## John Kieffer - Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function Systems

dmtcs:2983 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12) - https://doi.org/10.46298/dmtcs.2983
Asymptotics of Divide-And-Conquer Recurrences Via Iterated Function SystemsArticle

Authors: John Kieffer 1

• 1 Department of Electrical and Computer Engineering [Minneapolis]

Let $k≥2$ be a fixed integer. Given a bounded sequence of real numbers $(a_n:n≥k)$, then for any sequence $(f_n:n≥1)$ of real numbers satisfying the divide-and-conquer recurrence $f_n = (k-mod(n,k))f_⌊n/k⌋+mod(n,k)f_⌈n/k⌉ + a_n, n ≥k$, there is a unique continuous periodic function $f^*:\mathbb{R}→\mathbb{R}$ with period 1 such that $f_n = nf^*(\log _kn)+o(n)$. If $(a_n)$ is periodic with period $k, a_k=0$, and the initial conditions $(f_i:1 ≤i ≤k-1)$ are all zero, we obtain a specific iterated function system $S$, consisting of $k$ continuous functions from $[0,1]×\mathbb{R}$ into itself, such that the attractor of $S$ is $\{(x,f^*(x)): 0 ≤x ≤1\}$. Using the system $S$, an accurate plot of $f^*$ can be rapidly obtained.

Volume: DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)
Section: Proceedings
Published on: January 1, 2012
Imported on: January 31, 2017
Keywords: divide-and-conquer recurrences,iterated function systems,IFS attractor,self-affine functions,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]
Funding:
Source : OpenAIRE Graph
• Collaborative Research: Information Theory of Data Structures; Funder: National Science Foundation; Code: 0830457