The Maximum Agreement of Two Nested Phylogenetic Networks

  • Jesper Jansson
  • Wing-Kin Sung
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3341)

Abstract

Given a set \({\mathcal N}\) of phylogenetic networks, the maximum agreement phylogenetic subnetwork problem (MASN) asks for a subnetwork contained in every \(N_{i} \in {\mathcal N}\) with as many leaves as possible. MASN can be used to identify shared branching structure among phylogenetic networks or to measure their similarity. In this paper, we prove that the general case of MASN is NP-hard already for two phylogenetic networks, but that the problem can be solved efficiently if the two given phylogenetic networks exhibit a nested structure. We first show that the total number of nodes |V(N)| in any nested phylogenetic network N with n leaves and nesting depth d is O(n (d +1)). We then describe an algorithm for testing if a given phylogenetic network is nested, and if so, determining its nesting depth in O(|V(N)| · (d + 1)) time. Next, we present a polynomial-time algorithm for MASN for two nested phylogenetic networks N1, N2. Its running time is O(|V(N1)| · |V(N2)| · (d1 + 1) · (d2 + 1)), where d1 and d2 denote the nesting depths of N1 and N2, respectively. In contrast, the previously fastest algorithm for this problem runs in O(|V(N1)| · |V(N2)| · 4f) time, where f ≥ max{d1,d2}.

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References

  1. 1.
    Amir, A., Keselman, D.: Maximum agreement subtree in a set of evolutionary trees: Metrics and efficient algorithms. SIAM J. on Computing 26, 1656–1669 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bryant, D.: Building trees, hunting for trees, and comparing trees: theory and methods in phylogenetic analysis. PhD thesis, Univ. of Canterbury, New Zealand (1997)Google Scholar
  3. 3.
    Bryant, D., Moulton, V.: NeighborNet: An agglomerative method for the construction of planar phylogenetic networks. In: Guigó, R., Gusfield, D. (eds.) WABI 2002. LNCS, vol. 2452, pp. 375–391. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Choy, C., Jansson, J., Sadakane, K., Sung, W.-K.: Computing the maximum agreement of phylogenetic networks. In: Proc. of Computing: the 10th Australasian Theory Symposium (CATS 2004). ENTCS, vol. 91, pp. 134–147. Elsevier, Amsterdam (2004)Google Scholar
  5. 5.
    Cole, R., Farach-Colton, M., Hariharan, R., Przytycka, T., Thorup, M.: An O(n logn) algorithm for the maximum agreement subtree problem for binary trees. SIAM J. on Computing 30(5), 1385–1404 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cormen, T., Leiserson, C., Rivest, R.: Introduction to algorithms. MIT Press, Cambridge (1990)MATHGoogle Scholar
  7. 7.
    Farach, M., Przytycka, T., Thorup, M.: On the agreement of many trees. Information Processing Letters 55, 297–301 (1995)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Garey, M., Johnson, D.: Computers and Intractability – A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)MATHGoogle Scholar
  9. 9.
    Gąsieniec, L., Jansson, J., Lingas, A., Östlin, A.: On the complexity of constructing evolutionary trees. Journal of Combinatorial Optimization 3, 183–197 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gusfield, D., Eddhu, S., Langley, C.: Efficient reconstruction of phylogenetic networks with constrained recombination. In: Proc. of the Computational Systems Bioinformatics Conference (CSB2003), pp. 363–374 (2003)Google Scholar
  11. 11.
    Hein, J.: Reconstructing evolution of sequences subject to recombination using parsimony. Mathematical Biosciences 98(2), 185–200 (1990)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Huson, D.H., Dezulian, T., Klöpper, T.H., Steel, M.A.: Phylogenetic super-networks from partial trees. In: Jonassen, I., Kim, J. (eds.) WABI 2004. LNCS (LNBI), vol. 3240, pp. 388–399. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Jansson, J., Sung, W.-K.: Inferring a level-1 phylogenetic network from a dense set of rooted triplets. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 462–471. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Kao, M.-Y., Lam, T.-W., Sung, W.-K., Ting, H.-F.: An even faster and more unifying algorithm for comparing trees via unbalanced bipartite matchings. Journal of Algorithms 40(2), 212–233 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Li, W.-H.: Molecular Evolution. Sinauer Associates, Inc., Sunderland (1997)Google Scholar
  16. 16.
    Nakhleh, L., Sun, J., Warnow, T., Linder, C.R., Moret, B.M.E., Tholse, A.: Towards the development of computational tools for evaluating phylogenetic reconstruction methods. In: Proc. of the 8th Pacific Symposium on Biocomputing (PSB 2003), pp. 315–326 (2003)Google Scholar
  17. 17.
    Nakhleh, L., Warnow, T., Linder, C.R.: Reconstructing reticulate evolution in species – theory and practice. In: Proc. of the 8th Annual International Conf. on Research in Computational Molecular Biology (RECOMB 2004), pp. 337–346 (2004)Google Scholar
  18. 18.
    Posada, D., Crandall, K.A.: Intraspecific gene genealogies: trees grafting into networks. TRENDS in Ecology & Evolution 16(1), 37–45 (2001)CrossRefGoogle Scholar
  19. 19.
    Setubal, J., Meidanis, J.: Introduction to Comp. Molecular Biology. PWS (1997)Google Scholar
  20. 20.
    Steel, M., Warnow, T.: Kaikoura tree theorems: Computing the maximum agreement subtree. Information Processing Letters 48, 77–82 (1993)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Wang, L., Zhang, K., Zhang, L.: Perfect phylogenetic networks with recombination. Journal of Computational Biology 8(1), 69–78 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jesper Jansson
    • 1
  • Wing-Kin Sung
    • 1
  1. 1.School of ComputingNational University of SingaporeSingapore

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