Rotate Image

 

1. Clarify the problem: Before diving into solving the problem, let's clarify the requirements:

  • The input is a square matrix of integers.
  • We need to rotate the matrix in-place by 90 degrees clockwise.
  • The rotation should be performed in constant space without using any predefined functions.

2. Analyze the problem: To solve this problem, we can perform the rotation in-place by swapping elements layer by layer. We need to identify the layers in the matrix and swap the corresponding elements to achieve the rotation.

3. Design an algorithm: Here is the algorithm to solve the problem:

  1. Initialize start as 0 and end as the length of the matrix minus 1.
  2. While start is less than end:
    • For each element in the current layer:
      • Save the top element.
      • Move the left element to the top.
      • Move the bottom element to the left.
      • Move the right element to the bottom.
      • Move the saved top element to the right.
    • Increment start and decrement end.
  3. The matrix is now rotated by 90 degrees clockwise.

4. Explain your approach: The approach involves identifying the layers in the matrix and performing the rotation by swapping the corresponding elements layer by layer. By iterating through the layers and performing the necessary element swaps, we can achieve the desired rotation in-place.

5. Write clean and readable code:

python
def rotate(matrix):
    n = len(matrix)
    start = 0
    end = n - 1

    while start < end:
        for i in range(start, end):
            offset = i - start
            top = matrix[start][i]
            # Move left element to the top
            matrix[start][i] = matrix[end - offset][start]
            # Move bottom element to the left
            matrix[end - offset][start] = matrix[end][end - offset]
            # Move right element to the bottom
            matrix[end][end - offset] = matrix[i][end]
            # Move saved top element to the right
            matrix[i][end] = top

        start += 1
        end -= 1

6. Test your code: Let's test the code with the following test case:

python
# Example matrix:
# [[1, 2, 3],
#  [4, 5, 6],
#  [7, 8, 9]]

matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
rotate(matrix)
print(matrix)

The expected output is:

lua
[[7, 4, 1],
 [8, 5, 2],
 [9, 6, 3]]

7. Optimize if necessary: The provided solution has a time complexity of O(N^2), where N is the size of the matrix. We need to iterate through each element of the matrix to perform the rotation. Since we are performing the rotation in-place, the space complexity is O(1).

8. Handle error cases: The code assumes that the input is a valid square matrix. It does not explicitly handle cases where the input is not a square matrix or if the matrix is empty. Additional error handling can be implemented depending on the specific requirements.

9. Discuss complexity analysis:

  • Time complexity: The time complexity of the solution is O(N^2), where N is the size of the matrix. We need to iterate through each element of the matrix to perform the rotation.
  • Space complexity: The space complexity is O(1) since we are performing the rotation in-place without using any additional data structures.
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