Contents

Toggle## Divide and rule

**not**, into several smaller sub-problems, then recombine the partial solutions until the final solution is obtained.

## Principle

Unlike the dynamic programming, the results of the sub-problems are only useful for the result of the parent problem.

The divide and conquer algorithm can be broken down into three steps:

- Divide: the problem is split into size sub-problems
**b / w**. These sub-problems are of the same nature as their parent. We are therefore in the presence of**To**identical subproblems. When**b = 2**, we are talking about dichotomy. - To rule: the sub-problems are solved recursively.
- Recombine: the solutions of the subproblems are recombined to reconstruct the solution to the initial problem.

## Example

In order to better understand the construction of the algorithm, we will take the example of fast exponentiation. The principle of this algorithm is to calculate x^{not}.

In order to use the divide and conquer method, we need to find a mathematical program that allows us to rewrite our problem into equivalent subproblems. Since this is a recursive method, don't forget to include a final state. We are therefore in the presence of the following mathematical program:

The algorithm is as follows:

double power (double x, int n) {yew(n == 0) return 1;else{yew(n%2 == 0) return power (x * x, n / 2);elsereturn x * power (x * x, (n-1) / 2);end ifend if}

Its complexity is 0 (log n) - size of the complete binary tree at **not** tops.