# 48. Rotate Image

### 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](https://en.wikipedia.org/wiki/In-place_algorithm), which means you have to modify the input 2D matrix directly. DO NOT allocate another 2D matrix and do the rotation.

#### Example 1


![The matrix of Example 1 and its rotation](https://cdn.hashnode.com/res/hashnode/image/upload/v1685132902839/fd82dd16-5e51-4720-99e2-6b3aaaff45c5.jpeg align="center")


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

#### Example 2



![The matrix of Example 2 and its rotation](https://cdn.hashnode.com/res/hashnode/image/upload/v1685132938039/b6e39f3f-90bb-4cd2-800f-4b0d826d679c.jpeg align="center")


```plaintext
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 math behind

For any square matrix, the rotation 90 degrees clockwise can be performed in two steps:

1. Transpose the matrix.
    
2. Mirror the matrix vertically.
    

### Code

```cpp
#include <iostream>
#include <vector>
using namespace std;
void rotate(vector<vector<int>>& matrix) {
    const int n = matrix.size();
    // transpose
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
            swap(matrix[i][j], matrix[j][i]);
        }
    }
    // vertical mirror
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n / 2; j++ ) {
            swap(matrix[i][j], matrix[i][n - 1 - j]);
        }
    }
}
void printMatrix(vector<vector<int>>& matrix) {
    cout << "[";
    for (auto& row: matrix) {
        cout << "[";
        for (auto& a: row) {
            cout << a << ",";
        }
        cout << "],";
    }
    cout << "]\n";
}
int main() {
    vector<vector<int>> matrix{{1,2,3},{4,5,6},{7,8,9}};
    rotate(matrix);
    printMatrix(matrix);
    matrix = {{5,1,9,11},{2,4,8,10},{13,3,6,7},{15,14,12,16}};
    rotate(matrix);
    printMatrix(matrix);
}
```

```plaintext
Output:
[[7,4,1,],[8,5,2,],[9,6,3,],]
[[15,13,2,5,],[14,3,4,1,],[12,6,8,9,],[16,7,10,11,],]
```

#### Complexity

* Runtime: `O(n^2)`, where `n = matrix.length`.
    
* Extra space: `O(1)`.
    

### Implementation notes

1. The function [`std::swap`](https://en.cppreference.com/w/cpp/algorithm/swap) can be used to exchange two values.
    
2. When doing the transpose or mirroring, you could visit over one-half of the matrix.
    

---

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