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:
C++
// 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; } |
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Java
// 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 |
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Python3
# 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 |
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C#
// 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.. |
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PHP
<?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 ?> |
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Output:
100
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