k-means Requires Exponentially Many Iterations Even in the Plane

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. O(nkd)) is, in general, exponential in the number of points (when kd=Ω(n log n)). Recently, Arthur and Vassilvitskii [2] showed a super-polynomial worst-case analysis, improving the best known lower bound from Ω(n) to 2Ω(√n) with a construction in d=Ω(√n) dimensions. In [2] 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).