Master Method

Internal

• Recursive Algorithms Complexity

Overview

The master method or master theorem is a general mathematical tool for analyzing the running time of divide and conquer algorithms that divide a problem in equal subproblems and call themselves recursively. The fact that the subproblems are equal in size is important: master method does not apply to divide and conquer algorithms that divide the problem in unequal parts.

Master Method and Recurrences

is a method of analyzing running time of recursive algorithms by using the "master theorem". The master theorem applies to algorithms whose running times can be expressed as a recurrence equation. A generic form of a recurrence equation is:

T(n) = a⋅T(n/b) + f(n)

where a ≥ 1, b > 1 and f(n) is a function that approximates the non-recursive cost. Note that the above expression does include a boundary condition for n = 1. The reasons is that T(1) does not significantly change the solution to the recurrence, it may change it with a constant factor, but the order of growth will stay unchanged.

TODO CLRS page 93.

TODO integrate after class:

Integrate class https://www.coursera.org/learn/algorithms-divide-conquer/lecture/HkcdO/formal-statement

Analyzing Divide-and-Conquer Algorithms

TODO integrate after class:

For divide-and-conquer recursive problems, where solving the problem involves dividing the problem into a subproblems of size n/b of the original', applying the algorithm recursively a times on n/b-size problems and then combining the result, the run time can be expressed by the generic recurrence presented below. If the problem size is small enough (n ≤ c) the straightforward solution takes constant time, which is expressed as a constant function Θ(1).

       | Θ(1)                     if n ≤ c
T(n) = |
       | aT(n/b) + D(n) + C(n)    otherwise

where the D(n) is the cost of dividing the n-size problem into a subproblems, and the C(n) is the cost of combining the results.

TODO

• intuition
• definition
• proof
• Emphasize that is theta, not O

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