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

#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;
}

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