On Lloyd’s k-means Method∗

We present polynomial upper and lower bounds on the number of iterations performed by Lloyd’s method for k-means clustering. Our upper bounds are polynomial in the number of points, number of clusters, and the spread of the point set. We also present a lower bound, showing that in the worst case the k-means heuristic needs to perform Ω(n) iterations, for n points on the real line and two centers. Surprisingly, our construction spread is polynomial. This is the first construction showing that the k-means heuristic requires more than a polylogarithmic number of iterations. Furthermore, we present two alternative algorithms, with guaranteed performances, which are simple variants of Lloyd’s method. Results of our experimental studies on these algorithms are also presented.