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# Akra-Bazzi method for finding the time complexities

• Last Updated : 30 Jul, 2021

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:

1. 2. 3. Akra-Bazzi Method: This article explores another method for solving such recurrences that were developed by Mohammad Akra and Louay Bazzi in 1998. The Akra-Bazzi method can be applied to the recurrences of the following form: where, and are constants such that:

1. 2. 3. 4. 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,

=> => => => => => • Works for many divides and conquer algorithms.
• Has a lesser constraint over the format of the recurrence than Master’s Theorem.
• p can be calculated using numerical methods for complex recurrence relations.