What is the function of the Master Theorem in algorithm analysis?

The Master Theorem serves as a powerful tool for analyzing the time complexity of divide-and-conquer algorithms. It provides a way to determine the asymptotic behavior of recursive functions that fit a specific form. By applying the Master Theorem, you can solve recurrences that typically arise from algorithms that break a problem into smaller subproblems, solve those subproblems independently, and then combine their solutions to solve the original problem.

For example, in many algorithms, the running time can be expressed as a recurrence relation, such as T(n) = aT(n/b) + f(n), where 'a' is the number of subproblems, 'n/b' is the size of each subproblem, and 'f(n)' is the cost of dividing the problem and combining the results. The Master Theorem provides different cases to analyze such recurrences and derive their time complexities efficiently without the need for elaborate and often complex mathematical techniques like substitution or the use of the recursion tree method.

Its primary function is therefore to facilitate a quicker and more straightforward way to derive the performance characteristics of a significant class of algorithms, especially those that utilize the divide-and-conquer strategy.