Inferring a Level-1 Phylogenetic Network from a Dense Set of Rooted Triplets

  • Jesper Jansson
  • Wing-Kin Sung
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3106)


Given a set \({\cal T}\) of rooted triplets with leaf set L, we consider the problem of determining whether there exists a phylogenetic network consistent with \({\mathcal{T}}\), and if so, constructing one. If no restrictions are placed on the hybrid nodes in the solution, the problem is trivially solved in polynomial time by a simple sorting network-based construction. For the more interesting (and biologically more motivated) case where the solution is required to be a level-1 phylogenetic network, we present an algorithm solving the problem in O(n 6) time when \({\mathcal{T}}\) is dense (i.e., contains at least one rooted triplet for each cardinality three subset of L), where n = |L|. Note that the size of the input is Θ(n 3) if \({\mathcal{T}}\) is dense. We also give an O(n 5)-time algorithm for finding the set of all phylogenetic networks having a single hybrid node attached to exactly one leaf (and having no other hybrid nodes) that are consistent with a given dense set of rooted triplets.


Phylogenetic Network Unrooted Phylogenetic Tree Sorting Network Lower Common Ancestor Unordered Tree 
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.
    Aho, 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)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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
  3. 3.
    Chor, B., Hendy, M., Penny, D.: Analytic solutions for three-taxon MLMC trees with variable rates across sites. In: Gascuel, O., Moret, B.M.E. (eds.) WABI 2001. LNCS, vol. 2149, pp. 204–213. Springer, Heidelberg (2001)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), pp. 33–45. Elsevier, Amsterdam (2004)Google Scholar
  5. 5.
    Cormen, T., Leiserson, C., Rivest, R.: Introduction to Algorithms. The MIT Press, Massachusetts (1990)zbMATHGoogle Scholar
  6. 6.
    Ga̧sieniec, L., Jansson, J., Lingas, A., Östlin, A.: Inferring ordered trees from local constraints. In: Proc. of Computing: the 4th Australasian Theory Symposium (CATS 1998). Australian Computer Science Communications, vol. 20(3), pp. 67–76. Springer, Singapore (1998)Google Scholar
  7. 7.
    Ga̧sieniec, L., Jansson, J., Lingas, A., Östlin, A.: On the complexity of constructing evolutionary trees. Journal of Combinatorial Optimization 3, 183–197 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Gusfield, D., Eddhu, S., Langley, C.: Efficient reconstruction of phylogenetic networks with constrained recombination. In: Proc. of the Computational Systems Bioinformatics Conference (CSB 2003), pp. 363–374 (2003)Google Scholar
  9. 9.
    Hein, J.: Reconstructing evolution of sequences subject to recombination using parsimony. Mathematical Biosciences 98(2), 185–200 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fullydynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM 48(4), 723–760 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Jansson, J.: On the complexity of inferring rooted evolutionary trees. In: Proc. of the Brazilian Symp. on Graphs, Algorithms, and Combinatorics (GRACO 2001). Electronic Notes in Discrete Mathematics, vol. 7, pp. 121–125. Elsevier, Amsterdam (2001)Google Scholar
  13. 13.
    Jansson, J., Ng, J.H.-K., Sadakane, K., Sung, W.-K.: Rooted maximum agreement supertrees. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 499–508. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Jiang, T., Kearney, P., Li, M.: A polynomial time approximation scheme for inferring evolutionary trees from quartet topologies and its application. SIAM Journal on Computing 30(6), 1942–1961 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kannan, S., Lawler, E., Warnow, T.: Determining the evolutionary tree using experiments. Journal of Algorithms 21(1), 26–50 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    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
  17. 17.
    Nakhleh, L., Warnow, T., Linder, C.R.: Reconstructing reticulate evolution in species – theory and practice. In: Proc. of the 8th Annual International Conference on Research in Computational Molecular Biology (RECOMB 2004) (to appear)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.
    Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification 9(1), 91–116 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    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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