Advertisement

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

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

Abstract

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.

Keywords

Hierarchical classification Quasi-hierarchy Quasi-ultrametric Cluster analysis 

Notes

Acknowledgments

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

References

  1. 1.
    Ando, K., Inagaki, R., Shoji, K.: Efficient algorithms for subdominant cycle-complete cost functions and cycle-complete solutions. Discrete Appl. Math. 225, 1–10 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ando, K., Kato, S.: Reduction of ultrametric minimum cost spanning tree games to cost allocation games on rooted trees. J. Oper. Res. Soc. Japan 53, 62–68 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Benzécri, J.-P.: L’analyse des Données. Dunod, Paris (1973)zbMATHGoogle Scholar
  4. 4.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, London (2008)CrossRefGoogle Scholar
  5. 5.
    Diatta, J., Fichet, B.: Quasi-ultrametrics and their 2-ball hypergraphs. Discrete Math. 192, 87–102 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jardin, C.J., Jardin, N., Sibson, R.: The structure and construction of taxonomic hierarchies. Math. Biosci. 1, 173–179 (1967)CrossRefGoogle Scholar
  7. 7.
    Jardin, N., Sibson, R.: Mathematical Taxonomy. Wiley, New York (1971)zbMATHGoogle Scholar
  8. 8.
    Johnson, S.C.: Hierarchical clustering schemes. Psychometrika 32, 241–254 (1967)CrossRefGoogle Scholar
  9. 9.
    Milligan, G.W.: Ultrametric hierarchical clustering algorithms. Psychometrika 44, 343–346 (1979)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  11. 11.
    Trudeau, C.: A new stable and more responsive cost sharing solution for minimum cost spanning tree problems. Games Econ. Behav. 75, 402–412 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

Personalised recommendations