[leetcode]51. N-Queens

题目链接：https://leetcode.com/problems/n-queens/description/

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.."]
]

 方法一：

In this problem, we can go row by row, and in each position, we need to check if the column, the 45° diagonal and the 135° diagonal had a queen before.

class Solution {
public:
    std::vector<std::vector<std::string> > solveNQueens(int n) {
        std::vector<std::vector<std::string> > res;
        std::vector<std::string> nQueens(n, std::string(n, '.'));
        solveNQueens(res, nQueens, 0, n);
        return res;
    }
private:
    void solveNQueens(std::vector<std::vector<std::string> > &res, std::vector<std::string> &nQueens, int row, int &n) {
        if (row == n) {
            res.push_back(nQueens);
            return;
        }
        for (int col = 0; col != n; ++col)
            if (isValid(nQueens, row, col, n)) {
                nQueens[row][col] = 'Q';
                solveNQueens(res, nQueens, row + 1, n);
                nQueens[row][col] = '.';
            }
    }
    bool isValid(std::vector<std::string> &nQueens, int row, int col, int &n) {
        //check if the column had a queen before.
        for (int i = 0; i != row; ++i)
            if (nQueens[i][col] == 'Q')
                return false;
        //check if the 45° diagonal had a queen before.
        for (int i = row - 1, j = col - 1; i >= 0 && j >= 0; --i, --j)
            if (nQueens[i][j] == 'Q')
                return false;
        //check if the 135° diagonal had a queen before.
        for (int i = row - 1, j = col + 1; i >= 0 && j < n; --i, ++j)
            if (nQueens[i][j] == 'Q')
                return false;
        return true;
    }
};

方法二：

The number of columns is n, the number of 45° diagonals is 2 * n - 1, the number of 135° diagonals is also 2 * n - 1. When reach [row, col], the column No. is col, the 45° diagonal No. is row + col and the 135° diagonal No. is n - 1 + col - row. We can use three arrays to indicate if the column or the diagonal had a queen before, if not, we can put a queen in this position and continue.

NOTE: Please don't use vector<bool> flag to replace vector<int> flag in the following C++ code. In fact, vector<bool> is not a STL container. You should avoid to use it. You can also get the knowledge from here and here.

/**    | | |                / / /             \ \ \
  *    O O O               O O O               O O O
  *    | | |              / / / /             \ \ \ \
  *    O O O               O O O               O O O
  *    | | |              / / / /             \ \ \ \ 
  *    O O O               O O O               O O O
  *    | | |              / / /                 \ \ \
  *   3 columns        5 45° diagonals     5 135° diagonals    (when n is 3)
  */

The number of columns is n, the number of 45° diagonals is 2 * n - 1, the number of 135° diagonals is also 2 * n - 1. When reach [row, col], the column No. is col, the 45° diagonal No. is row + col and the 135° diagonal No. is n - 1 + col - row. We can use three arrays to indicate if the column or the diagonal had a queen before, if not, we can put a queen in this position and continue.

NOTE: Please don't use vector<bool> flag to replace vector<int> flag in the following C++ code. In fact, vector<bool> is not a STL container. You should avoid to use it. You can also get the knowledge from here and here.

/**    | | |                / / /             \ \ \
  *    O O O               O O O               O O O
  *    | | |              / / / /             \ \ \ \
  *    O O O               O O O               O O O
  *    | | |              / / / /             \ \ \ \ 
  *    O O O               O O O               O O O
  *    | | |              / / /                 \ \ \
  *   3 columns        5 45° diagonals     5 135° diagonals    (when n is 3)
  */

class Solution {
public:
    vector<vector<string> > solveNQueens(int n) {
        vector<vector<string>> res;
        vector<string> nQueens(n,string(n,'.'));
        vector<bool> flag_col(n,true),flag_45(2*n-1,true),flag_135(2*n-1,true);
        dfs(res,nQueens,flag_col,flag_45,flag_135,0,n);
        return res;
    }
private:
    void dfs(vector<vector<string>>& res,vector<string>& nQueens,
             vector<bool>& flag_col,vector<bool>& flag_45,vector<bool>& flag_135,int row,int &n)
    {
        if(row==n)
        {
            res.push_back(nQueens);
            return;
        }
        for(int col=0;col!=n;++col)
        {
            if(flag_col[col] && flag_45[row+col] && flag_135[n-1+col-row])
            {
                flag_col[col]=flag_45[row+col]=flag_135[n-1+col-row]=false;
                nQueens[row][col]='Q';
                dfs(res,nQueens,flag_col,flag_45,flag_135,row+1,n);
                nQueens[row][col]='.';
                flag_col[col]=flag_45[row+col]=flag_135[n-1+col-row]=true;
            }
        }
    }
};