Abstract
We show that adaptively sampled O(k) centers give a constant factor bi-criteria approximation for the k-means problem, with a constant probability. Moreover, these O(k) centers contain a subset of k centers which give a constant factor approximation, and can be found using LP-based techniques of Jain and Vazirani [JV01] and Charikar et al. [CGTS02]. Both these algorithms run in effectively O(nkd) time and extend the O(logk)-approximation achieved by the k-means++ algorithm of Arthur and Vassilvitskii [AV07].
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Aggarwal, A., Deshpande, A., Kannan, R. (2009). Adaptive Sampling for k-Means Clustering. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_2
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