Abstract

Techniques for solving divide-and-conquer recurrences are routinely taught to thousands of Computer Science students each year. The dominant approach to solving such recurrences is known as the Master Method [2]. Recently, Akra and Bazzi [1] discovered a surprisingly elegant generalization of the Master Method that yields a very simple formula for solving most divideand -conquer recurrences. In these notes, we provide a simple inductive proof of the Akra-Bazzi result and we extend the result to handle variations of divide-and-conquer recurrences that commonly arise in practice. 1 Introduction Divide-and-conquer recurrences are ubiquitous in the analysis of algorithms. Many methods are known for solving recurrences such as T (n) = ( 1 if n = 1 2T (dn=2e) +O(n) if n ? 1; but perhaps the most widely taught approach is the Master Method that is described in the seminal algorithms text by Cormen, Leiserson and Rivest [2]. The Master Method is fairly powerful and results in a closed form s...

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