Notes/Advanced Algorithms/Divide and Conquer.md

date	type
09.09.2024	theoretical

Divide and Conquer is a powerful algorithmic paradigm used to solve complex problems by breaking them down into smaller, more manageable sub-problems.

Steps in Divide and Conquer

1. Divide:

   • The problem is divided into smaller subproblems that are similar to the original but smaller in size.
   • This division continues until the subproblems become simple enough to be solved directly.

2. Conquer:

   • Solve the smaller subproblems recursively. If the subproblems are small enough (base cases), solve them directly without further recursion.

3. Combine:

   • Combine the solutions of the subproblems to get the solution to the original problem.

Example: Merge Sort

Merge Sort is a classic example of the Divide and Conquer approach:

1. Divide:

   • Split the unsorted array into two halves recursively until each subarray contains a single element.

2. Conquer:

   • Merge two sorted halves by comparing elements from each half in sorted order.

3. Combine:

   • Combine the sorted halves to form a single sorted array.

The time complexity of Merge Sort is \mathcal{O}(n \log n) because the array is repeatedly divided in half (\log n divisions) and each division requires a linear amount of work (\mathcal{O}(n)) to merge.

Other Examples

• Quick Sort: Uses Divide and Conquer by selecting a "pivot" element and partitioning the array into elements less than and greater than the pivot.
• Binary Search: Recursively divides a sorted array in half to search for an element, reducing the search space by half each time.
• Strassen's Matrix Multiplication: An algorithm that reduces the time complexity of matrix multiplication from \mathcal{O}(n^3) to approximately \mathcal{O}(n^{2.81}) by recursively breaking down matrices.