[leetcode]51. N-Queens
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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.."]]
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; } } }};
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