Determining the Consistency of Resolved Triplets and Fan Triplets

  • Jesper JanssonEmail author
  • Andrzej Lingas
  • Ramesh Rajaby
  • Wing-Kin Sung
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10229)


The \(\mathcal {R}^{+-} \mathcal {F}^{+-}\) Consistency problem takes as input two sets \(R^{+}\) and \(R^{-}\) of resolved triplets and two sets \(F^{+}\) and \(F^{-}\) of fan triplets, and asks for a distinctly leaf-labeled tree that contains all elements in \(R^{+} \cup F^{+}\) and no elements in \(R^{-} \cup F^{-}\) as embedded subtrees, if such a tree exists. This paper presents a detailed characterization of how the computational complexity of the problem changes under various restrictions. Our main result is an efficient algorithm for dense inputs satisfying \(R^{-} = \emptyset \) whose running time is linear in the size of the input and therefore optimal.


Phylogenetic tree Rooted triplets consistency Algorithm Computational complexity 



The authors would like to thank Sylvain Guillemot and Avraham Melkman for some discussions related to the topic of this paper. J.J. was partially funded by The Hakubi Project at Kyoto University and KAKENHI grant number 26330014.


  1. 1.
    Aho, A.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 J. Comput. 10(3), 405–421 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bininda-Emonds, O.R.P.: The evolution of supertrees. TRENDS Ecol. Evol. 19(6), 315–322 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bininda-Emonds, O.R.P., Cardillo, M., Jones, K.E., MacPhee, R.D.E., Beck, R.M.D., Grenyer, R., Price, S.A., Vos, R.A., Gittleman, J.L., Purvis, A.: The delayed rise of present-day mammals. Nature 446(7135), 507–512 (2007)Google Scholar
  4. 4.
    Bryant, D.: Building trees, hunting for trees, and comparing trees: theory and methods in phylogenetic analysis. Ph.D. thesis, University of Canterbury, Christchurch, New Zealand (1997)Google Scholar
  5. 5.
    Byrka, J., Gawrychowski, P., Huber, K.T., Kelk, S.: Worst-case optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks. J. Discrete Algorithms 8(1), 65–75 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Byrka, J., Guillemot, S., Jansson, J.: New results on optimizing rooted triplets consistency. Discrete Appl. Math. 158(11), 1136–1147 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chor, B., Hendy, M., Penny, D.: Analytic solutions for three taxon ML trees with variable rates across sites. Discrete Appl. Math. 155(6–7), 750–758 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Constantinescu, M., Sankoff, D.: An efficient algorithm for supertrees. J. Classif. 12(1), 101–112 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Jansson, J., Lingas, A., Lundell, E.-M.: The approximability of maximum rooted triplets consistency with fan triplets and forbidden triplets. In: Cicalese, F., Porat, E., Vaccaro, U. (eds.) CPM 2015. LNCS, vol. 9133, pp. 272–283. Springer, Cham (2015). doi: 10.1007/978-3-319-19929-0_23
  10. 10.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Associates, Inc., Sunderland (2004)Google Scholar
  11. 11.
    Garey, M., Johnson, D.: Computers and Intractability - A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  12. 12.
    Ga̧sieniec, L., Jansson, J., Lingas, A., Östlin, A.: On the complexity of constructing evolutionary trees. J. Comb. Optim. 3(2–3), 183–197 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    He, Y.J., Huynh, T.N.D., Jansson, J., Sung, W.-K.: Inferring phylogenetic relationships avoiding forbidden rooted triplets. J. Bioinform. Comput. Biol. 4(1), 59–74 (2006)CrossRefGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jansson, J., Lemence, R.S., Lingas, A.: The complexity of inferring a minimally resolved phylogenetic supertree. SIAM J. Comput. 41(1), 272–291 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jansson, J., Ng, J.H.-K., Sadakane, K., Sung, W.-K.: Rooted maximum agreement supertrees. Algorithmica 43(4), 293–307 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ng, M.P., Wormald, N.C.: Reconstruction of rooted trees from subtrees. Discrete Appl. Math. 69(1–2), 19–31 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Semple, C.: Reconstructing minimal rooted trees. Discrete Appl. Math. 127(3), 489–503 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Semple, C., Daniel, P., Hordijk, W., Page, R.D.M., Steel, M.: Supertree algorithms for ancestral divergence dates and nested taxa. Bioinformatics 20(15), 2355–2360 (2004)CrossRefGoogle Scholar
  21. 21.
    Snir, S., Rao, S.: Using max cut to enhance rooted trees consistency. IEEE/ACM Trans. Comput. Biol. Bioinform. 3(4), 323–333 (2006)CrossRefGoogle Scholar
  22. 22.
    Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. J. Classif. 9(1), 91–116 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sung, W.: Algorithms in Bioinformatics: A Practical Introduction. Chapman & Hall/CRC, Boca Raton (2010)zbMATHGoogle Scholar
  24. 24.
    Willson, S.J.: Constructing rooted supertrees using distances. Bull. Math. Biol. 66(6), 1755–1783 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zuckerman, D.: Linear degree extractors and the inapproximability of Max Clique and Chromatic Number. Theory Comput. 3(1), 103–128 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jesper Jansson
    • 1
    • 2
    Email author
  • Andrzej Lingas
    • 3
  • Ramesh Rajaby
    • 4
    • 5
  • Wing-Kin Sung
    • 4
    • 6
  1. 1.Laboratory of Mathematical Bioinformatics, ICRKyoto UniversityGokasho, Uji, KyotoJapan
  2. 2.Department of ComputingThe Hong Kong Polytechnic UniversityKowloonHong Kong, China
  3. 3.Department of Computer ScienceLund UniversityLundSweden
  4. 4.School of ComputingNational University of SingaporeSingaporeSingapore
  5. 5.NUS Graduate School for Integrative Sciences and EngineeringNational University of SingaporeSingaporeSingapore
  6. 6.Genome Institute of SingaporeSingaporeSingapore

Personalised recommendations