Hereditarily Optimal Realizations: Why are they Relevant in Phylogenetic Analysis, and how does one Compute them

  • Andreas Dress
  • Katharina T. Huber
  • Vincent Moulton
Conference paper


One of the main problems in phylogenetic analysis (where one is concerned with elucidating evolutionary patterns between present day species) is to find good approximations of genetic distances by weighted trees. As an aid to solving this problem, it might seem tempting to consider an optimal realization of the metric defined by the given distances — the guiding principle being that, in case the metric is tree-like, the optimal realization obtained will necessarily be that unique weighted tree that realizes this metric. Although optimal realizations of arbitrary distances are not generally trees, but rather weighted graphs, one could still hope to obtain an informative representation of the given metric, maybe even more informative than the best approximating tree. However, optimal realizations are not only difficult to compute, they may also be non-unique. In this note we focus on one possible way out of this dilemma: hereditarily optimal realizations. These are essentially unique, and can also be described in an explicit way. We define hereditarily optimal realizations, discuss some of their properties, and we indicate in particular why, due to recent results on the so-called T-construction of a metric space, it is a straight forward task to compute these realizations for a large class of phylogentically relevant metrics.

The author thanks the New Zealand Marsden Fund for its support.

The author thanks the Swedish Natural Science Research Council (NFR) for its support (grant# M12342-300).


Weighted Graph Weighted Tree Pendant Vertex Injective Hull Optimal Realization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Althöfer, I.: On optimal realizations of finite metric spaces by graphs, Discrete Comput. Geometry 3 (1988) 103–122zbMATHGoogle Scholar
  2. 2.
    Bandelt, H.-J., Dress, A.: A canonical decomposition theory for metrics on a finite set, Adv. in Math. 92 (1992) 47–105MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bandelt H.-J., Dress, A.: Split decomposition: a new and useful approach to phylogenetic analysis of distance data, Molecular Phylogenetics and Evolution 1 (3) (1992b) 242–252CrossRefGoogle Scholar
  4. 4.
    Bandelt, H.-J., Forster, P., Sykes, B., Richards, M.: Mitochondrial portraits of human population using median networks, Genetics 141 (October 1995) 743–753Google Scholar
  5. 5.
    Barthélémy, J., Guenoche A.: Trees and Proximity Representations, John Wiley & Sons, Chichester New York Brisbane Toronto Singapore, 1991zbMATHGoogle Scholar
  6. 6.
    Buneman, P.: The recovery of trees from measures of dissimilarity, In F. Hodson, Mathematics in the Archaeological and Historical Sciences, (pp.387–395), Edinburgh University Press, 1971Google Scholar
  7. 7.
    Chepoi, V., Fichet, B.: A note on circular decomposable metrics, Geometriae Dedicata 69 (3) (March 1998) 237–240MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Christopher, G., Farach, M., Trick, M.: The structure of circular decomposable metrics, Algorithms—ESA ’96 (Barcelona), Lecture Notes in Comput. Sci., 1136, Springer, Berlin, (1996) 486–500Google Scholar
  9. 9.
    Dress, A.: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984) 321–402MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dress, A., Hendy, M., Huber, K., Moulton, V.: On the number of vertices and edges in the Buneman Graph, Ann. Combin. 1 (1997) 329–337MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dress, A., Huber, K., Moulton, V.: Some variations on a theme by Buneman, Ann. Combin. 1 (1997) 339–352MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dress, A., Huber, K.T., Moulton, V.: A Comparison between two distinct continuous models in projective cluster theory: The median and the tight-span construction, Ann. Combin. 2 (1998) 299–311MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dress, A., Huber, K.T., Moulton, V.: An explicit computation of the infective hull of certain finite metric spaces in terms of their associated Buneman complex, Mid Sweden University Mathematics Department Report No. 6 (1999)Google Scholar
  14. 14.
    Dress, A., Huber, K.T., Moulton, V.: Hereditarily optimal realizations of consistent metrics, in preparationGoogle Scholar
  15. 15.
    Dress, A., Huber, K.T., Koolen, J., Moulton, V.: Six points suffice: How to check for metric consistency, Mid Sweden University Mathematics Department Report No. 8 (2000)Google Scholar
  16. 16.
    Dress, A., Huber, K.T., Lockhart, P., Moulton, V.: Lite Buneman networks: A technique for studying plant speciation, Mid Sweden University Mathematics Department Report No. 1 (1999)Google Scholar
  17. 17.
    Dress, A., Huson, D.: Computing phylogenetic networks from split systems, submitted (1999)Google Scholar
  18. 18.
    Dress, A., Huson, D., Moulton, V.: Analyzing and visualizing distance data using SplitsTree, Discrete Applied Mathematics 71 (1996) 95–110MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dress, A., Moulton, V., Terhalle, W.: T-theory: An Overview, Europ. J. Combinatorics 17 (1996) 161–175MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hillis, D., Moritz, C., Barbara, K.: Phylogenetic Inference, In Molecular Systematics, D.M. Hillis, (pp.407–514), Sinauer, 1996.Google Scholar
  21. 21.
    Huson, D.: SplitsTree: a program for analyzing and visualizing evolutionary data, Bioinformatics 14 (1) (1998) 68–73CrossRefGoogle Scholar
  22. 22.
    Imrich, W., Simoes-Pereira, J., Zamfirescu, C.: On optimal emdeddings of metrics in graphs, Journal of Combinatorial Theory, Series B, 36, No. 1, (1984) 1–15MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kalmanson K.: Edgeconvex circuits and the travelling salesman problem, Canadian Jour. Math., 27 (1975) 1000–1010MathSciNetzbMATHGoogle Scholar
  24. 24.
    Zaretsky, K.: Reconstruction of a tree from the distances between its pendant vertices, Uspekhi Math. Nauk (Russian Mathematical Surveys) 20 (1965) 90–92 (in Russian)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Andreas Dress
    • 1
  • Katharina T. Huber
    • 2
  • Vincent Moulton
    • 3
  1. 1.FSPM-StrukturbildungsprozesseUniversity of BielefeldBielefeldGermany
  2. 2.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand
  3. 3.FMIMid Sweden UniversitySundsvallSweden

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