k-means Requires Exponentially Many Iterations Even in the Plane

• Authors: Andrea Vattani
• Venue: Discrete & Computational Geometry - Special Issue: 25th Annual Symposium on Computational Geometry; Guest Editor: John Hershberger
• Year: 2011

The k-means algorithm is a well-known method for partitioning n points that lie in the d-dimensional space into k clusters. Its main features are simplicity and speed in practice. Theoretically, however, the best known upper bound on its running time (i.e., n O(kd)) is, in general, exponential in the number of points (when kd=Ω(n/log n)). Recently Arthur and Vassilvitskii (Proceedings of the 22nd Annual Symposium on Computational Geometry, pp. 144–153, 2006) showed a super-polynomial worst-case analysis, improving the best known lower bound from Ω(n) to $2^{\varOmega (\sqrt{n})}$ with a construction in $d=\varOmega (\sqrt{n})$ dimensions. In Arthur and Vassilvitskii (Proceedings of the 22nd Annual Symposium on Computational Geometry, pp. 144–153, 2006), they also conjectured the existence of super-polynomial lower bounds for any d≥2. Our contribution is twofold: we prove this conjecture and we improve the lower bound, by presenting a simple construction in the plane that leads to the exponential lower bound 2Ω(n).