Improved smoothed analysis of the k-means method

The k-means method is a widely used clustering algorithm. One of its distinguished features is its speed in practice. Its worst-case running-time, however, is exponential, leaving a gap between practical and theoretical performance. Arthur and Vassilvitskii [3] aimed at closing this gap, and they proved a bound of poly(nk, σ−1) on the smoothed running-time of the k-means method, where n is the number of data points and σ is the standard deviation of the Gaussian perturbation. This bound, though better than the worst-case bound, is still much larger than the running-time observed in practice. We improve the smoothed analysis of the k-means method by showing two upper bounds on the expected running-time of k-means. First, we prove that the expected running-time is bounded by a polynomial in n √ k and σ−1. Second, we prove an upper bound of kkd · poly(n, σ−1), where d is the dimension of the data space. The polynomial is independent of k and d, and we obtain a polynomial bound for the expected running-time for k, d ∈ O( √ log n/ log log n). Finally, we show that k-means runs in smoothed polynomial time for one-dimensional instances.