#   # 48. Rotate Image

## How to rotate a matrix

### 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]]
``````

#### 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

``````#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);
}
``````
``````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` 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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