# Python3 Program to Inplace rotate square matrix by 90 degrees | Set 1

• Last Updated : 30 Dec, 2021

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  9
Output:
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 13
The given matrix is rotated by 90 degree
in anti-clockwise direction.```

An approach that requires extra space is already discussed here.
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 (First
elements of 1st row, last row, 1st column
and last column) of first cycle in counter
clockwise.
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.

## Python3

 `# Python3 program to rotate a matrix by 90 degrees``N ``=` `4`` ` `# An Inplace function to rotate ``# N x N matrix by 90 degrees in``# anti-clockwise direction``def` `rotateMatrix(mat):``     ` `    ``# Consider all squares one by one``    ``for` `x ``in` `range``(``0``, ``int``(N ``/` `2``)):``         ` `        ``# Consider elements in group   ``        ``# of 4 in current square``        ``for` `y ``in` `range``(x, N``-``x``-``1``):``             ` `            ``# store current cell in temp variable``            ``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``def` `displayMatrix( mat ):``     ` `    ``for` `i ``in` `range``(``0``, N):``         ` `        ``for` `j ``in` `range``(``0``, N):``             ` `            ``print` `(mat[i][j], end ``=` `' '``)``        ``print` `("")``     ` `     ` ` ` ` ` `# Driver Code``mat ``=` `[[``0` `for` `x ``in` `range``(N)] ``for` `y ``in` `range``(N)]`` ` `# Test case 1``mat ``=` `[ [``1``, ``2``, ``3``, ``4` `],``        ``[``5``, ``6``, ``7``, ``8` `],``        ``[``9``, ``10``, ``11``, ``12` `],``        ``[``13``, ``14``, ``15``, ``16` `] ]``         ` `'''``# Test case 2``mat = [ [1, 2, 3 ],``        ``[4, 5, 6 ],``        ``[7, 8, 9 ] ]`` ` `# Test case 3``mat = [ [1, 2 ],``        ``[4, 5 ] ]``         ` `'''`` ` `rotateMatrix(mat)`` ` `# Print rotated matrix``displayMatrix(mat)`` ` ` ` `# This code is contributed by saloni1297`

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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