Linear Time Algorithms for Clustering Problems in Any Dimensions

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.

• Type: Conference
• Year: 2005
• Where: ICALP
• Authors: Amit Kumar, Yogish Sabharwal, Sandeep Sen
• Keywords: Algorithm, ICALP 2005, K-Means Algorithm, K-means Clustering Problems, Theoretical Computer Science