Improved analysis of Complete-Linkage clustering

Abstract

Complete-linkage clustering is a very popular method for computing hierarchical clusterings in practice, which is not fully understood theoretically. Given a finite set P ⊆ Rd of points, the complete-linkage method starts with each point from P in a cluster of its own and then iteratively merges two clusters from the current clustering that have the smallest diameter when merged into a single cluster. We study the problem of partitioning P into k clusters such that the largest diameter of the clusters is minimized and we prove that the complete-linkage method computes an O(1)-approximation for this problem for any metric that is induced by a norm, assuming that the dimension d is a constant. This improves the best previously known bound of O(log k) due to Ackermann et al. (Algorithmica, 2014). Our improved bound also carries over to the k-center and the discrete k-center problem.