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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 \(\varLambda (U) \subseteq L {\setminus } A\).

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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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  4. Siu-Ming Yiu
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Corresponding author

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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