The master theorem is a technique used in the analysis of recursive algorithms to determine their time complexity. It provides a framework for solving recurrence relations that arise in the divide-and-conquer algorithm design paradigm.

The master theorem is stated in terms of the following recurrence relation:

T(n) = a * T(n/b) + f(n)

where:

- T(n) represents the time complexity of the recursive algorithm for an input size of n.
- a represents the number of recursive subproblems generated in each recursive call.
- n/b represents the size of each subproblem.
- f(n) represents the time complexity of the work done outside the recursive calls, such as combining the subproblem solutions.

The master theorem provides a formula for computing the time complexity of the recurrence relation based on the values of a, b, and f(n). It is typically expressed in terms of three cases:

1. If f(n) = O(n^c) for some constant c < log_b(a), then T(n) = Theta(n^log_b(a)).

2. If f(n) = Theta(n^log_b(a) * log^k(n)), where k >= 0, and if a * f(n/b) <= c * f(n) for some constant c < 1 and sufficiently large n, then T(n) = Theta(n^log_b(a) * log^(k+1)(n)).

3. If f(n) = Omega(n^c) for some constant c > log_b(a), and if a * f(n/b) <= c * f(n) for some constant c > 1 and sufficiently large n, and if f(n) is sufficiently well-behaved, then T(n) = Theta(f(n)).

These cases cover different scenarios and provide a way to determine the time complexity of a recursive algorithm without explicitly solving the recurrence relation.

It's important to note that the master theorem is applicable to specific forms of recurrence relations, and not all recursive algorithms can be analyzed using this method. Additionally, when the recurrence relation does not fit into the defined cases of the master theorem, other techniques such as recursion tree or substitution method may be used for the analysis.

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