N-Queens

N-Queens

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 

This problem is kind of difficult to solve. The general thought is that for every row, we use a loop to put queen to each column. If the queens do not attack each other, entering next row recursively. This solution refers to the Code Gankers' blog http://blog.youkuaiyun.com/linhuanmars/article/details/20667175

1. keep an array to store the queen column position for every row. For example, for the 4-queens puzzle, we have two solutions. For the first solution, the array should be [1,3,0,2] and for the second solution, the array should be [2,0,3,1].

2. For the queen column position for each row, we need to check whether the position is valid to put the queen. For example, if the array is [1,1,0,2], it is invalid, because two queens appear at the same column. For another example, [2,3,3,1] is also invalid, because for the first row and second row, the two queens share the same diagonal. We need a private method to check whether the column position is valid or not.

public class Solution {
    public List<List<String>> solveNQueens(int n) {
        List<List<String>> ret = new ArrayList<List<String>>();
        
            
        helper(n,0,new int[n],ret);
        return ret;
    }
    
    
    private void helper(int n, int row, int[] columnForRow, List<List<String>> ret) {
        if (n == row) {
            ArrayList<String> list = new ArrayList<String>();;
            for (int i = 0; i < n; i++) {
                StringBuilder sb = new StringBuilder();
                for (int j = 0; j < n; j++) {
                    if (columnForRow[i] == j)
                        sb.append("Q");
                    
                    else
                        sb.append(".");
                }
                list.add(sb.toString());
            }
            
            ret.add(list);
            return;
        } else {
            for (int i = 0; i < n; i++) {
                columnForRow[row] = i;
                
                if (check(row, columnForRow)) {
                    helper(n, row + 1, columnForRow, ret);
                }
            }
        }
    }
    
    private boolean check(int row, int[] columnForRow) {
        for (int i = 0; i < row; i++) {
            if (columnForRow[i] == columnForRow[row] || Math.abs(columnForRow[i] - columnForRow[row]) == row - i)
                return false;
        }
        
        return true;
    }
}