Efficient Enumeration of the Directed Binary Perfect Phylogenies from Incomplete Data

  • Masashi Kiyomi
  • Yoshio Okamoto
  • Toshiki Saitoh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7276)


We study a character-based phylogeny reconstruction problem when an incomplete set of data is given. More specifically, we consider the situation under the directed perfect phylogeny assumption with binary characters in which for some species the states of some characters are missing. Our main object is to give an efficient algorithm to enumerate (or list) all perfect phylogenies that can be obtained when the missing entries are completed. While a simple branch-and-bound algorithm (B&B) shows a theoretically good performance, we propose another approach based on a zero-suppressed binary decision diagram (ZDD). Experimental results on randomly generated data exhibit that the ZDD approach outperforms B&B. We also prove that counting the number of phylogenetic trees consistent with a given data is #P-complete, thus providing an evidence that an efficient random sampling seems hard.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pe’er, I., Pupko, T., Shamir, R., Sharan, R.: Incomplete directed perfect phylogeny. SIAM J. Comput. 33, 590–607 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Camin, J.H., Sokal, R.R.: A method for deducing branching sequences in phylogeny. Evolution 19, 311–326 (1965)CrossRefGoogle Scholar
  3. 3.
    Gusfield, D., Frid, Y., Brown, D.: Integer Programming Formulations and Computations Solving Phylogenetic and Population Genetic Problems with Missing or Genotypic Data. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 51–64. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Agrawal, R., Srikant, R.: Fast algorithms for mining association rules in large databases. In: Bocca, J.B., Jarke, M., Zaniolo, C. (eds.) VLDB, pp. 487–499. Morgan Kaufmann (1994)Google Scholar
  5. 5.
    Minato, S.: Zero-suppressed BDDs for set manipulation in combinatorial problems. In: DAC, pp. 272–277. ACM Press (1993)Google Scholar
  6. 6.
    Knuth, D.E.: The Art of Computer Programming Volume 4, Fascicle 1, Bitwise Tricks & Techniques, Binary Decision Diagrams. Pearson Education, Inc., Boston (2009)zbMATHGoogle Scholar
  7. 7.
    Sinclair, A.: Algorithms for Random Generation & Counting: A Markov Chain Approach. Birkhäuser Boston, Boston Basel Berlin (1993)zbMATHCrossRefGoogle Scholar
  8. 8.
    Golumbic, M.C., Kaplan, H., Shamir, R.: Graph sandwich problems. J. Algorithms 19, 449–473 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kijima, S., Kiyomi, M., Okamoto, Y., Uno, T.: On listing, sampling, and counting the chordal graphs with edge constraints. Theor. Comput. Sci. 411, 2591–2601 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Heggernes, P., Mancini, F., Papadopoulos, C., Sritharan, R.: Strongly chordal and chordal bipartite graphs are sandwich monotone. J. Comb. Optim. 22, 438–456 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kiyomi, M., Okamoto, Y., Saitoh, T.: Efficient enumeration of the directed binary perfect phylogenies from incomplete data, arXiv:1203.3284 (2012)Google Scholar
  12. 12.
    Jansson, J.: Directed perfect phylogeny (binary characters). In: Kao, M.Y. (ed.) Encyclopedia of Algorithms, pp. 246–248. Springer, Heidelberg (2008)Google Scholar
  13. 13.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410–421 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hudson, R.R.: Generating samples under a Wright-Fisher neutral model of genetic variation. Bioinformatics 18, 337–338 (2002), CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Masashi Kiyomi
    • 1
  • Yoshio Okamoto
    • 2
  • Toshiki Saitoh
    • 3
  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan
  2. 2.Center for Graduate Education InitiativeJapan Advanced Institute of Science and TechnologyNomiJapan
  3. 3.ERATO Minato Discrete Structure Manipulation System ProjectJapan Technology and Science AgencySapporoJapan

Personalised recommendations