Randomizing the Number of Centers in k-means++

Vaclav Rozhon

Randomizing the Number of Centers in k-means++

Data Structures and Algorithms 2026-07-28 v1 Machine Learning Machine Learning

Authors: Vaclav Rozhon

Abstract

The $k$ -means++ algorithm is a standard and widely used seeding method for $k$ -means clustering, but for a fixed number $k$ of centers its worst-case expected approximation ratio is $\Theta(\log k)$ . We consider the same algorithm when an adversary first fixes the dataset and some $K$ ; the number of centers $k$ is then chosen uniformly from $\{K,\ldots,2K-1\}$ . We prove that $k$ -means++ is an $O(1)$ -approximation with constant probability in this budget-smoothed setup.

Keywords

k-means++ clustering approximation ratio randomization

Cite

@article{arxiv.2607.26202,
  title  = {Randomizing the Number of Centers in k-means++},
  author = {Vaclav Rozhon},
  journal= {arXiv preprint arXiv:2607.26202},
  year   = {2026}
}