A simple master Theorem for discrete divide and conquer recurrences

The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: $$X_{n}=a_n+\sum_{j=1}^m b_j X_{\lfloor p_j n \rfloor},$$ where the $p_i$'s belong to $(0,1)$. The main novelty of this work is there is no assumption of regularity or monotonicity for $(a_n)$. Then, this result can be applied to various sequences of random variables $(a_n)_{n\ge 0}$, for example such that $\sup_{n\ge 1}\mathbb{E}(|a_n|)<+\infty$

Esthetic Numbers and Lifting Restrictions on the Analysis of Summatory Functions of Regular Sequences

When asymptotically analysing the summatory function of a $q$-regular sequence in the sense of Allouche and Shallit, the eigenvalues of the sum of matrices of the linear representation of the sequence determine the "shape" (in particular the growth) of the asymptotic formula. Existing general results for determining the precise behavior (including the Fourier coefficients of the appearing fluctuations) have previously been restricted by a technical condition on these eigenvalues. The aim of this work is to lift these restrictions by providing a insightful proof based on generating functions for the main pseudo Tauberian theorem for all cases simultaneously. (This theorem is the key ingredient for overcoming convergence problems in Mellin--Perron summation in the asymptotic analysis.) One example is discussed in more detail: A precise asymptotic formula for the amount of esthetic numbers in the first~$N$ natural numbers is presented. Prior to this only the asymptotic amount of these numbers with a given digit-length was known.Comment: to appear in "2019 Proceedings of the Sixteenth Meeting on Analytic Algorithmics and Combinatorics (ANALCO)

Analisis Perbandingan Kinerja Metode Rekursif dan Metode Iteratif dalam Algoritma Linear Search

Salah satu algoritma pencarian data yang paling populer adalah algoritma linear search.&nbsp; Dalam proses pencarian data sebuah list menggunakan algoritma linear search dapat diterapkan dengan cara iteratif dan rekursif. Pandangan umum mengenai algoritma linear search adalah bahwa performa metode iteratif memiliki hasil yang sama dengan rekursif. Namun di beberapa penelitian menentang pernyataan tersebut yang mungkin tidak berlaku pada semua kasus. Dari analisis tersebut, penelitian ini berfokus pada perbandingan metode rekursif dan iteratif pada algoritma linear search untuk mengetahui algoritma mana yang paling sesuai, efisien dan efektif. Penelitian dilakukan menggunakan 3 studi kasus dengan masing-masing data sebanyak 1 juta, 10 juta, dan 100 juta. Penelitian berfokus pada hasil penggunaan memori dan waktu akses pada proses pencarian data menggunakan notasi Big-O dan bahasa pemrograman Python. Hasil penelitian menunjukkan bahwa algoritma linear search secara iteratif lebih efektif dan efisien dari pada rekursif. Meskipun kedua metode tersebut memiliki kompleksitas Big-O yang sama, namun hasil dari eksekusi program menunjukkan hasil yang berbeda. Dengan hasil algoritma linear search secara iteratif memiliki hasil waktu eksekusi dan penggunaan memori yang lebih unggul yaitu waktu akses dan penggunaan memori yang lebih sedikit dibanding metode rekursif.The linear search algorithm is one of the most popular data search algorithms. In the process of searching data for a list using a linear search algorithm, it can be applied in an iterative and recursive way. The general view of linear search algorithms is that the iterative methods perform the same as recursive methods. However, some studies contradict this statement which may not apply in all cases. From this analysis, this study focuses on the comparison of recursive and iterative methods on linear search algorithms to find out which algorithm is the most suitable, efficient, and effective. The research was conducted using 3 case studies with data of 1 million, 10 million, and 100 million respectively. The research focuses on the results of memory usage and access time in the data search process using Big-O notation and Python programming language. The results show that the iterative linear search algorithm is more effective and efficient than recursive. Although both methods have the same Big-O complexity, the results of program execution show different results. With the results of an iterative linear search algorithm, the results of the execution time and memory usage are superior, namely, the access time and memory usage are less than the recursive method

Rational series and asymptotic expansion for linear homogeneous divide-and-conquer recurrences

Among all sequences that satisfy a divide-and-conquer recurrence, the sequences that are rational with respect to a numeration system are certainly the most immediate and most essential. Nevertheless, until recently they have not been studied from the asymptotic standpoint. We show how a mechanical process permits to compute their asymptotic expansion. It is based on linear algebra, with Jordan normal form, joint spectral radius, and dilation equations. The method is compared with the analytic number theory approach, based on Dirichlet series and residues, and new ways to compute the Fourier series of the periodic functions involved in the expansion are developed. The article comes with an extended bibliography

Joint Spectral Radius, Dilation Equations, and Asymptotic Behavior of Radix-Rational Sequences

International audienceRadix-rational sequences are solutions of systems of recurrence equations based on the radix representation of the index. For each radix-rational sequence with complex values we provide an asymptotic expansion, essentially in the scale N^α log^l N. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence of first-order differences. The coefficients are Hölderian functions obtained through some dilation equations, which are usual in the domains of wavelets and refinement schemes. The proofs are ultimately based on elementary linear algebra.Les suites rationnelles dans une base de numération sont solutions de récurrences basées sur la représentation de l'indice dans la base de numération. Pour chacune de ces suites, à valeurs complexes, nous fournissons un développement asymptotique, essentiellement dans l'échelle N^α log^l N. La précision du développement dépend du rayon spectral de la représentation linéaire pour la suite des différences de la suite considérée. Les coefficients sont des fonctions höldériennes obtenues par des équations de dilatation, usuelles dans les théories des ondelettes et des schémas de raffinement. Les preuves sont basées sur l'algèbre linéaire élémentaire

A Master Theorem for Discrete Divide and Conquer Recurrences ∗

Divide-and-conquer recurrences are one of the most studied equationsin computerscience. Yet, discrete versions of these recurrences, namely T(n) = an + m∑ bjT (⌊pjn+δj⌋) j=1 for some known sequence an and given bj, pj and δj, present some challenges. The discrete nature of this recurrence (represented by the floor function) introduces certain oscillations not captured by the traditional Master Theorem, for example due to Akra and Bazzi who primary studied the continuous version of the recurrence. We apply powerful techniques such as Dirichlet series, Mellin-Perron formula, and (extended) Tauberian theorems of Wiener-Ikehara to provide a complete and precise solution to this basic computer science recurrence. We illustrate applicability of our results on several examples including a popular and fast arithmetic coding algorithm due to Boncelet for which we estimate its average redundancy. To the best of our knowledge, discrete divide and conquer recurrences were not studied in this generality and such detail; in particular, this allows us to compare the redundancy of Boncelet’s algorithm to the (asymptotically) optimal Tunstall scheme