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Faster Algorithms for Computing the R* Consensus Tree

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Abstract

The fastest known algorithms for computing the R* consensus tree of k rooted phylogenetic trees with n leaves each and identical leaf label sets run in \(O(n^{2} \sqrt{\log n})\) time when \(k = 2\) (Jansson and Sung in Algorithmica 66(2):329–345, 2013) and \(O(k n^{3})\) time when \(k \ge 3\) (Bryant in Bioconsensus, volume 61 of DIMACS series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, pp 163–184, 2003). This paper shows how to compute it in \(O(n^{2})\) time for \(k = 2, O(n^{2} \log ^{4/3} n)\) time for \(k = 3\), and \(O(n^{2} \log ^{k+2} n)\) time for unbounded k.

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Notes

  1. An example with \(k = 3\) for which Lemma 13 in [14] fails is: \(T_1 = (((a,b),c,d),(e,f));, T_2 = (((b,f),a,c),(d,e));, T_3 = (((a,c),b,e),(d,f));\) (here, trees are expressed using Newick notation; see http://evolution.genetics.washington.edu/phylip/newicktree.html). Then \(\mathcal {R}_{ maj } = \{ab|d,\, ab|e,\, ab|f,\, ac|d,\, ac|e,\, ac|f,\, bc|d,\, bc|e,\, bc|f\}\), and \(A = \{a,b,c\}\) is a strong cluster of \(\mathcal {R}_{ maj }\) by definition. However, condition (1) in Lemma 13 of [14] does not hold for \(i = 2\) as the subtree \(U = (b,f);\) of \(T_2\) rooted at a child of \( lca ^{T_2}(A)\) does not satisfy \(\varLambda (U) \subseteq A\) or \(\varLambda (U) \subseteq L {\setminus } A\).

References

  1. Bansal, M.S., Dong, J., Fernández-Baca, D.: Comparing and aggregating partially resolved trees. Theor. Comput. Sci. 412(48), 6634–6652 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Proceedings of the 4th Latin American Symposium on Theoretical Informatics (LATIN 2000). Volume 1776 of LNCS, pp. 88–94. Springer, Berlin (2000)

  3. Bender, M.A., Farach-Colton, M.: The level ancestor problem simplified. Theor. Comput. Sci. 321(1), 5–12 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bryant, D.: A classification of consensus methods for phylogenetics. In: Janowitz, M.F., Lapointe, F.-J., McMorris, F.R., Mirkin, B., Roberts, F.S. (eds.) Bioconsensus, Volume 61 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 163–184. American Mathematical Society (2003)

  5. Bryant, D., Berry, V.: A structured family of clustering and tree construction methods. Adv. Appl. Math. 27(4), 705–732 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chan, T.M., Pǎtraşcu, M.: Counting inversions, offline orthogonal range counting, and related problems. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 161–173. SIAM (2010)

  7. Cole, R., Farach-Colton, M., Hariharan, R., Przytycka, T., Thorup, M.: An \({O}(n \log n)\) algorithm for the maximum agreement subtree problem for binary trees. SIAM J. Comput. 30(5), 1385–1404 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cormen, T., Leiserson, C., Rivest, R.: Introduction to Algorithms. The MIT Press, Cambridge, MA (1990)

    MATH  Google Scholar 

  9. Degnan, J.H., DeGiorgio, M., Bryant, D., Rosenberg, N.A.: Properties of consensus methods for inferring species trees from gene trees. Syst. Biol. 58(1), 35–54 (2009)

    Article  Google Scholar 

  10. Ewing, G.B., Ebersberger, I., Schmidt, H.A., von Haeseler, A.: Rooted triple consensus and anomalous gene trees. BMC Evol. Biol. 8, 118 (2008)

    Article  Google Scholar 

  11. Felsenstein, J.: Inferring Phylogenies. Sinauer Associates Inc, Sunderland, MA (2004)

    Google Scholar 

  12. Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jansson, J., Sung, W.-K.: Constructing the R* consensus tree of two trees in subcubic time. Algorithmica 66(2), 329–345 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kannan, S., Warnow, T., Yooseph, S.: Computing the local consensus of trees. SIAM J. Comput. 27(6), 1695–1724 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee, C.-M., Hung, L.-J., Chang, M.-S., Shen, C.-B., Tang, C.-Y.: An improved algorithm for the maximum agreement subtree problem. Inf. Process. Lett. 94(5), 211–216 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Margush, T., McMorris, F.R.: Consensus \(n\)-trees. Bull. Math. Biol. 43(2), 239–244 (1981)

    MathSciNet  MATH  Google Scholar 

  18. Nakhleh, L., Warnow, T., Ringe, D., Evans, S.N.: A comparison of phylogenetic reconstruction methods on an Indo-European dataset. Trans. Philol. Soc. 103(2), 171–192 (2005)

    Article  Google Scholar 

  19. Semple, C., Steel, M.: Phylogenetics, Volume 24 of Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2003)

  20. Sung, W.-K.: Algorithms in Bioinformatics: A Practical Introduction. Chapman & Hall/CRC, London (2010)

    MATH  Google Scholar 

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Acknowledgments

J.J. was funded by The Hakubi Project at Kyoto University and KAKENHI Grant No. 26330014.

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Authors and Affiliations

  1. Laboratory of Mathematical Bioinformatics, Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan

    Jesper Jansson

  2. School of Computing, National University of Singapore, 13 Computing Drive, Singapore, 117417, Singapore

    Wing-Kin Sung

  3. Genome Institute of Singapore, 60 Biopolis Street, Genome, Singapore, 138672, Singapore

    Wing-Kin Sung

  4. Department of Computer Science and Engineering, University of Minnesota – Twin Cities, Minneapolis, MN, USA

    Hoa Vu

  5. Department of Computer Science, The University of Hong Kong, Pokfulam Road, Pokfulam, Hong Kong, China

    Siu-Ming Yiu

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  1. Jesper Jansson
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  2. Wing-Kin Sung
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  3. Hoa Vu
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  4. Siu-Ming Yiu
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Correspondence to Jesper Jansson.

Additional information

A preliminary version of this article appeared in Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC 2014), volume 8889 of Lecture Notes in Computer Science, pp. 414–425, Springer International Publishing Switzerland, 2014.

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Jansson, J., Sung, WK., Vu, H. et al. Faster Algorithms for Computing the R* Consensus Tree. Algorithmica 76, 1224–1244 (2016). https://doi.org/10.1007/s00453-016-0122-2

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