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)


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 \(A_{n} = \max\nolimits_{0 \leq i < n} \{ A_{i} + n^{k} \, p_{k}(\frac{i}{n}) \}\). 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 S n and S n , defined as S 0 = 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.


Recurrence Relation Asymptotic Limit Phylogenetic Network Binary Phylogenetic Tree Dichotomous Search 
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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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