Advertisement

The Complexity of Inferring a Minimally Resolved Phylogenetic Supertree

  • Jesper Jansson
  • Richard S. Lemence
  • Andrzej Lingas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6293)

Abstract

A recursive algorithm by Aho, Sagiv, Szymanski, and Ullman [1] forms the basis for many modern rooted supertree methods employed in Phylogenetics. However, as observed by Bryant [4], the tree output by the algorithm of Aho et al. is not always minimal; there may exist other trees which contain fewer nodes yet are still consistent with the input. In this paper, we prove strong polynomial-time inapproximability results for the problem of inferring a minimally resolved supertree from a given consistent set of rooted triplets (MinRS). We also present an exponential-time algorithm for solving MinRS exactly which is based on tree separators. It runs in 2 O(n logk) time when every node is required to have at most k children which are internal nodes and where n is the cardinality of the leaf label set of the input trees.

Keywords

Phylogenetic tree rooted triplet minimally resolved supertree NP-hardness tree separator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aho, A.V., Sagiv, Y., Szymanski, T.G., Ullman, J.D.: Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM Journal on Computing 10(3), 405–421 (1981)CrossRefGoogle Scholar
  2. 2.
    Bininda-Emonds, O.R.P.: The evolution of supertrees. TRENDS in Ecology and Evolution 19(6), 315–322 (2004)CrossRefPubMedGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T.: Exact graph coloring using inclusion-exclusion. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms, p. 289. Springer Science+Business Media, LLC, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Bryant, D.: Building Trees, Hunting for Trees, and Comparing Trees: Theory and Methods in Phylogenetic Analysis. PhD thesis, University of Canterbury, Christchurch, New Zealand (1997)Google Scholar
  5. 5.
    Byrka, J., Guillemot, S., Jansson, J.: New results on optimizing rooted triplets consistency. Discrete Applied Mathematics 158(11), 1136–1147 (2010)CrossRefGoogle Scholar
  6. 6.
    Chor, B., Hendy, M., Penny, D.: Analytic solutions for three taxon ML trees with variable rates across sites. Discrete Applied Mathematics 155(6-7), 750–758 (2007)CrossRefGoogle Scholar
  7. 7.
    Garey, M., Johnson, D.: Computers and Intractability – A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)Google Scholar
  8. 8.
    Gąsieniec, L., Jansson, J., Lingas, A., Östlin, A.: On the complexity of constructing evolutionary trees. Journal of Combinatorial Optimization 3(2-3), 183–197 (1999)CrossRefGoogle Scholar
  9. 9.
    Henzinger, M.R., King, V., Warnow, T.: Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology. Algorithmica 24(1), 1–13 (1999)CrossRefGoogle Scholar
  10. 10.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM 48(4), 723–760 (2001)CrossRefGoogle Scholar
  11. 11.
    Jansson, J., Ng, J.H.-K., Sadakane, K., Sung, W.-K.: Rooted maximum agreement supertrees. Algorithmica 43(4), 293–307 (2005)CrossRefGoogle Scholar
  12. 12.
    Kearney, P.: Phylogenetics and the quartet method. In: Jiang, T., Xu, Y., Zhang, M.Q. (eds.) Current Topics in Computational Molecular Biology, pp. 111–133. The MIT Press, Massachusetts (2002)Google Scholar
  13. 13.
    Otter, R.: The number of trees. Annals of Mathematics 49(3), 583–599 (1948)CrossRefGoogle Scholar
  14. 14.
    Page, R.D.M.: Modified mincut supertrees. In: Guigó, R., Gusfield, D. (eds.) WABI 2002. LNCS, vol. 2452, pp. 537–552. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Semple, C., Steel, M.: A supertree method for rooted trees. Discrete Applied Mathematics 105(1-3), 147–158 (2000)CrossRefGoogle Scholar
  16. 16.
    Snir, S., Rao, S.: Using Max Cut to enhance rooted trees consistency. IEEE/ACM Transactions on Computational Biology and Bioinformatics 3(4), 323–333 (2006)CrossRefPubMedGoogle Scholar
  17. 17.
    Zuckerman, D.: Linear degree extractors and the inapproximability of Max Clique and Chromatic Number. Theory of Computing 3(1), 103–128 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jesper Jansson
    • 1
  • Richard S. Lemence
    • 1
  • Andrzej Lingas
    • 2
  1. 1.Funded by the Special Coordination Funds for Promoting Science and TechnologyOchanomizu UniversityTokyoJapan
  2. 2.Department of Computer ScienceLund UniversityLundSweden

Personalised recommendations