Given a square matrix, turn it by 90 degrees in anti-clockwise direction without using any extra space.
Input: 1 2 3 4 5 6 7 8 9 Output: 3 6 9 2 5 8 1 4 7 Rotated the input matrix by 90 degrees 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 Rotated the input matrix by 90 degrees in anti-clockwise direction.
An approach that requires extra space is already discussed in a different article:
Inplace rotate square matrix by 90 degrees | Set 1
This post discusses the same problem with a different approach which is space-optimized.
Approach: The idea is to find the transpose of the matrix and then reverse the columns of the transposed matrix.
Here is an example to show how this works.
- To solve the given problem there are two tasks. 1st is finding the transpose and second is reversing the columns without using extra space
- A transpose of a matrix is when the matrix is flipped over its diagonal, i.e the row index of an element becomes the column index and vice versa. So to find the transpose interchange the elements at position (i, j) with (j, i). Run two loops, the outer loop from 0 to row count and inner loop from 0 to index of the outer loop.
- To reverse the column of the transposed matrix, run two nested loops, the outer loop from 0 to column count and inner loop from 0 to row count/2, interchange elements at (i, j) with (i, row[count-1-j]), where i and j are indices of inner and outer loop respectively.
4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13
Time complexity :O(R*C).
The matrix is traversed twice, so the complexity is O(R*C).
Space complexity :O(1).
The space complexity is constant as no extra space is required.
Implementation: Let’s see a method of Python numpy that can be used to arrive at the particular solution.
numpy implementation [[ 4 8 12 16] [ 3 7 11 15] [ 2 6 10 14] [ 1 5 9 13]]
Note: The above steps/programs do left (or anticlockwise) rotation. Let’s see how to do the right rotation or clockwise rotation. The approach would be similar. Find the transpose of the matrix and then reverse the rows of the transposed matrix.
This is how it is done.
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