  We study hierarchical clusterings of metric spaces that change over time.
This is a natural geometric primitive for the analysis of dynamic data sets.
Specifically, we introduce and study the problem of finding a temporally
coherent sequence of hierarchical clusterings from a sequence of unlabeled
point sets. We encode the clustering objective by embedding each point set into
an ultrametric space, which naturally induces a hierarchical clustering of the
set of points. We enforce temporal coherence among the embeddings by finding
correspondences between successive pairs of ultrametric spaces which exhibit
small distortion in the Gromov-Hausdorff sense. We present both upper and lower
bounds on the approximability of the resulting optimization problems.
