Additive Approximation for Near-Perfect Phylogeny Construction

  • Pranjal Awasthi
  • Avrim Blum
  • Jamie Morgenstern
  • Or Sheffet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

Abstract

We study the problem of constructing phylogenetic trees for a given set of species. The problem is formulated as that of finding a minimum Steiner tree on n points over the Boolean hypercube of dimension d. It is known that an optimal tree can be found in linear time [1] if the given dataset has a perfect phylogeny, i.e. cost of the optimal phylogeny is exactly d. Moreover, if the data has a near-perfect phylogeny, i.e. the cost of the optimal Steiner tree is d + q, it is known [2] that an exact solution can be found in running time which is polynomial in the number of species and d, yet exponential in q. In this work, we give a polynomial-time algorithm (in both d and q) that finds a phylogenetic tree of cost d + O(q 2). This provides the best guarantees known—namely, a (1 + o(1))-approximation—for the case \(\log(d) \ll q \ll \sqrt{d}\), broadening the range of settings for which near-optimal solutions can be efficiently found. We also discuss the motivation and reasoning for studying such additive approximations.

Keywords

Minimum Span Tree Steiner Tree Steiner Tree Problem Good Coordinate Minimum Steiner Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ding, Z., Filkov, V., Gusfield, D.: A Linear-Time Algorithm for the Perfect Phylogeny Haplotyping (PPH) Problem. In: Miyano, S., Mesirov, J., Kasif, S., Istrail, S., Pevzner, P.A., Waterman, M. (eds.) RECOMB 2005. LNCS (LNBI), vol. 3500, pp. 585–600. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Blelloch, G.E., Dhamdhere, K., Halperin, E., Ravi, R., Schwartz, R., Sridhar, S.: Fixed Parameter Tractability of Binary Near-Perfect Phylogenetic Tree Reconstruction. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 667–678. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Gusfield, D.: Algorithms on strings, trees, and sequences: computer science and computational biology. Cambridge University Press (1997)Google Scholar
  4. 4.
    Semple, C., Steel, M.: Phylogenetics. Oxford lecture series in mathematics and its applications. Oxford University Press (2003)Google Scholar
  5. 5.
    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. Science 307(5712), 1072–1079 (2005)CrossRefGoogle Scholar
  6. 6.
    The international hapmap project. Nature 426(6968), 789–796 (2003)Google Scholar
  7. 7.
    Alon, N., Chor, B., Pardi, F., Rapoport, A.: Approximate maximum parsimony and ancestral maximum likelihood. IEEE/ACM Trans. Comput. Biol. Bioinformatics 7, 183–187 (2010)CrossRefGoogle Scholar
  8. 8.
    Robins, G., Zelikovsky, A.: Improved steiner tree approximation in graphs. In: SODA, pp. 770–779. Society for Industrial and Applied Mathematics (2000)Google Scholar
  9. 9.
    Robins, G., Zelikovsky, A.: Improved steiner tree approximation in graphs (2000)Google Scholar
  10. 10.
    Robins, G., Zelikovsky, A.: Tighter bounds for graph steiner tree approximation. SIAM Journal on Discrete Mathematics 19, 122–134 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Misra, N., Blelloch, G., Ravi, R., Schwartz, R.: Generalized Buneman Pruning for Inferring the Most Parsimonious Multi-state Phylogeny. In: Berger, B. (ed.) RECOMB 2010. LNCS, vol. 6044, pp. 369–383. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Fernández-Baca, D., Lagergren, J.: A polynomial-time algorithm for near-perfect phylogeny. SIAM J. Comput. 32, 1115–1127 (2003)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)CrossRefGoogle Scholar
  14. 14.
    Foulds, L.R., Graham, R.L.: The Steiner problem in phylogeny is NP-complete. Adv. Appl. Math. 3 (1982)Google Scholar
  15. 15.
    Sridhar, S., Dhamdhere, K., Blelloch, G., Halperin, E., Ravi, R., Schwartz, R.: Algorithms for efficient near-perfect phylogenetic tree reconstruction in theory and practice. IEEE/ACM Trans. Comput. Biol. Bioinformatics 4, 561–571 (2007)CrossRefGoogle Scholar
  16. 16.
    Damaschke, P.: Parameterized enumeration, transversals, and imperfect phylogeny reconstruction. Theor. Comput. Sci. 351, 337–350 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Agarwala, R., Fernandez-Baca, D.: A polynomial-time algorithm for the perfect phylogeny problem when the number of character states is fixed. In: SFCS, pp. 140–147 (November 1993)Google Scholar
  18. 18.
    Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: An improved lp-based approximation for steiner tree. In: STOC. ACM (2010)Google Scholar
  19. 19.
    Bodlaender, H.L., Fellows, M.R., Warnow, T.: Two Strikes against Perfect Phylogeny. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 273–283. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  20. 20.
    Takahashi, H., Matsuyama, A.: An approximate solution for the steiner problem in graphs. Mathematica Japonica 24, 573–577 (1980)MathSciNetMATHGoogle Scholar
  21. 21.
    Berman, P., Ramaiyer, V.: Improved approximations for the steiner tree problem. In: SODA, pp. 325–334 (1992)Google Scholar
  22. 22.
    Prömel, H.J., Steger, A.: RNC-Approximation Algorithms for the Steiner Problem. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 559–570. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  23. 23.
    Karpinski, M., Zelikovsky, A.: New approximation algorithms for the steiner tree problems. Journal of Combinatorial Optimization 1, 47–65 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zelikovsky, A.: Better approximation bounds for the network and euclidean steiner tree problems. Technical report (1996)Google Scholar
  25. 25.
    Hougardy, S., Promel, H.J.: A 1.598 approximation algorithm for the steiner problem in graphs. In: SODA, pp. 448–453 (1999)Google Scholar
  26. 26.
    Borchers, A., Du, D.Z.: The k-steiner ratio in graphs. In: STOC, pp. 641–649. ACM (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pranjal Awasthi
    • 1
  • Avrim Blum
    • 1
  • Jamie Morgenstern
    • 1
  • Or Sheffet
    • 1
  1. 1.Carnegie Mellon University, PittsburghPittsburghUSA

Personalised recommendations