Asymptotic limits of a new type of maximization recurrence with an application to bioinformatics

  • Authors:

  • Kun-Mao Chao;An-Chiang Chu;Jesper Jansson;Richard S. Lemence;Alban Mancheron

  • Affiliations:

  • Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan;Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan;Ochanomizu University, Bunkyo-ku, Tokyo, Japan;Ochanomizu University, Bunkyo-ku, Tokyo, Japan,Institute of Mathematics, College of Science, University of the Philippines, Quezon City, Philippines;LIRMM/CNRS, Université Montpellier 2, Montpellier Cedex 5, France

  • Venue:
  • TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
  • Year:
  • 2012

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Abstract

We study the asymptotic behavior of a new type of maximization recurrence, defined as follows. Let k be a positive integer and p k (x ) a polynomial of degree k satisfying p k (0)=0. Define A 0 =0 and for n ≥1, let $A_{n} = \max\nolimits_{0 \leq i . We prove that $\lim_{n \rightarrow \infty} \frac{A_{n}}{n^k} \,=\, \sup \{ \frac{p_k(x)}{1-x^k}: 0 \leq x . We also consider two closely related maximization recurrences S n and S ′n , defined as S 0 =S ′0 =0, and for n ≥1, $S_{n} = \max\nolimits_{0 \leq i and $S'_{n} = \max\nolimits_{0 \leq i . 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.