# k-means Requires Exponentially Many Iterations Even in the Plane

@article{Vattani2011kmeansRE,
title={k-means Requires Exponentially Many Iterations Even in the Plane},
author={Andrea Vattani},
journal={Discrete \& Computational Geometry},
year={2011},
volume={45},
pages={596-616}
}
• Andrea Vattani
• Published 2011
• Computer Science, Mathematics
• Discrete & Computational Geometry
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., nO(kd)) is, in general, exponential in the number of points (when kd=Ω(n/log n)). Recently Arthur and Vassilvitskii (Proceedings of the 22nd Annual Symposium on Computational Geometry, pp. 144–153, 2006) showed a super-polynomial worst-case…
115 Citations
k-means requires exponentially many iterations even in the plane
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• 2012
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