We investigate variants of Lloyd’s heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd’s heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd’s heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd’s method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration. 1.

We present polynomial upper and lower bounds on the number of iterations performed by the k-means method (a.k.a. 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, the spread of the point set in this construction 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 performance, which are simple variants of the k-means method. Results of our experimental studies on these algorithms are also presented. 1

An essential step in designing a new computer architecture is the careful examination of different design options. It is critical that computer architects have efficient means by which they may estimate the impact of various design options on the overall machine. This task is complicated by the fact that different programs, and even different parts of the same program, may have distinct behaviors that interact with the hardware in different ways. Researchers use very detailed simulators to estimate processor performance, which models every cycle of an executing program. Unfortunately,

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# 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.