  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 $\sigma$.
