  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.
