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Totally Balanced Dissimilarities

  • François BruckerEmail author
  • Pascal Préa
  • Célia Châtel
Article
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Abstract

We show in this paper a bijection between totally balanced hypergraphs and so-called totally balanced dissimilarities. We give an efficient way (O(n3) where n is the number of elements) to (i) recognize if a given dissimilarity is totally balanced and (ii) approximate it if it is not the case. We also introduce a new kind of dissimilarity which generalizes chordal graphs and allows a polynomial number of clusters that can be easily computed and interpreted.

Keywords

Dissimilarities Totally balanced hypergraphs Binary matrices 

Notes

References

  1. Anstee, R.P. (1983). Hypergraphs with no special cycles. Combinatorica, 3, 141–146.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Antsee, R.P., & Farber, M. (1984). Characterizations of totally balanced matrices. Journal of Algorithms, 5, 215–230.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bandelt, H.-J., & Dress, A.W.M. (1989). Weak hierarchies associated with similarity measures – an additive clustering technique. Bulletin of Mathematical Biology, 51, 133–166.MathSciNetzbMATHGoogle Scholar
  4. Barthélemy, J.-P., & Brucker, F. (2008). Binary clustering. Journal of Discrete Applied Mathematics, 156, 1237–1250.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bertrand, P. (2000). Set systems and dissimilarities. European Journal of Combinatorics, 21, 727–743.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bertrand, P., & Diatta, J. (2014) In Aleskerov, F, Goldengorin, B, Pardalo, P.M. (Eds.), Weak hierarchies: a central clustering structure clusters, orders, and trees: methods and applications. Berlin: Springer. chapter 14.Google Scholar
  7. Brucker, F. (2005). From hypertrees to arboreal quasi-ultrametrics. Discrete Applied Mathematics, 147, 3–26.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Brucker, F., & Gély, A. (2009). Parsimonious cluster systems. Advances in Data Analysis and Classification, 3, 189–204.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Brucker, F., & Gély, A. (2010). Crown-free lattices and their related graphs. Order, 28, 443–454.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Brucker, F., & Préa, P. (2015). Totally balanced formal concept representations. Proceedings of ICFCA, 215, 169–182.zbMATHGoogle Scholar
  11. Diatta, J., & Fichet, B. (1994) In Diday, E., Lechevallier, Y., Schader, M., Bertrand, P., Burtschy, B. (Eds.), From Asprejan hierarchies and Bandelt-Dress weak-hierarchies to quasi-hierarchies, new approaches in classification and data analysis, (pp. 111–118). Berlin: Springer.Google Scholar
  12. Dirac, G.A. (1961). On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25(1961), 71–76.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Farber, M. (1983). Characterizations of strongly chordal graphs. Discrete Mathematics, 43, 173–189.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Henley, N.M. (1969). A psychological study of the semantics of animal terms. Journal of Verbal Learning and Verbal Behaviour, 8, 176–184.CrossRefGoogle Scholar
  15. Lehel, J. (1985). A characterization of totally balanced hypergraphs. Discrete Mathematics, 57, 59–65.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Lovasz, L. (1968). Graphs and set systems, Beiträge zur Graphentheorie H. Sachs, & et al. (Eds.) Teubner, Leipzig.Google Scholar
  17. Lubiw, A. (1987). Doubly lexical orderings of matrices. SIAM Journal on Computing, 16, 854–879.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Sokal, R., & Michener, D. (1968). A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin, 38, 1409–1438.Google Scholar
  19. Spinrad, J. (1993). Doubly lexical ordering of dense 0-1 matrices. Information Processing Letters, 45, 229–235.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Spinrad, J. (2003). Efficient graph representations, American Mathematical Society, 2003.Google Scholar

Copyright information

© The Classification Society 2019

Authors and Affiliations

  1. 1.École Centrale MarseilleLIf, CNRS UMR 7279Marseille Cedex 20France

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