Computing the Rooted Triplet Distance between Galled Trees by Counting Triangles

  • Jesper Jansson
  • Andrzej Lingas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7354)

Abstract

We consider a generalization of the rooted triplet distance between two phylogenetic trees to two phylogenetic networks. We show that if each of the two given phylogenetic networks is a so-called galled tree with n leaves then the rooted triplet distance can be computed in o(n2.688) time. Our upper bound is obtained by reducing the problem of computing the rooted triplet distance to that of counting monochromatic and almost- monochromatic triangles in an undirected, edge-colored graph. To count different types of colored triangles in a graph efficiently, we extend an existing technique based on matrix multiplication and obtain several new related results that may be of independent interest.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bansal, M.S., Dong, J., Fernández-Baca, D.: Comparing and aggregating partially resolved trees. Theoretical Computer Science 412(48), 6634–6652 (2011)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chan, H.-L., Jansson, J., Lam, T.-W., Yiu, S.-M.: Reconstructing an ultrametric galled phylogenetic network from a distance matrix. Journal of Bioinformatics and Computational Biology 4(4), 807–832 (2006)CrossRefGoogle Scholar
  4. 4.
    Choy, C., Jansson, J., Sadakane, K., Sung, W.-K.: Computing the maximum agreement of phylogenetic networks. Theoretical Computer Science 335(1), 93–107 (2005)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. Journal of Symbolic Computation 9, 251–280 (1990)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Critchlow, D.E., Pearl, D.K., Qian, C.: The triples distance for rooted bifurcating phylogenetic trees. Systematic Biology 45(3), 323–334 (1996)CrossRefGoogle Scholar
  7. 7.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Associates, Inc., Sunderland (2004)Google Scholar
  8. 8.
    Gusfield, D., Eddhu, S., Langley, C.: Optimal, efficient reconstruction of phylogenetic networks with constrained recombination. Journal of Bioinformatics and Computational Biology 2(1), 173–213 (2004)CrossRefGoogle Scholar
  9. 9.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing 13(2), 338–355 (1984)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Huson, D.H., Rupp, R., Scornavacca, C.: Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press (2010)Google Scholar
  11. 11.
    van Iersel, L., Kelk, S.: Constructing the Simplest Possible Phylogenetic Network from Triplets. Algorithmica 60(2), 207–235 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Jansson, J., Nguyen, N.B., Sung, W.-K.: Algorithms for Combining Rooted Triplets into a Galled Phylogenetic Network. SIAM Journal on Computing 35(5), 1098–1121 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Morrison, D.: Introduction to Phylogenetic Networks. RJR Productions (2011)Google Scholar
  14. 14.
    Nakhleh, L., Warnow, T., Ringe, D., Evans, S.N.: A comparison of phylogenetic reconstruction methods on an Indo-European dataset. Transactions of the Philological Society 103(2), 171–192 (2005)CrossRefGoogle Scholar
  15. 15.
    Nielsen, J., Kristensen, A.K., Mailund, T., Pedersen, C.N.S.: A sub-cubic time algorithm for computing the quartet distance between two general trees. Algorithms for Molecular Biology 6, Article 15 (2011)Google Scholar
  16. 16.
    Stothers, A.J.: On the Complexity of Matrix Multiplication. PhD thesis, University of Edinburgh (2010)Google Scholar
  17. 17.
    Tarjan, R.E.: Applications of path compression on balanced trees. Journal of the ACM 26(4), 690–715 (1979)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Wang, L., Ma, B., Li, M.: Fixed topology alignment with recombination. Discrete Applied Mathematics 104(1-3), 281–300 (2000)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Vassilevska, V., Williams, R., Yuster, R.: Finding Heaviest H-Subgraphs in Real Weighted Graphs, with Applications. ACM Transactions on Algorithms 6(3), Article 44 (2010)Google Scholar
  20. 20.
    Vassilevska Williams, V.: Breaking the Coppersmith-Winograd barrier. UC Berkely and Stanford University (2011) (manuscript)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jesper Jansson
    • 1
  • Andrzej Lingas
    • 2
  1. 1.Laboratory of Mathematical Bioinformatics, Institute for Chemical ResearchKyoto UniversityUjiJapan
  2. 2.Department of Computer ScienceLund UniversityLundSweden

Personalised recommendations