A bad instance for k-means++

Authors:
Tobias Brunsch;Heiko Röglin
Affiliations:
Department of Computer Science, University of Bonn, Germany;Department of Computer Science, University of Bonn, Germany
Venue:
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Year:
2011

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Abstract

k-means++ is a seeding technique for the k-means method with an expected approximation ratio of O(log k), where k denotes the number of clusters. Examples are known on which the expected approximation ratio of k-means++ is Ω(log k), showing that the upper bound is asymptotically tight. However, it remained open whether k-means++ yields an O(1)-approximation with probability 1/poly(k) or even with constant probability. We settle this question and present instances on which k-means++ achieves an approximation ratio of (2/3-ε) ċ log k only with exponentially small probability.