# 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
Conference paper
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 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.

## Keywords

Recurrence Relation Asymptotic Limit Phylogenetic Network Binary Phylogenetic Tree Dichotomous Search
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.

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## 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