Analyzing the Time Complexity

To analyze the time complexity of the N-Queens solution, let's consider the optimized approach using backtracking. In this approach, we place queens on the chessboard one by one, starting from the first row and moving column by column.

In the given C++ code snippet, we have a variable n that represents the number of queens (size of the chessboard). We iterate through each row and column to place the queens on the board.

The time complexity of this approach can be calculated as follows:

• For each row, we have to consider all the columns to place a queen. So, the inner loop runs n times for each row.
• As there are n rows, the total number of iterations for placing the queens is n^2.

Therefore, the time complexity of the N-Queens solution using backtracking is O(n^2).

It's important to note that the time complexity of the N-Queens problem is dependent on the size of the chessboard (number of queens). As the size increases, the time complexity grows exponentially.

Feel free to try different values of n in the code snippet and observe how the number of iterations and time taken change.

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#include <iostream>

using namespace std;

​

int main() {

  // Analyzing the time complexity

  // of the N-Queens solution

​

  int n = 8; // Number of queens

​

  // Place queens on the board

  int count = 0;

  for (int i = 0; i < n; i++) {

    // Place queen on row i, column 0

    // ... code ...

​

    for (int j = 0; j < n; j++) {

      // Place queen on row i, column j

      // ... code ...

    }

  }

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  cout << "Number of possible solutions: " << count << endl;

​

  return 0;

}