Order preserving hierarchical agglomerative clustering

Partial orders and directed acyclic graphs are commonly recurring data structures that arise naturally in numerous domains and applications and are used to represent ordered relations between entities in the domains. Examples are task dependencies in a project plan, transaction order in distributed ledgers and execution sequences of tasks in computer programs, just to mention a few. We study the problem of order preserving hierarchical clustering of this kind of ordered data. That is, if we have $a < b$ in the original data and denote their respective clusters by $[a]$ and $[b]$, then we shall have $[a] < [b]$ in the produced clustering. The clustering is similarity based and uses standard linkage functions, such as single- and complete linkage, and is an extension of classical hierarchical clustering. To achieve this, we define the output from running classical hierarchical clustering on strictly ordered data to be partial dendrograms; sub-trees of classical dendrograms with several connected components. We then construct an embedding of partial dendrograms over a set into the family of ultrametrics over the same set. An optimal hierarchical clustering is defined as the partial dendrogram corresponding to the ultrametric closest to the original dissimilarity measure, measured in the p-norm. Thus, the method is a combination of classical hierarchical clustering and ultrametric fitting. A reference implementation is employed for experiments on both synthetic random data and real world data from a database of machine parts. When compared to existing methods, the experiments show that our method excels both in cluster quality and order preservation.