# Print maximum sum square sub-matrix of given size

Given an N x N matrix, find a k x k submatrix where k <= N and k >= 1, such that sum of all the elements in submatrix is maximum. The input matrix can contain zero, positive and negative numbers.

For example consider below matrix, if k = 3, then output should print the sub-matrix enclosed in blue. We strongly recommend you to minimize your browser and try this yourself first.

A Simple Solution is to consider all possible sub-squares of size k x k in our input matrix and find the one which has maximum sum. Time complexity of above solution is O(N2k2).

We can solve this problem in O(N2) time. This problem is mainly an extension of this problem of printing all sums. The idea is to preprocess the given square matrix. In the preprocessing step, calculate sum of all vertical strips of size k x 1 in a temporary square matrix stripSum[][]. Once we have sum of all vertical strips, we can calculate sum of first sub-square in a row as sum of first k strips in that row, and for remaining sub-squares, we can calculate sum in O(1) time by removing the leftmost strip of previous subsquare and adding the rightmost strip of new square.

Below is C++ implementation of above idea.

 `// An efficient C++ program to find maximum sum ` `// sub-square matrix ` `#include ` `using` `namespace` `std; ` ` `  `// Size of given matrix ` `#define N 5 ` ` `  `// A O(n^2) function to the maximum sum sub- ` `// squares of size k x k in a given square ` `// matrix of size n x n ` `void` `printMaxSumSub(``int` `mat[][N], ``int` `k) ` `{ ` `    ``// k must be smaller than or equal to n ` `    ``if` `(k > N) ``return``; ` ` `  `    ``// 1: PREPROCESSING ` `    ``// To store sums of all strips of size k x 1 ` `    ``int` `stripSum[N][N]; ` ` `  `    ``// Go column by column ` `    ``for` `(``int` `j=0; j max_sum) ` `        ``{ ` `            ``max_sum = sum; ` `            ``pos = &(mat[i]); ` `        ``} ` ` `  `        ``// Calculate sum of remaining squares in ` `        ``// current row by removing the leftmost ` `        ``// strip of previous sub-square and adding ` `        ``// a new strip ` `        ``for` `(``int` `j=1; j max_sum) ` `            ``{ ` `                ``max_sum = sum; ` `                ``pos = &(mat[i][j]); ` `            ``} ` `        ``} ` `    ``} ` ` `  `    ``// Print the result matrix ` `    ``for` `(``int` `i=0; i

Output:

```Maximum sum 3 x 3 matrix is
8 6 7
4 4 4
5 5 5```

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

This article is attributed to GeeksforGeeks.org

Matrix Matrix

code

load comments