Linear Time Algorithms for Clustering Problems in Any Dimensions

  • Amit Kumar
  • Yogish Sabharwal
  • Sandeep Sen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3580)


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.


Approximation Algorithm Cluster Problem Linear Time Algorithm Constant Probability Random Sampling Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Amit Kumar
    • 1
  • Yogish Sabharwal
    • 2
  • Sandeep Sen
    • 3
  1. 1.Dept of Comp Sc & EnggIndian Institute of TechnologyNew DelhiIndia
  2. 2.IBM India Research Lab, Block-IIIT Delhi, Hauz KhasNew DelhiIndia
  3. 3.Dept of Comp Sc & EnggIndian Institute of TechnologyKharagpurIndia

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