Fast Exact k-Means, k-Medians and Bregman Divergence Clustering in 1D

The $k$-Means clustering problem on $n$ points is NP-Hard for any dimension $d\ge 2$, however, for the 1D case there exist exact polynomial time algorithms. Previous literature reported an $O(kn^2)$ time dynamic programming algorithm that uses $O(kn)$ space. We present a new algorithm computing the optimal clustering in only $O(kn)$ time using linear space. For $k = \Omega(\lg n)$, we improve this even further to $n 2^{O(\sqrt{ \lg \lg n \lg k})}$ time. We generalize the new algorithm(s) to work for the absolute distance instead of squared distance and to work for any Bregman Divergence as well.