Maximum sum of hour glass in matrix

Given a 2D matrix, the task is to find the maximum sum of an hourglass.

An hour glass is made of 7 cells in following form.

t u v
x
x y z

Examples: 

Input : mat[][] = [[1, 1, 1, 0, 0],
[0, 1, 0, 0, 0],
[1, 1, 1, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]]
Output : 7
Explanation: Below is the hour glass with maximum sum:
1 1 1
1
1 1 1

Input : mat[][] = [[0, 3, 0, 0, 0],
[0, 1, 0, 0, 0],
[1, 1, 1, 0, 0],
[0, 0, 2, 4, 4],
[0, 0, 0, 2, 4]]
Output : 11
Explanation: Below is the hour glass with maximum sum:
1 0 0
4
0 2 4

Approach:

The idea is to traverse through the entire matrix and check each possible position where a complete hourglass pattern is possible.

If we count the total number of hourglasses in a matrix of size R x C, we can say that the count is equal to the count of possible top left cells in an hourglass. The number of top-left cells in an hourglass is equal to (R-2)*(C-2). Therefore, in a matrix total number of an hourglass is (R-2)*(C-2).

mat[][] = 2 3 0 0 0
0 1 0 0 0
1 1 1 0 0
0 0 2 4 4
0 0 0 2 0
Possible hour glass are :
2 3 0 3 0 0 0 0 0
1 0 0
1 1 1 1 1 0 1 0 0

0 1 0 1 0 0 0 0 0
1 1 0
0 0 2 0 2 4 2 4 4

1 1 1 1 1 0 1 0 0
0 2 4
0 0 0 0 0 2 0 2 0

Step by step approach:

1. Check if the matrix dimensions are at least 3×3 to fit an hourglass. If not, return -1.
2. Iterate through the matrix, stopping 3 rows and 3 columns before the end.
3. At each position, sum the 7 cells forming the hourglass pattern (top row, middle element, bottom row).
4. Compare the current sum with the maximum sum found so far and update if needed.

// C++ program to find Maximum
// sum of hour glass in matrix
#include <bits/stdc++.h>
using namespace std;

int maxHourGlassSum(vector<vector<int>> &mat) {
    int n = mat.size(), m = mat[0].size();
    
    // If matrix is too small 
    // to contain an hourglass
    if (n < 3 || m < 3)
        return -1;
    
    int res = INT_MIN;
    
    for (int i = 0; i <= n - 3; i++) {
        for (int j = 0; j <= m - 3; j++) {
            
            // Calculate sum of current hourglass
            int sum = mat[i][j] + mat[i][j+1] + mat[i][j+2] +
                                  mat[i+1][j+1] +
                      mat[i+2][j] + mat[i+2][j+1] + mat[i+2][j+2];
            
            // Update result if current hourglass sum is greater
            res = max(res, sum);
        }
    }
    
    return res;
}

int main() {
    vector<vector<int>> mat = {
        {1, 1, 1, 0, 0},
        {0, 1, 0, 0, 0},
        {1, 1, 1, 0, 0},
        {0, 0, 0, 0, 0},
        {0, 0, 0, 0, 0}
    };
    cout << maxHourGlassSum(mat);
    return 0;
}

// Java program to find Maximum
// sum of hour glass in matrix
class GfG {

    static int maxHourGlassSum(int[][] mat) {
        int n = mat.length, m = mat[0].length;
        
        // If matrix is too small 
        // to contain an hourglass
        if (n < 3 || m < 3)
            return -1;
        
        int res = Integer.MIN_VALUE;
        
        for (int i = 0; i <= n - 3; i++) {
            for (int j = 0; j <= m - 3; j++) {
                
                // Calculate sum of current hourglass
                int sum = mat[i][j] + mat[i][j+1] + mat[i][j+2] +
                                      mat[i+1][j+1] +
                          mat[i+2][j] + mat[i+2][j+1] + mat[i+2][j+2];
                
                // Update result if current hourglass sum is greater
                res = Math.max(res, sum);
            }
        }
        
        return res;
    }

    public static void main(String[] args) {
        int[][] mat = {
            {1, 1, 1, 0, 0},
            {0, 1, 0, 0, 0},
            {1, 1, 1, 0, 0},
            {0, 0, 0, 0, 0},
            {0, 0, 0, 0, 0}
        };
        System.out.println(maxHourGlassSum(mat));
    }
}

# Python program to find Maximum
# sum of hour glass in matrix

def maxHourGlassSum(mat):
    n = len(mat)
    m = len(mat[0])
    
    # If matrix is too small 
    # to contain an hourglass
    if n < 3 or m < 3:
        return -1
    
    res = float('-inf')
    
    for i in range(n - 2):
        for j in range(m - 2):
            
            # Calculate sum of current hourglass
            sum = mat[i][j] + mat[i][j+1] + mat[i][j+2] + \
                               mat[i+1][j+1] + \
                  mat[i+2][j] + mat[i+2][j+1] + mat[i+2][j+2]
            
            # Update result if current hourglass sum is greater
            res = max(res, sum)
    
    return res

if __name__ == "__main__":
    mat = [
        [1, 1, 1, 0, 0],
        [0, 1, 0, 0, 0],
        [1, 1, 1, 0, 0],
        [0, 0, 0, 0, 0],
        [0, 0, 0, 0, 0]
    ]
    print(maxHourGlassSum(mat))

// C# program to find Maximum
// sum of hour glass in matrix
using System;

class GfG {

    static int maxHourGlassSum(int[][] mat) {
        int n = mat.Length, m = mat[0].Length;
        
        // If matrix is too small 
        // to contain an hourglass
        if (n < 3 || m < 3)
            return -1;
        
        int res = int.MinValue;
        
        for (int i = 0; i <= n - 3; i++) {
            for (int j = 0; j <= m - 3; j++) {
                
                // Calculate sum of current hourglass
                int sum = mat[i][j] + mat[i][j+1] + mat[i][j+2] +
                                      mat[i+1][j+1] +
                          mat[i+2][j] + mat[i+2][j+1] + mat[i+2][j+2];
                
                // Update result if current hourglass sum is greater
                res = Math.Max(res, sum);
            }
        }
        
        return res;
    }

    static void Main(string[] args) {
        int[][] mat = new int[][] {
            new int[] {1, 1, 1, 0, 0},
            new int[] {0, 1, 0, 0, 0},
            new int[] {1, 1, 1, 0, 0},
            new int[] {0, 0, 0, 0, 0},
            new int[] {0, 0, 0, 0, 0}
        };
        Console.WriteLine(maxHourGlassSum(mat));
    }
}

// JavaScript program to find Maximum
// sum of hour glass in matrix

function maxHourGlassSum(mat) {
    let n = mat.length, m = mat[0].length;
    
    // If matrix is too small 
    // to contain an hourglass
    if (n < 3 || m < 3)
        return -1;
    
    let res = Number.MIN_SAFE_INTEGER;
    
    for (let i = 0; i <= n - 3; i++) {
        for (let j = 0; j <= m - 3; j++) {
            
            // Calculate sum of current hourglass
            let sum = mat[i][j] + mat[i][j+1] + mat[i][j+2] +
                                mat[i+1][j+1] +
                      mat[i+2][j] + mat[i+2][j+1] + mat[i+2][j+2];
            
            // Update result if current hourglass sum is greater
            res = Math.max(res, sum);
        }
    }
    
    return res;
}

const mat = [
    [1, 1, 1, 0, 0],
    [0, 1, 0, 0, 0],
    [1, 1, 1, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0]
];

console.log(maxHourGlassSum(mat));

7

Time complexity: O(n*m), where n is the number of rows and m is the number of columns in the matrix.
Auxiliary Space: O(1)