Abstract
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).
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References
Althöfer, I.: On optimal realizations of finite metric spaces by graphs, Discrete Comput. Geometry 3 (1988) 103–122
Bandelt, H.-J., Dress, A.: A canonical decomposition theory for metrics on a finite set, Adv. in Math. 92 (1992) 47–105
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) 242–252
Bandelt, H.-J., Forster, P., Sykes, B., Richards, M.: Mitochondrial portraits of human population using median networks, Genetics 141 (October 1995) 743–753
Barthélémy, J., Guenoche A.: Trees and Proximity Representations, John Wiley & Sons, Chichester New York Brisbane Toronto Singapore, 1991
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, 1971
Chepoi, V., Fichet, B.: A note on circular decomposable metrics, Geometriae Dedicata 69 (3) (March 1998) 237–240
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) 486–500
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) 321–402
Dress, A., Hendy, M., Huber, K., Moulton, V.: On the number of vertices and edges in the Buneman Graph, Ann. Combin. 1 (1997) 329–337
Dress, A., Huber, K., Moulton, V.: Some variations on a theme by Buneman, Ann. Combin. 1 (1997) 339–352
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) 299–311
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)
Dress, A., Huber, K.T., Moulton, V.: Hereditarily optimal realizations of consistent metrics, in preparation
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)
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)
Dress, A., Huson, D.: Computing phylogenetic networks from split systems, submitted (1999)
Dress, A., Huson, D., Moulton, V.: Analyzing and visualizing distance data using SplitsTree, Discrete Applied Mathematics 71 (1996) 95–110
Dress, A., Moulton, V., Terhalle, W.: T-theory: An Overview, Europ. J. Combinatorics 17 (1996) 161–175
Hillis, D., Moritz, C., Barbara, K.: Phylogenetic Inference, In Molecular Systematics, D.M. Hillis, (pp.407–514), Sinauer, 1996.
Huson, D.: SplitsTree: a program for analyzing and visualizing evolutionary data, Bioinformatics 14 (1) (1998) 68–73
Imrich, W., Simoes-Pereira, J., Zamfirescu, C.: On optimal emdeddings of metrics in graphs, Journal of Combinatorial Theory, Series B, 36, No. 1, (1984) 1–15
Kalmanson K.: Edgeconvex circuits and the travelling salesman problem, Canadian Jour. Math., 27 (1975) 1000–1010
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)
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Dress, A., Huber, K.T., Moulton, V. (2001). Hereditarily Optimal Realizations: Why are they Relevant in Phylogenetic Analysis, and how does one Compute them. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds) Algebraic Combinatorics and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59448-9_8
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DOI: https://doi.org/10.1007/978-3-642-59448-9_8
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