TY - UNPB

T1 - Worst-Case and Smoothed Analysis of the Hartigan-Wong Method for k-Means Clustering

AU - Manthey, Bodo

AU - van Rhijn, Jesse

PY - 2023

Y1 - 2023

N2 - We analyze the running time of the Hartigan-Wong method, an old algorithm for the k-means clustering problem. First, we construct an instance on the line on which the method can take $2\Omega(n)$ steps to converge, demonstrating that the Hartigan-Wong method has exponential worst-case running time even when k-means is easy to solve. As this is in contrast to the empirical performance of the algorithm, we also analyze the running time in the framework of smoothed analysis. In particular, given an instance of n points in d dimensions, we prove that the expected number of iterations needed for the Hartigan-Wong method to terminate is bounded by $k^{12kd} \cdot \poly(n, k, d, 1/\sigma)$ when the points in the instance are perturbed by independent d-dimensional Gaussian random variables of mean 0 and standard deviation $\sigma$.

AB - We analyze the running time of the Hartigan-Wong method, an old algorithm for the k-means clustering problem. First, we construct an instance on the line on which the method can take $2\Omega(n)$ steps to converge, demonstrating that the Hartigan-Wong method has exponential worst-case running time even when k-means is easy to solve. As this is in contrast to the empirical performance of the algorithm, we also analyze the running time in the framework of smoothed analysis. In particular, given an instance of n points in d dimensions, we prove that the expected number of iterations needed for the Hartigan-Wong method to terminate is bounded by $k^{12kd} \cdot \poly(n, k, d, 1/\sigma)$ when the points in the instance are perturbed by independent d-dimensional Gaussian random variables of mean 0 and standard deviation $\sigma$.

M3 - Preprint

BT - Worst-Case and Smoothed Analysis of the Hartigan-Wong Method for k-Means Clustering

PB - ArXiv.org

ER -