Asymptotic Limits of a New Type of Maximization Recurrence with an Application to Bioinformatics

  • Kun-Mao Chao
  • An-Chiang Chu
  • Jesper Jansson
  • Richard S. Lemence
  • Alban Mancheron
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7287)

Abstract

We study the asymptotic behavior of a new type of maximization recurrence, defined as follows. Let k be a positive integer and pk(x) a polynomial of degree k satisfying pk(0) = 0. Define A0 = 0 and for n ≥ 1, let \(A_{n} = \max\nolimits_{0 \leq i < n} \{ A_{i} + n^{k} \, p_{k}(\frac{i}{n}) \}\). We prove that \(\lim_{n \rightarrow \infty} \frac{A_{n}}{n^k} \,=\, \sup \{ \frac{p_k(x)}{1-x^k}: 0 \leq x <1\}\). We also consider two closely related maximization recurrences Sn and Sn, defined as S0 = S0 = 0, and for n ≥ 1, \(S_{n} = \max\nolimits_{0 \leq i < n} \{ S_{i} + \frac{i(n-i)(n-i-1)}{2} \}\) and \(S'_{n} = \max\nolimits_{0 \leq i < n} \{ S'_{i} + {n-i \choose 3} + 2i { n-i \choose 2} + (n-i){ i \choose 2} \}\). We prove that \(\lim\nolimits_{n \rightarrow \infty} \frac{S_{n}}{n^3} = \frac{2\sqrt{3}-3}{6} \approx 0.077350...\) and \(\lim\nolimits_{n \rightarrow \infty} \frac{S'_{n}}{3{n \choose 3}} = \frac{2(\sqrt{3}-1)}{3} \approx 0.488033...\), resolving an open problem from Bioinformatics about rooted triplets consistency in phylogenetic networks.

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References

  1. 1.
    Bininda-Emonds, O.R.P.: The evolution of supertrees. Trends in Ecology and Evolution 19(6), 315–322 (2004)CrossRefGoogle Scholar
  2. 2.
    Byrka, J., Gawrychowski, P., Huber, K.T., Kelk, S.: Worst-case optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks. Journal of Discrete Algorithms 8(1), 65–75 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Massachusetts (2009)MATHGoogle Scholar
  4. 4.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Associates, Inc., Sunderland (2004)Google Scholar
  5. 5.
    Fredman, M.L., Knuth, D.E.: Recurrence relations based on minimization. Journal of Mathematical Analysis and Applications 48(2), 534–559 (1974)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gąsieniec, L., Jansson, J., Lingas, A., Östlin, A.: On the complexity of constructing evolutionary trees. Journal of Combinatorial Optimization 3(2-3), 183–197 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gusfield, D., Eddhu, S., Langley, C.: Efficient reconstruction of phylogenetic networks with constrained recombination. In: Proceedings of the Computational Systems Bioinformatics Conference (CSB2 2003), pp. 363–374 (2003)Google Scholar
  8. 8.
    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)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hwang, H.-K., Tsai, T.-H.: An asymptotic theory for recurrence relations based on minimization and maximization. Theoretical Computer Science 290(3), 1475–1501 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Huson, D.H., Rupp, R., Scornavacca, C.: Phylogenetic Networks: Concepts, Algorithms and Applications. Cambridge University Press (2010)Google Scholar
  11. 11.
    Jansson, J., Nguyen, N., Sung, W.: Algorithms for combining rooted triplets into a galled phylogenetic network. SIAM Journal on Computing 35(5), 1098–1121 (2006)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kapoor, S., Reingold, E.M.: Recurrence relations based on minimization and maximization. Journal of Mathematical Analysis and Applications 109(2), 591–604 (1985)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Li, Z., Reingold, E.M.: Solution of a divide-and-conquer maximin recurrence. SIAM Journal on Computing 18(6), 1188–1200 (1989)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Morrison, D.: Introduction to Phylogenetic Networks. RJR Productions (2011)Google Scholar
  15. 15.
    Saha, A., Wagh, M.D.: Minmax recurrences in analysis of algorithms. In: Proceedings of Southeastcon 1993. IEEE (1993)Google Scholar
  16. 16.
    Wang, L., Ma, B., Li, M.: Fixed topology alignment with recombination. Discrete Applied Mathematics 104(1-3), 281–300 (2000)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kun-Mao Chao
    • 1
  • An-Chiang Chu
    • 1
  • Jesper Jansson
    • 2
  • Richard S. Lemence
    • 2
    • 3
  • Alban Mancheron
    • 4
  1. 1.Department of Computer Science and Information EngineeringNational Taiwan UniversityTaipeiTaiwan
  2. 2.Ochanomizu UniversityBunkyo-kuJapan
  3. 3.Institute of Mathematics, College of ScienceUniversity of the PhilippinesQuezon CityPhilippines
  4. 4.LIRMM/CNRSUniversité Montpellier 2Montpellier Cedex 5France

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