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Print maximum sum square sub-matrix of given size
• Difficulty Level : Hard
• Last Updated : 13 Aug, 2019

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 the implementation of above idea.

## C++

 `// 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

## Java

 `// An efficient Java program to find maximum sum ``// sub-square matrix `` ` `// Class to store the position of start of ``// maximum sum in matrix``class` `Position {``    ``int` `x;``    ``int` `y;`` ` `    ``// Constructor``    ``Position(``int` `x, ``int` `y) {``        ``this``.x = x;``        ``this``.y = y;``    ``}`` ` `    ``// Updates the position if new maximum sum``    ``// is found``    ``void` `updatePosition(``int` `x, ``int` `y) {``        ``this``.x = x;``        ``this``.y = y;``    ``}`` ` `    ``// returns the current value of X``    ``int` `getXPosition() {``        ``return` `this``.x;``    ``}`` ` `    ``// returns the current value of y``    ``int` `getYPosition() {``        ``return` `this``.y;``    ``}``}`` ` `class` `Gfg {``    ``// Size of given matrix``    ``static` `int` `N;`` ` `    ``// 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``    ``static` `void` `printMaxSumSub(``int``[][] mat, ``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 = ``new` `int``[N][N];`` ` `        ``// Go column by column``        ``for` `(``int` `j = ``0``; j < N; j++) {`` ` `            ``// Calculate sum of first k x 1 rectangle``            ``// in this column``            ``int` `sum = ``0``;``            ``for` `(``int` `i = ``0``; i < k; i++)``                ``sum += mat[i][j];``            ``stripSum[``0``][j] = sum;`` ` `            ``// Calculate sum of remaining rectangles``            ``for` `(``int` `i = ``1``; i < N - k + ``1``; i++) {``                ``sum += (mat[i + k - ``1``][j] - mat[i - ``1``][j]);``                ``stripSum[i][j] = sum;``            ``}``        ``}`` ` `        ``// max_sum stores maximum sum and its``        ``// position in matrix``        ``int` `max_sum = Integer.MIN_VALUE;``        ``Position pos = ``new` `Position(-``1``, -``1``);`` ` `        ``// 2: CALCULATE SUM of Sub-Squares using stripSum[][]``        ``for` `(``int` `i = ``0``; i < N - k + ``1``; i++) {`` ` `            ``// Calculate and print sum of first subsquare``            ``// in this row``            ``int` `sum = ``0``;``            ``for` `(``int` `j = ``0``; j < k; j++)``                ``sum += stripSum[i][j];`` ` `            ``// Update max_sum and position of result``            ``if` `(sum > max_sum) {``                ``max_sum = sum;``                ``pos.updatePosition(i, ``0``);``            ``}`` ` `            ``// 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 < N - k + ``1``; j++) {``                ``sum += (stripSum[i][j + k - ``1``] - stripSum[i][j - ``1``]);`` ` `                ``// Update max_sum and position of result``                ``if` `(sum > max_sum) {``                    ``max_sum = sum;``                    ``pos.updatePosition(i, j);``                ``}``            ``}``        ``}`` ` `        ``// Print the result matrix``        ``for` `(``int` `i = ``0``; i < k; i++) {``            ``for` `(``int` `j = ``0``; j < k; j++) {``                ``System.out.print(mat[i + pos.getXPosition()][j + pos.getYPosition()] + ``" "``);``            ``}``            ``System.out.println();``        ``}``    ``}`` ` `    ``// Driver program to test above function``    ``public` `static` `void` `main(String[] args) {``        ``N = ``5``;``        ``int``[][] mat = { { ``1``, ``1``, ``1``, ``1``, ``1` `}, ``                ``{ ``2``, ``2``, ``2``, ``2``, ``2` `}, ``                ``{ ``3``, ``8``, ``6``, ``7``, ``3` `}, ``                ``{ ``4``, ``4``, ``4``, ``4``, ``4` `},``            ``{ ``5``, ``5``, ``5``, ``5``, ``5` `} };``    ``int` `k = ``3``;`` ` `        ``System.out.println(``"Maximum sum 3 x 3 matrix is"``);``        ``printMaxSumSub(mat, k);``    ``}``}`` ` `// This code is contributed by Vivek Kumar Singh`

## C#

 `// An efficient C# program to find maximum sum ``// sub-square matrix ``using` `System;`` ` `// Class to store the position of start of ``// maximum sum in matrix ``class` `Position ``{ ``    ``int` `x; ``    ``int` `y; `` ` `    ``// Constructor ``    ``public` `Position(``int` `x, ``int` `y)``    ``{ ``        ``this``.x = x; ``        ``this``.y = y; ``    ``} `` ` `    ``// Updates the position if new maximum sum ``    ``// is found ``    ``public` `void` `updatePosition(``int` `x, ``int` `y)``    ``{ ``        ``this``.x = x; ``        ``this``.y = y; ``    ``} `` ` `    ``// returns the current value of X ``    ``public` `int` `getXPosition()``    ``{ ``        ``return` `this``.x; ``    ``} `` ` `    ``// returns the current value of y ``    ``public` `int` `getYPosition() ``    ``{ ``        ``return` `this``.y; ``    ``} ``} `` ` `class` `GFG ``{ ``     ` `    ``// Size of given matrix ``    ``static` `int` `N; `` ` `    ``// 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 ``    ``static` `void` `printMaxSumSub(``int``[,] mat, ``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 = ``new` `int``[N, N]; `` ` `        ``// Go column by column ``        ``for` `(``int` `j = 0; j < N; j++) ``        ``{ `` ` `            ``// Calculate sum of first k x 1 rectangle ``            ``// in this column ``            ``int` `sum = 0; ``            ``for` `(``int` `i = 0; i < k; i++) ``                ``sum += mat[i, j]; ``            ``stripSum[0, j] = sum; `` ` `            ``// Calculate sum of remaining rectangles ``            ``for` `(``int` `i = 1; i < N - k + 1; i++) ``            ``{ ``                ``sum += (mat[i + k - 1, j] - ``                        ``mat[i - 1, j]); ``                ``stripSum[i, j] = sum; ``            ``} ``        ``} `` ` `        ``// max_sum stores maximum sum and its ``        ``// position in matrix ``        ``int` `max_sum = ``int``.MinValue; ``        ``Position pos = ``new` `Position(-1, -1); `` ` `        ``// 2: CALCULATE SUM of Sub-Squares using stripSum[,] ``        ``for` `(``int` `i = 0; i < N - k + 1; i++)``        ``{ `` ` `            ``// Calculate and print sum of first subsquare ``            ``// in this row ``            ``int` `sum = 0; ``            ``for` `(``int` `j = 0; j < k; j++) ``                ``sum += stripSum[i, j]; `` ` `            ``// Update max_sum and position of result ``            ``if` `(sum > max_sum) ``            ``{ ``                ``max_sum = sum; ``                ``pos.updatePosition(i, 0); ``            ``} `` ` `            ``// 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 < N - k + 1; j++) ``            ``{ ``                ``sum += (stripSum[i, j + k - 1] - ``                        ``stripSum[i, j - 1]); `` ` `                ``// Update max_sum and position of result ``                ``if` `(sum > max_sum)``                ``{ ``                    ``max_sum = sum; ``                    ``pos.updatePosition(i, j); ``                ``} ``            ``} ``        ``} `` ` `        ``// Print the result matrix ``        ``for` `(``int` `i = 0; i < k; i++) ``        ``{ ``            ``for` `(``int` `j = 0; j < k; j++) ``            ``{ ``                ``Console.Write(mat[i + pos.getXPosition(),``                                  ``j + pos.getYPosition()] + ``" "``); ``            ``} ``            ``Console.WriteLine(); ``        ``} ``    ``} `` ` `    ``// Driver Code``    ``public` `static` `void` `Main(String[] args) ``    ``{ ``        ``N = 5; ``        ``int``[,] mat = {{ 1, 1, 1, 1, 1 }, ``                      ``{ 2, 2, 2, 2, 2 }, ``                        ``{ 3, 8, 6, 7, 3 }, ``                      ``{ 4, 4, 4, 4, 4 }, ``                      ``{ 5, 5, 5, 5, 5 }}; ``        ``int` `k = 3; `` ` `        ``Console.WriteLine(``"Maximum sum 3 x 3 matrix is"``); ``        ``printMaxSumSub(mat, k); ``    ``} ``} `` ` `// This code is contributed by Princi Singh`

Output:

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

This article is contributed by Aditya Goel. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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