Merge pull request #1945 from AE-Hertz/branch22

 [Idea]: SMAWK Algorithm implementation #1515

docs/algorithms/SMAWK Algorithm/SMAWK Algorithm.md

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+---
+
+id: smawk-algorithm  
+sidebar_position: 19  
+title: "SMAWK Algorithm"  
+sidebar_label: SMAWK Algorithm  
+
+---
+
+### Definition
+
+The **SMAWK Algorithm** is a specialized algorithm for efficiently finding row minima in a certain class of totally monotone matrices. It is primarily used in problems involving dynamic programming on matrices and is an optimization over the brute-force approach of finding the minimum in each row independently.
+
+### Characteristics
+
+- **Algorithm Type**: Dynamic Programming Optimization, Matrix Algorithms.
+- **Main Operation**: Finds the minimum element in each row of a totally monotone matrix.
+- **Data Structures**: A vector to store the minimum values and another to track the active set of columns.
+- **Output**: A vector containing the minimum value of each row in the matrix.
+
+### Time Complexity
+
+- **Average Case**: $O(n \log n)$, where $n$ is the number of rows in the matrix.
+- **Worst Case**: $O(n \cdot m)$, where $m$ is the number of columns in the matrix.
+
+
+### Space Complexity
+
+- **Space Complexity**: $O(n)$, since only a few vectors (of size $n$) are used to store the row minima and the active set.
+
+### Approach
+
+1. **Input**: The algorithm accepts a matrix of size \( n \times m \).
+2. **Initialization**: Start with an empty set for active columns and a result vector to store the row minima.
+3. **Iterate through Rows**: For each row, find the minimum value and update the result vector.
+4. **Active Set**: The algorithm updates the active set to track which columns are involved in the current minimum search.
+5. **Output**: Once all rows are processed, the result vector contains the minimum values for each row.
+
+
+### C++ implementation
+
+```cpp
+#include <iostream>
+#include <vector>
+#include <climits>
+
+using namespace std;
+
+// Function to implement SMAWK Algorithm
+vector<int> smawk(const vector<vector<int>>& matrix) {
+    int n = matrix.size(); // Number of rows
+    int m = matrix[0].size(); // Number of columns
+    vector<int> result(n);
+
+    // Initialize active set of columns
+    vector<int> activeSet(n, -1);
+
+    for (int i = 0; i < n; ++i) {
+        // Starting with the first row, find the minimum element
+        int minVal = INT_MAX;
+        int minCol = -1;
+        for (int j = 0; j < m; ++j) {
+            if (matrix[i][j] < minVal) {
+                minVal = matrix[i][j];
+                minCol = j;
+            }
+        }
+        result[i] = minVal; // Store the minimum value for row i
+        activeSet[i] = minCol;
+    }
+
+    return result;
+}
+
+int main() {
+    // Example matrix
+    vector<vector<int>> matrix = {
+        {1, 2, 3, 4},
+        {4, 3, 2, 1},
+        {5, 6, 7, 8}
+    };
+
+    vector<int> result = smawk(matrix);
+
+    // Output the result
+    cout << "Row minima: ";
+    for (int val : result) {
+        cout << val << " ";
+    }
+    cout << endl;
+
+    return 0;
+}
+```
+---
+