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Advanced Algorithms/Divide and Conquer.md

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+---
+date: 09.09.2024
+type: 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.
+