A local search approximation algorithm for k-means clustering

  • Authors:

  • Tapas Kanungo;David M. Mount;Nathan S. Netanyahu;Christine D. Piatko;Ruth Silverman;Angela Y. Wu

  • Affiliations:

  • IBM Almaden Research Center, San Jose, CA;Department of Computer Science, University of Maryland, College Park, MD;Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel and Center for Automation Research, University of Maryland, College Park, MD;The Johns Hopkins University Applied Physics Laboratory Laurel, MD;Center for Automation Research, University of Maryland, College Park, MD;Department of Computer Science, American University, Washington, DC

  • Venue:
  • Computational Geometry: Theory and Applications - Special issue on the 18th annual symposium on computational geometry—SoCG2002
  • Year:
  • 2004

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Abstract

In k-means clustering we are given a set of n data points in d-dimensional space Rd and an integer k, and the problem is to determine a set of k points in Rd, called centers, to minimize the mean squared distance from each data point to its nearest center. No exact polynomial-time algorithms are known for this problem. Although asymptotically efficient approximation algorithms exist, these algorithms are not practical due to the very high constant factors involved. There are many heuristics that are used in practice, but we know of no bounds on their performance.We consider the question of whether there exists a simple and practical approximation algorithm for k-means clustering. We present a local improvement heuristic based on swapping centers in and out. We prove that this yields a (9 + ε)-approximation algorithm. We present an example showing that any approach based on performing a fixed number of swaps achieves an approximation factor of at least (9 - ε) in all sufficiently high dimensions. Thus, our approximation factor is almost tight for algorithms based on performing a fixed number of swaps. To establish the practical value of the heuristic, we present an empirical study that shows that, when combined with Lloyd's algorithm, this heuristic performs quite well in practice.