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.
Formulation The Akra–Bazzi method applies to recurrence formulas of the form:
T ( x ) = g ( x ) + ∑ i = 1 k a i T ( b i x + h i ( x ) ) for x ≥ x 0 . {\displaystyle T(x)=g(x)+\sum _{i=1}^{k}a_{i}T(b_{i}x+h_{i}(x))\qquad {\text{for }}x\geq x_{0}.}
The conditions for usage are:
sufficient base cases are provided
a i {\displaystyle a_{i}} and b i {\displaystyle b_{i}} are constants for all i {\displaystyle i}
a i > 0 {\displaystyle a_{i}>0} for all i {\displaystyle i}
0 < b i < 1 {\displaystyle 0<b_{i}<1} for all i {\displaystyle i}
| g ′ ( x ) | ∈ O ( x c ) {\displaystyle \left|g'(x)\right|\in O(x^{c})} , where c is a constant and O notates Big O notation
| h i ( x ) | ∈ O ( x ( log x ) 2 ) {\displaystyle \left|h_{i}(x)\right|\in O\left({\frac {x}{(\log x)^{2}}}\right)} for all i {\displaystyle i}
x 0 {\displaystyle x_{0}} is a constant The asymptotic behavior of T ( x ) {\displaystyle T(x)} is found by determining the value of p {\displaystyle p} for which ∑ i = 1 k a i b i p = 1 {\displaystyle \sum _{i=1}^{k}a_{i}b_{i}^{p}=1} and plugging that value into the equation:
T ( x ) ∈ Θ ( x p ( 1 + ∫ 1 x g ( u ) u p + 1 d u ) ) {\displaystyle T(x)\in \Theta \left(x^{p}\left(1+\int _{1}^{x}{\frac {g(u)}{u^{p+1}}}du\right)\right)}
… excerpt ends here. Continue reading the full article.
