A Smoothed Analysis of the k-Means Method

Clustering is a fundamental problem in computer science with applications ranging from biology to information retrieval and data compression. In a clustering problem, a set of objects, usually represented as points in a high-dimensional space R, is to be partitioned such that objects in the same group share similar properties. The k-means method is a traditional clustering algorithm, originally conceived by Lloyd [1982]. It begins with an arbitrary clustering based on k centers in R, and then repeatedly makes local improvements until the clustering stabilizes. The algorithm is greedy and as such, it offers virtually no accuracy guarantees. However, it is both very simple and very fast, which makes it appealing in practice. Indeed, one recent survey of data mining techniques states that the k-means method “is by far the most popular clustering algorithm used in scientific and industrial applications” [Berkhin 2002]. However, theoretical analysis has long been at stark contrast with what is observed in practice. In particular, it was recently shown that the worst-case running time of the k-means method is 2 even on two-dimensional instances [Vattani pear]. Conversely, the only upper bounds known for the general case are k and n. Both upper bounds