Rotate a matrix by 90 degree in clockwise direction without using any extra space

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

Examples: 

Input:
1 2 3 
4 5 6
7 8 9  
Output:
7 4 1 
8 5 2
9 6 3

Input:
1 2
3 4
Output:
3 1
4 2 

Method 1

Approach: The approach is similar to Inplace rotate square matrix by 90 degrees | Set 1. The only thing that is different is to print the elements of the cycle in a clockwise direction i.e. An N x N matrix will have floor(N/2) square cycles. 
For example, a 3 X 3 matrix will have 1 cycle. The cycle is formed by its 1st row, last column, last row, and 1st column. 
For each square cycle, we swap the elements involved with the corresponding cell in the matrix in the clockwise direction. We just need a temporary variable for this.

Explanation:

Let size of row and column be 3. 
During first iteration – 
a[i][j] = Element at first index (leftmost corner top)= 1.
a[j][n-1-i]= Rightmost corner top Element = 3.
a[n-1-i][n-1-j] = Rightmost corner bottom element = 9.
a[n-1-j][i] = Leftmost corner bottom element = 7.
Move these elements in the clockwise direction. 
During second iteration – 
a[i][j] = 2.
a[j][n-1-i] = 6.
a[n-1-i][n-1-j] = 8.
a[n-1-j][i] = 4. 
Similarly, move these elements in the clockwise direction. 

Below is the implementation of the above approach: 

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

Complexity Analysis:

Time Complexity – O(n*n)

Auxiliary Space – O(1)

Method 2:

Approach: The approach is based on the pattern made by indices after rotating the matrix. Consider the following illustration to have a clear insight into it.

Consider a 3 x 3 matrix having indices (i, j) as follows. 

00 01 02 
10 11 12 
20 21 22

After rotating the matrix by 90 degrees in clockwise direction, indices transform into
20 10 00  current_row_index = 0, i = 2, 1, 0 
21 11 01 current_row_index = 1, i = 2, 1, 0 
22 12 02  current_row_index = 2, i = 2, 1, 0

Observation: In any row, for every decreasing row index i, there exists a constant column index j, such that j = current_row_index. 

This pattern can be printed using 2 nested loops.
(This pattern of writing indices is achieved by writing the exact indices of the desired elements of 
where they actually existed in the original array.)  

Below is the implementation of the above approach:

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

Complexity Analysis:

Time Complexity – O(n*n)

Auxiliary Space – O(1)

Method 3:

 Approach: The Approach is to rotate the given matrix two times, first time with respect to the Main diagonal, next time rotate the resultant matrix with respect to the middle column, Consider the following illustration to have a clear insight into it.

 Rotate square matrix 90 degrees in a clockwise direction

Below is the implementation of the above approach:

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

Complexity Analysis:

Time Complexity – O(n*n)

Auxiliary Space – O(1)

Method 4:

Approach: This approach is similar to method 3 the only difference is that in first rotation we rotate about the Secondary Diagonal and after that about the Middle row.

 Rotate square matrix 90 degrees in a clockwise direction

Below is the implementation of the above approach:

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

Complexity Analysis:

Time Complexity – O(n*n)

Auxiliary Space – O(1)

Method 5: 

Approach: We first transpose the given matrix, and then reverse the content of individual rows to get the resultant 90 degree clockwise rotated matrix.

1  2  3                                                1  4  7                                                                 7  4  1

4  5  6        ——Transpose——>    2  5  8         —-Reverse individual rows—->    8  5  2     (Resultant matrix)

7  8  9                                                3  6  9                                                                 9  6  3

Below is the implementation of the above approach:

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

Complexity Analysis:

Time Complexity – O(n*n)

Auxiliary Space – O(1)