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Data Structures

## Square Matrix | ( Find sum of all sub-squares of size k x k )

Given an n x n square matrix, find sum of all sub-squares of size k x k where k is smaller than or equal to n.

Examples

**Input:**

n = 5, k = 3

arr[][] = { {1, 1, 1, 1, 1},

{2, 2, 2, 2, 2},

{3, 3, 3, 3, 3},

{4, 4, 4, 4, 4},

{5, 5, 5, 5, 5},

};

**Output:**

18  18  18

27  27  27

36  36  36

**Input:**

n = 3, k = 2

arr[][] = { {1, 2, 3},

{4, 5, 6},

{7, 8, 9},

};

**Output:**

12  16

24  28


A Simple Solution is to one by one pick starting point (leftmost-topmost corner) of all possible sub-squares. Once the starting point is picked, calculate sum of sub-square starting with the picked starting point.

Following is C++ implementation of this idea.

// A simple C++ program to find sum of all subsquares of size k x k
#include <iostream>
using namespace std;

// Size of given matrix
#define n 5

// A simple function to find sum of all sub-squares of size k x k
// in a given square matrix of size n x n
void printSumSimple(int mat[][n], int k)
{
// k must be smaller than or equal to n
if (k > n) return;

// row number of first cell in current sub-square of size k x k
for (int i=0; i<n-k+1; i++)
{
// column of first cell in current sub-square of size k x k
for (int j=0; j<n-k+1; j++)
{
// Calculate and print sum of current sub-square
int sum = 0;
for (int p=i; p<k+i; p++)
for (int q=j; q<k+j; q++)
sum += mat[p][q];
cout << sum << "  ";
}

// Line separator for sub-squares starting with next row
cout << endl;
}
}

// Driver program to test above function
int main()
{
int mat[n][n] = {{1, 1, 1, 1, 1},
{2, 2, 2, 2, 2},
{3, 3, 3, 3, 3},
{4, 4, 4, 4, 4},
{5, 5, 5, 5, 5},
};
int k = 3;
printSumSimple(mat, k);
return 0;
}


Output:

  18  18  18

27  27  27

36  36  36

Time complexity of above solution is O(k2n2). We can solve this problem in O(n2) time using a Tricky Solution . 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.

Following is C++ implementation of this idea.

// An efficient C++ program to find sum of all subsquares of size k x k
#include <iostream>
using namespace std;

// Size of given matrix
#define n 5

// A O(n^2) function to find sum of all sub-squares of size k x k
// in a given square matrix of size n x n
void printSumTricky(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<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;
}
}

// 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];
cout << sum << "  ";

// Calculate sum of remaining squares in current row by
// removing the leftmost strip of previous sub-square and
for (int j=1; j<n-k+1; j++)
{
sum += (stripSum[i][j+k-1] - stripSum[i][j-1]);
cout << sum << "  ";
}

cout << endl;
}
}

// Driver program to test above function
int main()
{
int mat[n][n] = {{1, 1, 1, 1, 1},
{2, 2, 2, 2, 2},
{3, 3, 3, 3, 3},
{4, 4, 4, 4, 4},
{5, 5, 5, 5, 5},
};
int k = 3;
printSumTricky(mat, k);
return 0;
}


Output:

  18  18  18

27  27  27

36  36  36