Bulletin of Mathematical Biology

, Volume 76, Issue 10, pp 2517–2541 | Cite as

Reconstructing a Phylogenetic Level-1 Network from Quartets

  • J. C. M. KeijsperEmail author
  • R. A. Pendavingh
Original Article


We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on \(n\) taxa from the set of all quartets containing a certain fixed taxon, in \(O(n^3)\) time. We also present a more general method which can handle more diverse quartet data, but which takes \(O(n^6)\) time. Both methods proceed by solving a certain system of linear equations over the two-element field \(\mathrm{GF}(2)\). For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our \(O(n^6)\) algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an \(O(n^2)\)-sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set.


Phylogenetic networks Cyclic ordering Quartets Level-1 networks Supernetwork method Polynomial time algorithm 



The authors would like to thank Steven Kelk for initiating this research by showing us the question of Gambette et al. (2012).We also thank two anonymous referees for their helpful comments.


  1. Artin M (1991) Algebra. Prentice Hall Inc, Englewood Cliffs ISBN 0-13-004763-5Google Scholar
  2. Bandelt H-J, Dress A (1986) Reconstructing the shape of a tree from observed dissimilarity data. Adv Appl Math 7(3):309–343 ISSN 0196-8858MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bandelt H-J, Dress A (1993) A relational approach to split decomposition. In: Studies in classification, data analysis and knowledge organization, pp 123–131Google Scholar
  4. Bard GV (2009) Algebraic cryptanalysis. Springer, DordrechtCrossRefzbMATHGoogle Scholar
  5. Berry V, Gascuel O (2000) Inferring evolutionary trees with strong combinatorial evidence. Theor Comput Sci 240(2):271–298 ISSN 0304-3975. Computing and combinatorics (Shanghai, 1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Coppersmith D (1994) Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm. Math Comput 62(205):333–350 ISSN 0025-5718MathSciNetzbMATHGoogle Scholar
  7. Erdös PL, Steel MA, Székely LA, Warnow TJ (1999) A few logs suffice to build (almost) all trees. I. Random Struct Algorithms 14(2):153–184 ISSN 1042-9832CrossRefzbMATHGoogle Scholar
  8. Gambette P, Berry V, Paul C (2012) Quartets and unrooted phylogenetic networks. JBCB 10(4):1250004.1–1250004.23.doi: 10.1142/S0219720012500047
  9. Grünewald S, Huber KT, Wu Q (2008) Two novel closure rules for constructing phylogenetic super-networks. Bull Math Biol 70(7):1906–1924MathSciNetCrossRefzbMATHGoogle Scholar
  10. Grünewald S, Moulton V, Spillner A (2009) Consistency of the QNet algorithm for generating planar split networks from weighted quartets. Discrete Appl Math 157(10):2325–2334 ISSN 0166-218XMathSciNetCrossRefzbMATHGoogle Scholar
  11. Heyting A (1980) Axiomatic projective geometry. Bibliotheca Mathematica [Mathematics Library], 2nd edn. V. Wolters-Noordhoff Scientific Publications Ltd., Groningen ISBN 0-444-85431-2Google Scholar
  12. Huntington EV (1924) A new set of postulates for betweenness, with proof of complete independence. Trans Am Math Soc 26(2):257–282 ISSN 0002-9947MathSciNetCrossRefzbMATHGoogle Scholar
  13. Huson DH, Rupp R, Scornavacca C (2010) Phylogenetic networks: concepts, algorithms and applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Jansson J, Sung W-K (2006) Inferring a level-1 phylogenetic network from a dense set of rooted triplets. Theor Comput Sci 363(1):60–68 ISSN 0304-3975MathSciNetCrossRefzbMATHGoogle Scholar
  15. Jansson J, Nguyen NB, Sung W-K (2006) Algorithms for combining rooted triplets into a galled phylogenetic network. SIAM J Comput 35(5):1098–1121 ISSN 0097-5397MathSciNetCrossRefzbMATHGoogle Scholar
  16. Semple C, Steel M (2003) Phylogenetics, volume 24 of Oxford lecture series in mathematics and its applications. Oxford University Press, Oxford ISBN 0-19-850942-1Google Scholar
  17. Semple C, Steel M (2004) Cyclic permutations and evolutionary trees. Adv Appl Math 32(4):669–680 ISSN 0196-8858MathSciNetCrossRefzbMATHGoogle Scholar
  18. Sonke W (2013) Fylogenetica. 2013a. url:
  19. Sonke W (2013) Reconstructing a level-1-network from quartets. 2013b. url:
  20. Steel M (1992) The complexity of reconstructing trees from qualitative characters and subtrees. J Classif 9(1):91–116 ISSN 0176-4268MathSciNetCrossRefzbMATHGoogle Scholar
  21. Stein W et al. (2014) Sage Mathematics Software, (Version 6.2). The Sage Development Team, 2014. url:
  22. To T-H, Habib M (2009) Level-k phylogenetic networks are constructable from a dense triplet set in polynomial time. In Proceedings of the 20th annual symposium on combinatorial pattern matching, CPM ’09. Berlin, Heidelberg, 2009. Springer, pp 275–288. ISBN 978-3-642-02440-5Google Scholar
  23. Van Iersel L, Kelk S (2011) Constructing the simplest possible phylogenetic network from triplets. Algorithmica 60(2):207–235 ISSN 0178-4617Google Scholar
  24. Van Iersel L, Keijsper J, Kelk S, Stougie L, Hagen F, Boekhout T (2009) Constructing level-2 phylogenetic networks from triplets. IEE/ACM Trans Comp Biol Bioinform 6(4):667–681Google Scholar
  25. Wiedemann DH (1986) Solving sparse linear equations over finite fields. IEEE Trans Inform Theory 32(1):54–62 ISSN 0018-9448MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations