51. N-Queens（回溯-搜索）

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.

Example:

Input: 4
Output: [
 [".Q..",  // Solution 1
  "...Q",
  "Q...",
  "..Q."],

 ["..Q.",  // Solution 2
  "Q...",
  "...Q",
  ".Q.."]
]
Explanation: There exist two distinct solutions to the 4-queens puzzle as shown above.

深度优先的思想，如果当前位置与其他位置相互攻击，则减掉该节点，然后回溯。

class Solution {
public:
    vector<vector<string>> res;
    void backtrack(vector<string>& board,int level,int n) {
        if(level == n) {
            res.push_back(board);
        }
        for(int i = 0; i < n; ++i) {
            if (valid(board, level, i,n)) {
                board[level][i] = 'Q';
                backtrack(board,level+1,n);
                board[level][i] = '.';
            }
        }
    }
    bool valid(vector<string>& board, int level, int index,int n) {
        // 检查列
        for (int i = level-1; i >= 0; --i) {
            if (board[i][index] == 'Q') {
                return false;
            }
        }
        // 检查45度角 
        for (int li = level -1 , ii = index -1 ;ii >= 0 && li >= 0;--li,--ii) {
            if (board[li][ii]=='Q') {
                return false;
            }
        }
        // 检查135度角
        for (int li = level - 1, ii = index + 1 ; li >=0 && ii < n;--li,++ii) {
            if (board[li][ii] == 'Q') {
                return false;
            }
        }
        return true;
    }
    vector<vector<string>> solveNQueens(int n) {
        vector<string> board = vector<string>(n,string(n,'.'));
        backtrack(board,0,n);
        return res;
    }
};

 

public class Solution {
    private List<List<String>> res = new ArrayList<List<String>>();
    public List<List<String>> solveNQueens(int n) {
        char[][] board = new char[n][n];
        for(int i = 0; i < n; i++)
            for(int j = 0; j < n; j++)
                board[i][j] = '.';
        dfs(board, 0);
        return res;
    }
    
    private void dfs(char[][] board, int col) {
        if(col==board.length){
            res.add(construct(board));
            return;
        }
        for(int i = 0;i<board.length;i++){
            if(validate(board,i,col)){
            board[i][col] = 'Q';
            dfs(board,col+1);
            board[i][col] = '.';
            }
        }
    }
    
    private boolean validate(char[][] board, int x, int y) {
       for(int i = 0;i<board.length;i++)
           for(int j = 0;j<y;j++)
               if(board[i][j]=='Q'&&(x==i||x+j==y+i||x+y==i+j))
                   return false;
        return true;
    }
    
    private List<String> construct(char[][] board) {
        List<String> res = new ArrayList<String>();
        for(int i = 0; i < board.length; i++) {
            String s = new String(board[i]);
            res.add(s);
        }
        return res;
    }
}