Sum of all Submatrices of a Given Matrix

Given a NxN 2-D matrix, the task to find the sum of all the submatrices.

Examples:

Input :  arr[] = {{1, 1},
                  {1, 1}};
Output : 16
Explanation: 
Number of sub-matrices with 1 elements = 4
Number of sub-matrices with 2 elements = 4
Number of sub-matrices with 3 elements = 0
Number of sub-matrices with 4 elements = 1

Since all the entries are 1, the sum becomes
sum = 1 * 4 + 2 * 4 + 3 * 0 + 4 * 1 = 16

Input : arr[] = {{1, 2, 3},
                 {4, 5, 6},
                 {7, 8, 9}}
Output : 500

Simple Solution: A naive solution is to generate all the possible submatrices and sum up all of them.
The time complexity of this approach will be O(n6).

Efficient Solution : For each element of the matrix, let us try to find the number of sub-matrices, the element will lie in.
This can be done in O(1) time. Let us suppose the index of an element be (X, Y) in 0 based indexing, then the number of submatrices (Sx, y) for this element will be in can be given by the formula Sx, y = (X + 1) * (Y + 1) * (N – X) * (N – Y) . This formula works, because we just have to choose two different positions on the matrix that will create a submatrix that envelopes the element. Thus, for each element, ‘sum’ can be updated as sum += (Sx, y) * Arrx, y.

Below is the implementation of the above approach:

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program to find the sum of all
// possible submatrices of a given Matrix
  
#include <iostream>
#define n 3
using namespace std;
  
// Function to find the sum of all
// possible submatrices of a given Matrix
int matrixSum(int arr[][n])
{
    // Variable to store
    // the required sum
    int sum = 0;
  
    // Nested loop to find the number 
    // of submatrices, each number belongs to
    for (int i = 0; i < n; i++)
        for (int j = 0; j < n; j++) {
  
            // Number of ways to choose
            // from top-left elements
            int top_left = (i + 1) * (j + 1);
  
            // Number of ways to choose
            // from bottom-right elements
            int bottom_right = (n - i) * (n - j);
            sum += (top_left * bottom_right * arr[i][j]);
        }
  
    return sum;
}
  
// Driver Code
int main()
{
    int arr[][n] = { { 1, 1, 1 },
                     { 1, 1, 1 },
                     { 1, 1, 1 } };
  
    cout << matrixSum(arr);
  
    return 0;
}
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java program to find the sum of all 
// possible submatrices of a given Matrix 
class GFG
{
  
    static final int n = 3;
  
    // Function to find the sum of all 
    // possible submatrices of a given Matrix 
    static int matrixSum(int arr[][]) 
    {
        // Varialbe to store 
        // the required sum 
        int sum = 0;
  
        // Nested loop to find the number 
        // of submatrices, each number belongs to 
        for (int i = 0; i < n; i++)
        {
            for (int j = 0; j < n; j++) 
            {
  
                // Number of ways to choose 
                // from top-left elements 
                int top_left = (i + 1) * (j + 1);
  
                // Number of ways to choose 
                // from bottom-right elements 
                int bottom_right = (n - i) * (n - j);
                sum += (top_left * bottom_right * arr[i][j]);
            }
        }
  
        return sum;
    }
  
    // Driver Code 
    public static void main(String[] args) 
    {
        int arr[][] = {{1, 1, 1},
        {1, 1, 1},
        {1, 1, 1}};
  
        System.out.println(matrixSum(arr));
    }
}
  
// This code contributed by Rajput-Ji
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python3 program to find the sum of all 
# possible submatrices of a given Matrix 
n = 3
  
# Function to find the sum of all 
# possible submatrices of a given Matrix 
def matrixSum(arr) : 
      
    # Variable to store the required sum 
    sum = 0
  
    # Nested loop to find the number of
    # submatrices, each number belongs to 
    for i in range(n) : 
        for j in range(n) :
  
            # Number of ways to choose 
            # from top-left elements 
            top_left = (i + 1) * (j + 1); 
  
            # Number of ways to choose 
            # from bottom-right elements 
            bottom_right = (n - i) * (n - j); 
            sum += (top_left * bottom_right * 
                                  arr[i][j]); 
  
    return sum
  
# Driver Code 
if __name__ == "__main__" :
    arr = [[ 1, 1, 1 ], 
           [ 1, 1, 1 ], 
           [ 1, 1, 1 ]]; 
  
    print(matrixSum(arr)) 
      
# This code is contributed by Ryuga
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# program to find the sum of all 
// possible submatrices of a given Matrix 
using System;
  
class GFG
{
static int n = 3;
  
// Function to find the sum of all 
// possible submatrices of a given Matrix 
static int matrixSum(int [,]arr) 
{
    // Varialbe to store the 
    // required sum 
    int sum = 0;
  
    // Nested loop to find the number of  
    // submatrices, each number belongs to 
    for (int i = 0; i < n; i++)
    {
        for (int j = 0; j < n; j++) 
        {
  
            // Number of ways to choose 
            // from top-left elements 
            int top_left = (i + 1) * (j + 1);
  
            // Number of ways to choose 
            // from bottom-right elements 
            int bottom_right = (n - i) * (n - j);
            sum += (top_left * bottom_right * arr[i, j]);
        }
    }
  
    return sum;
}
  
// Driver Code 
public static void Main() 
{
    int [,]arr = {{1, 1, 1},
    {1, 1, 1},
    {1, 1, 1}};
  
    Console.WriteLine(matrixSum(arr));
}
}
  
// This code contributed by vt_m..
chevron_right

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
// PHP program to find the sum of all
// possible submatrices of a given Matrix
  
// Function to find the sum of all
// possible submatrices of a given Matrix
function matrixSum($arr)
{
    $n = 3;
      
    // Variable to store the required sum
    $sum = 0;
  
    // Nested loop to find the number 
    // of submatrices, each number belongs to
    for ($i = 0; $i < $n; $i++)
        for ($j = 0; $j < $n; $j++)
        {
  
            // Number of ways to choose
            // from top-left elements
            $top_left = ($i + 1) * ($j + 1);
  
            // Number of ways to choose
            // from bottom-right elements
            $bottom_right = ($n - $i) * ($n - $j);
            $sum += ($top_left * $bottom_right
                                 $arr[$i][$j]);
        }
  
    return $sum;
}
  
// Driver Code
$arr = array(array(1, 1, 1),
             array(1, 1, 1),
             array(1, 1, 1));
  
echo matrixSum($arr);
  
// This code is contributed
// by Akanksha Rai
?>
chevron_right

Output:
100

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.





Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.



Article Tags :