Inplace rotate square matrix by 90 degrees

Given a square matrix, turn it by 90 degrees in anti-clockwise direction without using any extra space.

Examples :

`Input:Matrix: 1  2  3 4  5  6 7  8  9Output: 3  6  9  2  5  8  1  4  7 The given matrix is rotated by 90 degree in anti-clockwise direction.Input: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 Output: 4  8 12 16  3  7 11 15  2  6 10 14  1  5  9 13The given matrix is rotated by 90 degree in anti-clockwise direction.`

Approach: To solve the question without any extra space, rotate the array in form of squares, dividing the matrix into squares or cycles. For example,
A 4 X 4 matrix will have 2 cycles. The first cycle is formed by its 1st row, last column, last row and 1st column. The second cycle is formed by 2nd row, second-last column, second-last row and 2nd column. The idea is for each square cycle, swap the elements involved with the corresponding cell in the matrix in anti-clockwise direction i.e. from top to left, left to bottom, bottom to right and from right to top one at a time using nothing but a temporary variable to achieve this.

Demonstration:

`First Cycle (Involves Red Elements) 1  2  3 4  5  6  7 8  9 10 11 12  13 14 15 16 Moving first group of four elements (Firstelements of 1st row, last row, 1st column and last column) of first cycle in counterclockwise.  4  2  3 16 5  6  7 8  9 10 11 12  1 14  15 13  Moving next group of four elements of first cycle in counter clockwise  4  8  3 16  5  6  7  15   2  10 11 12  1  14  9 13 Moving final group of four elements of first cycle in counter clockwise  4  8 12 16  3  6  7 15  2 10 11 14  1  5  9 13 Second Cycle (Involves Blue Elements) 4  8 12 16  3  6 7  15  2  10 11 14  1  5  9 13 Fixing second cycle 4  8 12 16  3  7 11 15  2  6 10 14  1  5  9 13`

Algorithm:

1. There is N/2 squares or cycles in a matrix of side N. Process a square one at a time. Run a loop to traverse the matrix a cycle at a time, i.e loop from 0 to N/2- 1, loop counter is i
2. Consider elements in group of 4 in current square, rotate the 4 elements at a time. So the number of such groups in a cycle is N - 2*i.
3. So run a loop in each cycle from x to N - x- 1, loop counter is y
4. The elements in the current group is (x, y), (y, N-1-x), (N-1-x, N-1-y), (N-1-y, x), now rotate the these 4 elements, i.e (x, y) <- (y, N-1-x), (y, N-1-x)<- (N-1-x, N-1-y), (N-1-x, N-1-y)<- (N-1-y, x), (N-1-y, x)<- (x, y)
5. Print the matrix.
// C++ program to rotate a matrix
// by 90 degrees
#include <bits/stdc++.h>
#define N 4
using namespace std;
void displayMatrix(
int mat[N][N]);
// An Inplace function to
// rotate a N x N matrix
// by 90 degrees in
// anti-clockwise direction
void rotateMatrix(int mat[][N])
{
// Consider all squares one by one
for (int x = 0; x < N / 2; x++) {
// Consider elements in group
// of 4 in current square
for (int y = x; y < N - x - 1; y++) {
// Store current cell in
// temp variable
int temp = mat[x][y];
// Move values from right to top
mat[x][y] = mat[y][N - 1 - x];
// Move values from bottom to right
mat[y][N - 1 - x]
= mat[N - 1 - x][N - 1 - y];
// Move values from left to bottom
mat[N - 1 - x][N - 1 - y]
= mat[N - 1 - y][x];
// Assign temp to left
mat[N - 1 - y][x] = temp;
}
}
}
// Function to print the matrix
void displayMatrix(int mat[N][N])
{
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++)
printf("%2d ", mat[i][j]);
printf("\n");
}
printf("\n");
}
/* Driver program to test above functions */
int main()
{
// Test Case 1
int mat[N][N] = {
{ 1, 2, 3, 4 },
{ 5, 6, 7, 8 },
{ 9, 10, 11, 12 },
{ 13, 14, 15, 16 }
};
// Tese Case 2
/* int mat[N][N] = {
{1, 2, 3},
{4, 5, 6},
{7, 8, 9}
};
*/
// Tese Case 3
/*int mat[N][N] = {
{1, 2},
{4, 5}
};*/
// displayMatrix(mat);
rotateMatrix(mat);
// Print rotated matrix
displayMatrix(mat);
return 0;
}

Output :
` 4  8 12 16  3  7 11 15  2  6 10 14  1  5  9 13 `

Complexity Analysis:

• Time Complexity: O(n*n), where n is side of array.
A single traversal of the matrix is needed.
• Space Complexity: O(1).
As a constant space is needed

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