  As a staple of data analysis and unsupervised learning, the problem of
private clustering has been widely studied under various privacy models.
Centralized differential privacy is the first of them, and the problem has also
been studied for the local and the shuffle variation. In each case, the goal is
to design an algorithm that computes privately a clustering, with the smallest
possible error. The study of each variation gave rise to new algorithms: the
landscape of private clustering algorithms is therefore quite intricate.
  In this paper, we show that a 20-year-old algorithm can be slightly modified
to work for any of these models. This provides a unified picture: while
matching almost all previously known results, it allows us to improve some of
them and extend it to a new privacy model, the continual observation setting,
where the input is changing over time and the algorithm must output a new
solution at each time step.
