# On the Solution of Linear Recurrence Equations

@article{Akra1998OnTS, title={On the Solution of Linear Recurrence Equations}, author={Mohamad A. Akra and Louay Bazzi}, journal={Computational Optimization and Applications}, year={1998}, volume={10}, pages={195-210} }

AbstractIn this article, we present a general solution for linear divide-and-conquer recurrences of the form
$$u_n = \sum\limits_{i = 1}^k {a_i u} $$
⌊
$$\frac{n}{{b_i }}$$
⌋ + g(n) Our approach handles more cases than the Master method does {1}. We achieve this advantage by defining a new transform - the Order transform - which has useful properties for providing asymptotic answers (compared to other transforms which supply exact answers). This transform helps in mapping the sequence under…

## 52 Citations

A Simple Master Theorem For Discrete Divide And Conquer Recurrences

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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…

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Divide-and-conquer recurrences of the form f(n) = f (⌊ n/2⌋ ) + f ( ⌈ n/2⌉ ) + g(n) (n⩾ 2), with g(n) and f(1) given, appear very frequently in the analysis of computer algorithms and related areas.…

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Powerful techniques such as Dirichlet series, Mellin-Perron formula, and (extended) Tauberian theorems of Wiener-Ikehara are applied to provide a complete and precise solution to this basic computer science recurrence.

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A method for obtaining tail-bounds for random variables satisfying certain probabilistic recurrences that arise in the analysis of randomized parallel divide and conquer algorithms, and shows that in some cases, the work-recurrence can be bounded under simpler assumptions than Karp's.

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Notes on Better Master Theorems for Divide-and-Conquer Recurrences

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A simple inductive proof of the Akra-Bazzi result is provided and the result is extended to handle variations of divide-and-conquer recurrences that commonly arise in practice.

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