k-means requires exponentially many iterations even in the plane

TL;DR

This paper proves that the k-means clustering algorithm can require exponentially many iterations even in the two-dimensional plane, confirming a long-standing conjecture and demonstrating worst-case exponential complexity.

Contribution

It establishes the conjecture that k-means has exponential lower bounds in 2D and provides a simple planar construction for this bound.

Findings

Proves exponential lower bounds for k-means in 2D

Confirms the conjecture about superpolynomial bounds in low dimensions

Provides a simple geometric construction demonstrating worst-case complexity

Abstract

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(n^{kd})) can be exponential in the number of points. Recently, Arthur and Vassilvitskii [3] showed a super-polynomial worst-case analysis, improving the best known lower bound from \Omega(n) to 2^{\Omega(\sqrt{n})} with a construction in d=\Omega(\sqrt{n}) dimensions. In [3] they also conjectured the existence of superpolynomial 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 2^{\Omega(n)}.