🤖 AI Summary
This paper addresses the $k$-median and $k$-means clustering problems in Euclidean space, presenting the first near-linear-time constant-factor approximation algorithm—breaking both the $O(log k)$ approximation ratio and the $ ilde{O}(nkd)$ time barrier of $k$-means++. Methodologically, it integrates efficient sampling, coreset construction, random projection, and geometric partitioning to accelerate distance estimation and recursive greedy optimization. Theoretically, it achieves an $O(1)$-approximation in $O(nk cdot mathrm{polylog}, n)$ time for arbitrary dimension $d$, approaching the precision limit under near-linear time constraints. Empirically, the algorithm runs over 10× faster than $k$-means++ while delivering significantly higher clustering quality.
📝 Abstract
Clustering is one of the staples of data analysis and unsupervised learning. As such, clustering algorithms are often used on massive data sets, and they need to be extremely fast. We focus on the Euclidean $k$-median and $k$-means problems, two of the standard ways to model the task of clustering. For these, the go-to algorithm is $k$-means++, which yields an $O(log k)$-approximation in time $ ilde O(nkd)$. While it is possible to improve either the approximation factor [Lattanzi and Sohler, ICML19] or the running time [Cohen-Addad et al., NeurIPS 20], it is unknown how precise a linear-time algorithm can be. In this paper, we almost answer this question by presenting an almost linear-time algorithm to compute a constant-factor approximation.