N-Queens 解答

Question

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

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

Each solution contains a distinct board configuration of the n-queens' placement, where 'Q' and '.' both indicate a queen and an empty space respectively.

For example,
There exist two distinct solutions to the 4-queens puzzle:

[
 [".Q..",  // Solution 1
  "...Q",
  "Q...",
  "..Q."],

 ["..Q.",  // Solution 2
  "Q...",
  "...Q",
  ".Q.."]
]

Solution

N-queens问题是NP问题。基本思想其实是用brute-force列举出所有可能然后检查。

这里我们可以用一维数组来表示棋盘。A[rowNum] = colNum代表(rowNum, colNum)处是queen。

用迭代的方法进行层层遍历。

到第x行时，说明前(x - 1)行的queen放置都合法。

那我们就遍历前(x - 1)行的结果，如果当前第x行放置的点：

1. 列的值与之前的重复 --> false

2. （行 ＋ 列）或 （行 － 列）的值和之前的 （行 ＋ 列）或（行 － 列）的值相同 --> false

注意对于这题，因为是用一维数组表示棋盘，所以我们不用恢复迭代时设置的queen的位置值。

（参考：CodeGanker, 小莹子）

public class Solution {
    public List<List<String>> solveNQueens(int n) {
        List<List<String>> result = new ArrayList<List<String>>();
        // Here, we use a 1-D array to represent queen positions
        // queen[1] = 2 means that on (1, 2), there is a queen
        int[] queen = new int[n];
        helper(n, 0, result, queen);
        return result;
    }
    
    private void helper(int n, int rowNum, List<List<String>> result, int[] queen) {
        if (rowNum == n) {
            List<String> oneResult = new ArrayList<String>();
            // Current chess board is valid
            for (int i = 0; i < n; i++) {
                StringBuilder sb = new StringBuilder(n);
                int tmp = n;
                while (tmp > 0) {
                    sb.append(".");
                    tmp--;
                }
                int column = queen[i];
                sb.replace(column, column + 1, "Q");
                oneResult.add(sb.toString());
            }
            result.add(oneResult);
            return;
        }
        
        for (int i = 0; i < n; i++) {
            queen[rowNum] = i;
            if (check(rowNum, queen))
                helper(n, rowNum + 1, result, queen);
        }
    }
    
    private boolean check(int rowNum, int[] queen) {
        int colNum = queen[rowNum];
        for (int i = 0; i < rowNum; i++) {
            if (queen[i] == colNum || (queen[i] + i == rowNum + colNum) || (queen[i] - i == colNum - rowNum))
                return false;
        }
        return true;
    }
}