Sub-dominant theory in numerical taxonomy

Authors:
François Brucker
Affiliations:
ENST Bretagne, Département IASC, Brest Cedex, France
Venue:
Discrete Applied Mathematics - Special issue: Discrete mathematics & data mining II (DM & DM II)
Year:
2006

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Abstract

Sub-dominant theory provides efficient tools for clustering. However, it classically works only for ultrametrics and ad hoc extensions like Jardine and Sibson's 2-ultrametrics. In this paper we study the extension of the notion of sub-dominant to other distance models in classification accounting for overlapping clusters.We prove that a given dissimilarity admits one and only one lower-maximal quasi-ultrametric and one and only one lower-maximal weak k-ultrametric. In addition, we also prove the existence of (several) lower-maximal strongly Robinsonian dissimilarities. The construction of the lower-maximal weak k-ultrametric (for k = 2) and quasi-ultrametric can be performed in polynomial time.