Skip to content

N-Queens II⚓︎

Link

Description⚓︎

The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other.

Given an integer n, return the number of distinct solutions to the n-queens puzzle.

Example 1:

  • Input: n = 4
  • Output: 2
  • Explanation: There are two distinct solutions to the 4-queens puzzle as shown.

Example 2:

  • Input: n = 1
  • Output: 1

Constraints:

  • 1 <= n <= 9

Solution⚓︎

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
class Solution {
private:
    int res;
    vector<bool> dg, udg, col;
    int _n;
    void backtracking(int row) {
        if (row == _n) {
            res++;
            return;
        }
        for (int i = 0; i < _n; i++) {
            if (!col[i] && !dg[row - i + _n] && !udg[row + i]) {
                col[i] = dg[row - i + _n] = udg[row + i] = true;
                backtracking(row + 1);
                col[i] = dg[row - i + _n] = udg[row + i] = false;
            }
        }
    }
public:
    int totalNQueens(int n) {
        _n = n;
        col = vector<bool>(n, false);
        dg = udg = vector<bool>(2 * n, false);
        res = 0;
        backtracking(0);
        return res;
    }
};
  • Time complexity: \(O(N!)\);
  • Space complexity: \(O(N)\).

Another way of writing:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
class Solution {
private:
    vector<bool> col, dg, udg;
    int _n;
    int backtracking(int row) {
        if (row == _n) return 1;
        int res = 0;
        for (int i = 0; i < _n; i++) {
            if (!col[i] && !dg[row - i + _n] && !udg[row + i]) {
                col[i] = dg[row - i + _n] = udg[row + i] = true;
                res += backtracking(row + 1);
                col[i] = dg[row - i + _n] = udg[row + i] = false;
            }
        }
        return res;
    }
public:
    int totalNQueens(int n) {
        col = vector<bool>(n);
        dg = udg = vector<bool>(2 * n);
        _n = n;
        return backtracking(0);
    }
};