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 , where 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.
We show a worst-case lower bound and a smoothed upper bound on the number of iterations performed by the Iterative Closest Point (ICP) algorithm. First proposed by Besl and McKay, the algorithm is widely used in computational geometry, where it is known for its simplicity and its observed speed. The theoretical study of ICP was initiated by Ezra, Sharir, and Efrat, who showed that the worst-case running time to align two sets of @math points in @math is between @math and @math . We substantially tighten this gap by improving the lower bound to @math . To help reconcile this bound with the algorithm's observed speed, we also show that the smoothed complexity of ICP is polynomial, independent of the dimensionality of the data. Using similar methods, we improve the best known smoothed upper bound for the popular k-means method to @math , once again independent of the dimension.