Directed Binary Hierarchies and Directed Ultrametrics

Abstract

Directed binary hierarchies have been introduced in order to give a graphical reduced representation of a family of association rules. This type of structure extends the classical binary hierarchical classification in a very specific way. In this paper an accurate formalization of this new structure is studied. A directed hierarchy is defined as a set of ordered pairs of subsets of the initial individual set satisfying specific conditions. A new notion of directed ultrametricity is studied. The main result consists in establishing a bijective correspondence between a directed ultrametric space and a directed binary hierarchy. Finally, an algorithm is proposed in order to transform a directed ultrametric structure into a graphical representation associated with a directed binary hierarchy.