# Akra-Bazzi method for finding the time complexities

• Difficulty Level : Basic
• Last Updated : 18 May, 2022

Master’s theorem is a popular method to solve time complexity recurrences of the form:

With constraints over a, b and f(n). The recurrence relation form limits the usability of the Master’s theorem. Following are three recurrences that cannot be solved directly using master’s theorem:

Akra-Bazzi Method: This article explores another method for solving such recurrences that were developed by Mohammad Akra and Louay Bazzi in 1996. The Akra-Bazzi method can be applied to the recurrences of the following form:

where,   and  are constants such that:

Next, find p such that

Then

Examples
Let’s consider the three recurrences discussed above and solve them using the method:

Example 1.

Here

1. a1 = 3
2. b1
3. a2 = 2
4. b2
5. b1 and b2 are in the range (0, 1)
6. g(n) = \theta(n) which is O(nc), here c can be 1.

In this problem h1(n) and h2(n) are not present.
Here p=1 satisfies

Finally,

=>

=>

=>

=>

Example 2.

Here

1. a =
2. b =
3. g(n) =
4. b is in the range (0, 1)
5. g(n) = \theta(n^2) which is in O(nc), here c can be 1.

In this problem h(n) is not present.
Here p= – 1 satisfies

Finally,

=>

=>

=>

=>

=>

Example 3.

Here

1. a = 9
2. b =
3. g(n) = \theta(n)
4. b is in the range(0, 1)
5. g(n) =  which is O(nc), here c can be 1.
6. h(n) =   which is

Here p=2 satisfies

Finally,

=>

=>

=>

=>

=>

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