48. Rotate Image

48. Rotate Image

Difficulty: Medium

Topics: Array, Math, Matrix

You are given an n x n 2D matrix representing an image, rotate the image by 90 degrees (clockwise).

You have to rotate the image in-place, which means you have to modify the input 2D matrix directly. DO NOT allocate another 2D matrix and do the rotation.

Example 1:

• Input: matrix = [[1,2,3],[4,5,6],[7,8,9]]
• Output: [[7,4,1],[8,5,2],[9,6,3]]

Example 2:

• Input: matrix = [[5,1,9,11],[2,4,8,10],[13,3,6,7],[15,14,12,16]]
• Output: [[15,13,2,5],[14,3,4,1],[12,6,8,9],[16,7,10,11]]

Constraints:

• n == matrix.length == matrix[i].length
• 1 <= n <= 20
• -1000 <= matrix[i][j] <= 1000

Solution:

The solution rotates an n x n matrix 90 degrees clockwise in-place without using extra memory. It achieves this by a two-step process: first transposing the matrix (swap rows and columns), then reversing each row. This approach is a standard and efficient solution for this problem.

Approach:

• Transpose the matrix — Convert all matrix[i][j] to matrix[j][i] by swapping elements across the diagonal.
• Reverse each row — After transposition, reversing the order of elements in each row effectively completes the 90° clockwise rotation.
• In-place operations — Both steps modify the original matrix directly, satisfying the problem constraint.

Let's implement this solution in PHP: 48. Rotate Image

<?php
/**
 * @param Integer[][] $matrix
 * @return NULL
 */
function rotate(array &$matrix) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
echo rotate([[1,2,3],[4,5,6],[7,8,9]]) . "\n";                                  // Output: [[7,4,1],[8,5,2],[9,6,3]]
echo rotate([[5,1,9,11],[2,4,8,10],[13,3,6,7],[15,14,12,16]]) . "\n";           // Output: [[15,13,2,5],[14,3,4,1],[12,6,8,9],[16,7,10,11]]
?>

Explanation:

• Transpose brings rows into columns, which is part of the rotation but in the wrong order.
  • In a 3×3 matrix, first row [1, 2, 3] becomes first column [1, 4, 7] after transpose.
• Reversing each row arranges these columns horizontally in the correct order for clockwise rotation.
  • After transpose, first row is [1, 4, 7]. After reversal → [7, 4, 1], which matches final first row.
• For a matrix of size n, the inner loop for transpose starts at i + 1 to avoid double-swapping.
• The reversal step uses array_reverse for simplicity, but a manual swap loop would also work and be just as efficient.

Complexity

• Time Complexity: O(n²) — Each element is accessed once during transpose (n²/2 swaps) and once during row reversal (n × n/2 swaps).
• Space Complexity: O(1) — Only a constant amount of extra space is used.

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