Sublinear-Time Approximation for Clustering Via Random Sampling

In this paper we present a novel analysis of a random sampling approach for three clustering problems in metric spaces: k-median, min-sum k-clustering, and balanced k-median. For all these problems we consider the following simple sampling scheme: select a small sample set of points uniformly at random from V and then run some approximation algorithm on this sample set to compute an approximation of the best possible clustering of this set. Our main technical contribution is a significantly strengthened analysis of the approximation guarantee by this scheme for the clustering problems. The main motivation behind our analyses was to design sublinear-time algorithms for clustering problems. Our second contribution is the development of new approximation algorithms for the aforementioned clustering problems. Using our random sampling approach we obtain for the first time approximation algorithms that have the running time independent of the input size, and depending on k and the diameter of the metric space only.