Akra–Bazzi method

Akra–Bazzi method   (Redirected from Akra–Bazzi theorem) Jump to navigation Jump to search This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi.[1] Contents 1 Formulation 2 Example 3 Significance 4 See also 5 References 6 External links Formulation The Akra–Bazzi method applies to recurrence formulas of the form[1] {displaystyle T(x)=g(x)+sum _{i=1}^{k}a_{i}T(b_{i}x+h_{i}(x))qquad {text{for }}xgeq x_{0}.} The conditions for usage are: sufficient base cases are provided {displaystyle a_{i}} and {displaystyle b_{i}} are constants for all {displaystyle i} {displaystyle a_{i}>0} for all {displaystyle i} {displaystyle 03} . In applying the Akra–Bazzi method, the first step is to find the value of {displaystyle p} for which {displaystyle {frac {7}{4}}left({frac {1}{2}}right)^{p}+left({frac {3}{4}}right)^{p}=1} . In this example, {displaystyle p=2} . Then, using the formula, the asymptotic behavior can be determined as follows:[3] {displaystyle {begin{aligned}T(x)&in Theta left(x^{p}left(1+int _{1}^{x}{frac {g(u)}{u^{p+1}}},duright)right)\&=Theta left(x^{2}left(1+int _{1}^{x}{frac {u^{2}}{u^{3}}},duright)right)\&=Theta (x^{2}(1+ln x))\&=Theta (x^{2}log x).end{aligned}}} Significance The Akra–Bazzi method is more useful than most other techniques for determining asymptotic behavior because it covers such a wide variety of cases. Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in the merge sort, the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as {displaystyle T(1)=0} and {displaystyle T(n)=Tleft(leftlfloor {frac {1}{2}}nrightrfloor right)+Tleft(leftlceil {frac {1}{2}}nrightrceil right)+n-1} for integers {displaystyle n>0} , and can thus be computed using the Akra–Bazzi method to be {displaystyle Theta (nlog n)} .

See also Master theorem (analysis of algorithms) Asymptotic complexity References ^ Jump up to: a b Akra, Mohamad; Bazzi, Louay (May 1998). "On the solution of linear recurrence equations". Computational Optimization and Applications. 10 (2): 195–210. doi:10.1023/A:1018373005182. ^ "Proof and application on few examples" (PDF). ^ Cormen, Thomas; Leiserson, Charles; Rivest, Ronald; Stein, Clifford (2009). Introduction to Algorithms. MIT Press. ISBN 978-0262033848. External links O Método de Akra-Bazzi na Resolução de Equações de Recorrência (in Portuguese) Categories: Asymptotic analysisTheorems in discrete mathematicsRecurrence relationsBazzi family