A Fast Approximation Scheme for Low-Dimensional k-Means

Vincent Cohen-Addad (1)

1 RO - Recherche Opérationnelle

We consider the popular k-means problem in d-dimensional Euclidean space. Recently Friggstad, Reza-pour, Salavatipour [FOCS’16] and Cohen-Addad, Klein, Mathieu [FOCS’16] showed that the standard local search algorithm yields a (1 + ε)-approximation in time (n · k)1/∊O(d), giving the first polynomial-time approximation scheme for the problem in low-dimensional Euclidean space. While local search achieves optimal approximation guarantees, it is not competitive with the state-of-the-art heuristics such as the famous k-means++ and D2-sampling algorithms. In this paper, we aim at bridging the gap between theory and practice by giving a (1 + ε)-approximation algorithm for low-dimensional k-means running in time n · k · (log n)(d∊−1)O(d), and so matching the running time of the k-means++ and D2-sampling heuristics up to polylogarithmic factors. We speed-up the local search approach by making a non-standard use of randomized dissections that allows to find the best local move efficiently using a quite simple dynamic program. We hope that our techniques could help design better local search heuristics for geometric problems.

Domaines: Algorithme et structure de données [cs.DS]

Vincent Cohen-Addad. A Fast Approximation Scheme for Low-Dimensional k-Means. SODA 2018 - 29h Annual ACM-SIAM Symposium on Discrete Algorithms, Jan 2018, New Orleans, LA, United States. pp.430-440, ⟨10.1137/1.9781611975031.29⟩. ⟨hal-02169544⟩