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Optimal algorithms for computing edge weights in planar split networks

  • Vincent Moulton
  • Andreas Spillner
Article

Abstract

In phylogenetics, biologists commonly compute split networks when trying to better understand evolutionary data. These graph-theoretical structures represent collections of weighted bipartitions or splits of a finite set, and provide a means to display conflicting evolutionary signals. The weights associated to the splits are used to scale the edges in the network and are often computed using some distance matrix associated with the data. In this paper we present optimal polynomial time algorithms for three basic problems that arise in this context when computing split weights for planar split-networks. These generalize algorithms that have been developed for special classes of split networks (namely, trees and outer-labeled planar networks). As part of our analysis, we also derive a Crofton formula for full flat split systems, structures that naturally arise when constructing planar split-networks.

Keywords

Split network Flat split system Crofton formula 

Mathematics Subject Classification (2000)

68R01 

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References

  1. 1.
    Bandelt, H.J., Dress, A.: A canonical decomposition theory for metrics on a finite set. Adv. Math. 92, 47–105 (1992) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bandelt, H.J., Dress, A.: Split decomposition: a new and useful approach to phylogenetic analysis of distance data. Mol. Phylogenet. Evol. 1, 242–252 (1992) CrossRefGoogle Scholar
  3. 3.
    Bryant, D.: Personal communication Google Scholar
  4. 4.
    Bryant, D., Dress, A.: Linearly independent split systems. Eur. J. Comb. 28, 1814–1831 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bryant, D., Moulton, V.: NeighborNet: an agglomerative method for the construction of phylogenetic networks. Mol. Biol. Evol. 21, 255–265 (2004) CrossRefGoogle Scholar
  6. 6.
    Bryant, D., Waddell, P.: Rapid evaluation of least-squares and minimum-evolution criteria on phylogenetic trees. Mol. Biol. Evol. 15, 1346–1359 (1998) CrossRefGoogle Scholar
  7. 7.
    Cavalli-Sforza, L., Edwards, A.: Phylogenetic analysis models and estimation procedures. Evolution 32, 550–570 (1967) CrossRefGoogle Scholar
  8. 8.
    Chepoi, V., Fichet, B.: A note on circular decomposable metrics. Geom. Dedic. 69, 237–240 (1998) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dress, A., Huson, D.: Constructing split graphs. IEEE/ACM Trans. Comput. Biol. Bioinform. 1, 109–115 (2004) CrossRefGoogle Scholar
  10. 10.
    Dress, A., Koolen, J., Moulton, V.: 4n−10. Ann. Comb. 8, 463–471 (2004) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Edelsbrunner, H.: Algorithms in combinatorial geometry. Springer, Berlin (1987) MATHGoogle Scholar
  12. 12.
    Felsner, S.: Geometric Graphs and Arrangements. Vieweg, Wiesbaden (2004) MATHCrossRefGoogle Scholar
  13. 13.
    Fitch, W., Margoliash, E.: Construction of phylogenetic trees. Science 155, 279–284 (1967) CrossRefGoogle Scholar
  14. 14.
    Golub, G., van Loan, C.: Matrix computation. Johns Hopkins University Press, Baltimore (1996) Google Scholar
  15. 15.
    Huson, D., Bryant, D.: Application of phylogenetic networks in evolutionary studies. Mol. Biol. Evol. 23, 254–267 (2006) CrossRefGoogle Scholar
  16. 16.
    Lawson, C., Hanson, R.: Solving least squares problems. Prentice Hall, New York (1974) MATHGoogle Scholar
  17. 17.
    Makarenkov, V., Lapointe, F.J.: A weighted least-squares approach for inferring phylogenies from incomplete distance matrices. Bioinformatics 20, 2113–2121 (2004) CrossRefGoogle Scholar
  18. 18.
    Spillner, A., Nguyen, B., Moulton, V.: Constructing and drawing planar split networks. In: IEEE/ACM Transactions on Computational Biology and Bioinformatics, 04 Aug. 2011. IEEE computer Society Digital Library. IEEE Computer Society, Los Alamitos (2011). http://doi.ieeecomputersociety.org/10.1109/TCBB.2011.115 Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2011

Authors and Affiliations

  1. 1.School of Computing SciencesUniversity of East AngliaNorwichUK
  2. 2.Department of Mathematics and Computer ScienceUniversity of GreifswaldGreifswaldGermany

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