Accelerating K-Means ++ Algorithm

For choosing initial cluster centers, $k$-means++ is an important algorithm. Given $n$ data points in $\mathbb{R}^{d}$, the current state-of-the-art algorithm runs in $\widetilde{O}(k)$ iterations, and each iteration takes $\widetilde{O}(n d k)$ time. The overall running time is thus $\widetilde{O}\left(n d k^{2}\right)$. In this work, we present a new algorithm that can solve the $k$ means++ problem with nearly optimal running time. We propose a new algorithm that only takes $\widetilde{O}\left(n d+n k^{2}\right)$ time in total.