Analysis of Agglomerative Clustering

TL;DR

This paper provides a theoretical analysis of the agglomerative complete linkage clustering algorithm, showing it approximates the diameter $k$-clustering problem within a factor of $O( ext{log} k)$ for constant dimensions and general metrics.

Contribution

It offers the first theoretical approximation guarantees for agglomerative complete linkage clustering for diameter $k$-clustering and related problems, extending beyond Euclidean metrics.

Findings

Achieves $O( ext{log} k)$-approximation for diameter $k$-clustering

Extends analysis to any metric based on a norm

Provides approximation guarantees for $k$-center and discrete $k$-center problems

Abstract

The diameter $k$-clustering problem is the problem of partitioning a finite subset of $\mathbb{R}^d$ into $k$ subsets called clusters such that the maximum diameter of the clusters is minimized. One early clustering algorithm that computes a hierarchy of approximate solutions to this problem (for all values of $k$) is the agglomerative clustering algorithm with the complete linkage strategy. For decades, this algorithm has been widely used by practitioners. However, it is not well studied theoretically. In this paper, we analyze the agglomerative complete linkage clustering algorithm. Assuming that the dimension $d$ is a constant, we show that for any $k$ the solution computed by this algorithm is an $O(\log k)$-approximation to the diameter $k$-clustering problem. Our analysis does not only hold for the Euclidean distance but for any metric that is based on a norm. Furthermore, we analyze the closely related $k$-center and discrete $k$-center problem. For the corresponding agglomerative algorithms, we deduce an approximation factor of $O(\log k)$ as well.