation of the Gaussian perturbations. This means that if an arbitrary input data set is randomly perturbed, then the k-means method will run in expected polynomial time on that input set.
Cite
@article{arxiv.0904.1113,
title = {k-Means has Polynomial Smoothed Complexity},
author = {David Arthur and Bodo Manthey and Heiko Röglin},
journal= {arXiv preprint arXiv:0904.1113},
year = {2009}
}
Comments
Full version of FOCS 2009 paper. The argument has been improved and the restriction to at least three dimensions could be dropped