# 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 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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© 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