Rotate Image medium

Problem Statement

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]]

Steps and Explanation

The key to solving this problem efficiently and in-place is to understand the relationship between the indices of elements before and after rotation. We can achieve the rotation in four steps:

1. Transpose: We first transpose the matrix. Transposing a matrix swaps rows and columns. This means element at matrix[i][j] becomes matrix[j][i].

2. Reverse Columns: After transposing, we reverse each row (column in the original matrix). This completes the 90-degree clockwise rotation.

Let's illustrate with Example 1:

Original Matrix:

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

Step 1: Transpose

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

Step 2: Reverse Columns (rows after transpose)

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

Code (C#)

using System;

public class Solution {
    public void Rotate(int[][] matrix) {
        int n = matrix.Length;

        // Transpose the matrix
        for (int i = 0; i < n; i++) {
            for (int j = i; j < n; j++) {
                int temp = matrix[i][j];
                matrix[i][j] = matrix[j][i];
                matrix[j][i] = temp;
            }
        }

        // Reverse each row
        for (int i = 0; i < n; i++) {
            Array.Reverse(matrix[i]);
        }
    }
}

Complexity Analysis

• Time Complexity: O(n^2), where n is the dimension of the square matrix. We iterate through the matrix twice (once for transposition and once for reversal).

• Space Complexity: O(1). The rotation is done in-place; we don't use any extra space proportional to the input size. We only use a constant amount of extra space for temporary variables.