Abstract
We generalize the k-means algorithm presented by the authors [14] and show that the resulting algorithm can solve a larger class of clustering problems that satisfy certain properties (existence of a random sampling procedure and tightness). We prove these properties for the k-median and the discrete k-means clustering problems, resulting in O(2(k/ε)O(1) dn) time (1+ε)-approximation algorithms for these problems. These are the first algorithms for these problems linear in the size of the input (nd for n points in d dimensions), independent of dimensions in the exponent, assuming k and ε to be fixed. A key ingredient of the k-median result is a (1+ε)-approximation algorithm for the 1-median problem which has running time O(2(1/ε)O(1) d). The previous best known algorithm for this problem had linear running time.
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Kumar, A., Sabharwal, Y., Sen, S. (2005). Linear Time Algorithms for Clustering Problems in Any Dimensions. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_111
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DOI: https://doi.org/10.1007/11523468_111
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