Exact and Approximation Algortihms for Clustering

In this paper we present a n O(k 1 1=d ) time algorithm for solving the k-center problem in R d , under L 1 and L 2 metrics. The algorithm extends to other metrics, and can be used to solve the discrete k-center problem, as well. We also describe a simple (1 + )-approximation algorithm for the k-center problem, with running time O(n log k) + (k= ) O(k 1 1=d ) . Finally, we present a n O(k 1 1=d ) time algorithm for solving the L-capacitated k-center problem, provided that L = (n=k 1 1=d ) or L = O(1). We conclude with a simple approximation algorithm for the L-capacitated k-center problem. The work on this paper was partially supported by a National Science Foundation Grant CCR-93{01259, by an Army Research O ce MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award and matching funds from Xerox Corporation, and by a grant from the U.S.-Israeli Binational Science Foundation. y Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA z Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA