Characterizing Cycle-Complete Dissimilarities in Terms of Associated Indexed 2-Hierarchies

  • Kazutoshi AndoEmail author
  • Kazuya Shoji
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)


2-ultrametrics are a generalization of the ultrametrics and it is known that there is a one-to-one correspondence between the set of 2-ultrametrics and the set of indexed 2-hierarchies (which are a generalization of indexed hierarchies). Cycle-complete dissimilarities, recently introduced by Trudeau, are a generalization of ultrametrics and form a subset of the 2-ultrametrics; therefore the set of cycle-complete dissimilarities corresponds to a subset of the indexed 2-hierarchies. In this study, we characterize this subset as the set of indexed acyclic 2-hierarchies, which in turn allows us to characterize the cycle-complete dissimilarities. In addition, we present an O\((n^2\log n)\) time algorithm that, given an arbitrary cycle-complete dissimilarities of order n, finds the corresponding indexed acyclic 2-hierarchy.


Hierarchical classification Quasi-hierarchy Quasi-ultrametric Cluster analysis 



The authors are grateful to the anonymous referees for useful comments which improved the presentation of the original version of this paper.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of EngineeringShizuoka UniversityHamamatsuJapan
  2. 2.Graduate School of Integrated Science and TechnologyShizuoka UniversityHamamatsuJapan

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