Skip to main content
SpringerLink
Log in

A comparison between two distinct continuous models in projective cluster theory: The median and the tight-span construction

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

It is possible to consider two variants of cluster theory: Inaffine cluster theory, one considers collections ofsubsets of a given setX of objects or states, whereas inprojective cluster theory, one considers collections ofsplits (orbipartitions) of that set. In both contexts, it can be desirable to produce acontinuous model, that is, a spaceT encompassing the given setX which represents in a well-specified and more or less parsimonious way all possibleintermediate objects ortransition states compatible with certain restrictions derived from the given collection of subsets or splits. We investigate an interesting and intriguing relationship between two such constructions that appear in the context of projective cluster theory: TheBuneman construction and thetight-span (or justT)construction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H.-J. Bandelt, Phylogenetic networks, Verh. naturwiss. Ver. Hamburg34 (1991) 51–57.

    Google Scholar 

  2. H.-J. Bandelt and A. Dress, A canonical decomposition theory for metrics on a finite set, Adv. Math.92 (1992) 47–105.

    Article  MathSciNet  Google Scholar 

  3. H.-J. Bandelt and A. Dress, A new and useful approach to phylogenetic analysis, Molecular Phylogenetics and Evolution1 (1992) 242–252.

    Article  Google Scholar 

  4. J. Barthélemy, From copair hypergraphs to median graphs with latent vertices Discrete Math.76 (1989) 9–28.

    Article  MathSciNet  Google Scholar 

  5. J. Barthélemy and A. Guenoche, Trees and Proximity Representations, John Wiley, 1991.

  6. P. Buneman, The recovery of trees from measures of dissimilarity, In: F. Hodson et al., Mathematics in the Archeological and Historical Sciences, Edinburgh University Press, 1971, pp. 387–395.

  7. A. Dress, Towards a theory of holistic clustering, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 37, Amer. Math. Soc., 1997, pp. 271–289.

  8. A. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math.53 (1984) 321–402.

    Article  MathSciNet  Google Scholar 

  9. A. Dress, M. Hendy, K.T. Huber, and V. Moulton, On the number of vertices and edges of the Buneman graph, Ann. Combin.1 (1997) 329–337.

    Article  MathSciNet  Google Scholar 

  10. A. Dress, K. T. Huber, P. Lockhart, and V. Moulton, Lite Buneman networks: A technique for studying plant speciation, Mid Sweden University, Department of Mathematics Reports1 (1999).

  11. A. Dress, K.T. Huber, and V. Moulton, Some variations on a theme by Buneman, Ann. Combin.1 (1997) 339–352.

    Article  MathSciNet  Google Scholar 

  12. A. Dress, D. Huson, and V. Moulton, Analyzing and visualizing distance data with the splitstree graph, Discrete Appl. Math.71 (1996) 95–110.

    Article  MathSciNet  Google Scholar 

  13. A. Dress, V. Moulton, and W. Terhalle,T-theory: An Overview, Europ. J. Combin.17 (1996) 161–175.

    Article  MathSciNet  Google Scholar 

  14. K. T. Huber, AT-theoretical approach to phylogenetic analysis and cluster analysis, Ph.D. Thesis, Bielefeld, 1997.

  15. R. Webster, Convexity, Oxford University Press, 1994.

Download references

Author information

Authors and Affiliations

  1. FSPM-Strukturbildungsprozesse, University of Bielefeld, D-33501, Bielefeld, Germany

    A. W. M. Dress

  2. Institute of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North, New Zealand

    K. T. Huber

  3. FMI, Mid Sweden University, S 851-70, Sundsvall, Sweden

    V. Moulton

Authors
  1. A. W. M. Dress
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. T. Huber
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Moulton
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dress, A.W.M., Huber, K.T. & Moulton, V. A comparison between two distinct continuous models in projective cluster theory: The median and the tight-span construction. Annals of Combinatorics 2, 299–311 (1998). https://doi.org/10.1007/BF01608527

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01608527

AMS Subject Classification

Keywords

Search

Navigation