Directed binary hierarchies and directed ultrametrics

Israël-César Lerman 1 Pascale Kuntz 2
1 SYMBIOSE - Biological systems and models, bioinformatics and sequences
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
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 in a very specific way that underlying binary hierarchical classification. In this paper an accurate formalization of this new structure is studied. A binary directed hierarchy is defined as a set of ordered pairs of subsets of the initial individual set satisfying specific conditions. New notion of directed ultrametricity is studied. The main result consists of establishing a bijective correspondence between a directed ultrametric space and a directed binary hierarchy. Moreover, an algorithm is proposed in order to transform a directed ultrametric structure into a graphical representation associated with a directed binary hierarchy.
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Israël-César Lerman, Pascale Kuntz. Directed binary hierarchies and directed ultrametrics. [Research Report] RR-6815, INRIA. 2009, pp.27. ⟨inria-00361727v2⟩

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