Fixed Parameter Tractability of Binary Near-Perfect Phylogenetic Tree Reconstruction

  • Guy E. Blelloch
  • Kedar Dhamdhere
  • Eran Halperin
  • R. Ravi
  • Russell Schwartz
  • Srinath Sridhar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)

Abstract

We consider the problem of finding a Steiner minimum tree in a hypercube. Specifically, given n terminal vertices in an m dimensional cube and a parameter q, we compute the Steiner minimum tree in time O(72q + 8qnm2), under the assumption that the length of the minimum Steiner tree is at most m + q.

This problem has extensive applications in taxonomy and biology. The Steiner tree problem in hypercubes is equivalent to the phylogeny (evolutionary tree) reconstruction problem under the maximum parsimony criterion, when each taxon is defined over binary states. The taxa, character set and mutation of a phylogeny correspond to terminal vertices, dimensions and traversal of a dimension in a Steiner tree. Phylogenetic trees that mutate each character exactly once are called perfect phylogenies and their size is bounded by the number of characters. When a perfect phylogeny consistent with the data set exists it can be constructed in linear time. However, real data sets often do not admit perfect phylogenies. In this paper, we consider the problem of reconstructing near-perfect phylogenetic trees (referred to as BNPP). A near-perfect phylogeny relaxes the perfect phylogeny assumption by allowing at most q additional mutations. We show for the first time that the BNPP problem is fixed parameter tractable (FPT) and significantly improve the previous asymptotic bounds.

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References

  1. 1.
    Agarwala, R., Fernandez-Baca, D.: A Polynomial-Time Algorithm for the Perfect Phylogeny Problem when the Number of Character States is Fixed. SIAM Journal on Computing 23 (1994)Google Scholar
  2. 2.
    Bodlaender, H., Fellows, M., Warnow, T.: Two Strikes Against Perfect Phylogeny. In: proc. International Colloquium on Automata, Languages and Programming (1992)Google Scholar
  3. 3.
    Bodlaender, H., Fellows, M., Hallett, M., Wareham, H., Warnow, T.: The Hardness of Perfect Phylogeny, Feasible Register Assignment and Other Problems on Thin Colored Graphs. In: Theoretical Computer Science (2000)Google Scholar
  4. 4.
    Bonet, M., Steel, M., Warnow, T., Yooseph, S.: Better Methods for Solving Parsimony and Compatibility. Journal of Computational Biology 5(3) (1992)Google Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science (1999)Google Scholar
  6. 6.
    Day, W.H., Sankoff, D.: Computational Complexity of Inferring Phylogenies by Compatibility. Systematic Zoology (1986)Google Scholar
  7. 7.
    Damaschke, P.: Parameterized Enumeration, Transversals, and Imperfect Phylogeny Reconstruction. In: proc. International Workshop on Parameterized and Exact Computation (2004)Google Scholar
  8. 8.
    Eskin, E., Halperin, E., Karp, R.M.: Efficient Reconstruction of Haplotype Structure via Perfect Phylogeny. Journal of Bioinformatics and Computational Biology (2003)Google Scholar
  9. 9.
    Fernandez-Baca, D., Lagergren, J.: A Polynomial-Time Algorithm for Near-Perfect Phylogeny. SIAM Journal on Computing 32 (2003)Google Scholar
  10. 10.
    Foulds, L.R., Graham, R.L.: The Steiner problem in Phylogeny is NP-complete. Advances in Applied Mathematics (3) (1982)Google Scholar
  11. 11.
    Ganapathy, G., Ramachandran, V., Warnow, T.: Better Hill-Climbing Searches for Parsimony. In: Workshop on Algorithms in Bioinformatics (2003)Google Scholar
  12. 12.
    Gusfield, D.: Efficient Algorithms for Inferring Evolutionary Trees. Networks 21 (1991)Google Scholar
  13. 13.
    Gusfield, D.: Algorithms on Strings, Trees and Sequences. Cambridge University Press, Cambridge (1999)Google Scholar
  14. 14.
    Gusfield, D., Bansal, V.: A Fundamental Decomposition Theory for Phylogenetic Networks and Incompatible Characters. In: proc. Research in Computational Molecular Biology (2005)Google Scholar
  15. 15.
    Gusfield, D., Eddhu, S., Langley, C.: Efficient Reconstruction of Phylogenetic Networks with Constrained Recombination. In: Proc. IEEE Computer Society Bioinformatics Conference (2003)Google Scholar
  16. 16.
    Hinds, D.A., Stuve, L.L., Nilsen, G.B., Halperin, E., Eskin, E., Ballinger, D.G., Frazer, K.A., Cox, D.R.: Whole Genome Patterns of Common DNA Variation in Three Human Populations. In: Science (2005), http://www.perlegen.com
  17. 17.
    The International HapMap Consortium. The International HapMap Project. Nature 426 (2003)Google Scholar
  18. 18.
    Kannan, S., Warnow, T.: A Fast Algorithm for the Computation and Enumeration of Perfect Phylogenies. SIAM Journal on Computing 26 (1997)Google Scholar
  19. 19.
    Promel, H.J., Steger, A.: The Steiner Tree Problem: A Tour Through Graphs Algorithms and Complexity. Vieweg Verlag (2002)Google Scholar
  20. 20.
    Sridhar, S., Dhamdhere, K., Blelloch, G.E., Halperin, E., Ravi, R., Schwartz, R.: Simple Reconstruction of Binary Near-Perfect Phylogenetic Trees. In: proc. International Workshop on Bioinformatics Research and Applications (2006)Google Scholar
  21. 21.
    Semple, C., Steel, M.: Phylogenetics. Oxford University Press, Oxford (2003)MATHGoogle Scholar
  22. 22.
    Steel, M.A.: The Complexity of Reconstructing Trees from Qualitative Characters and Subtrees. J. Classification 9 (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Guy E. Blelloch
    • 1
  • Kedar Dhamdhere
    • 2
  • Eran Halperin
    • 3
  • R. Ravi
    • 4
  • Russell Schwartz
    • 5
  • Srinath Sridhar
    • 1
  1. 1.Computer Science Department, CMU 
  2. 2.Google IncMountain View
  3. 3.ICSIBerkeley
  4. 4.Tepper School of Business, CMU 
  5. 5.Dept Biological Sciences, CMU 

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