解数独

编写一个程序，通过填充空格来解决数独问题。

数独的解法需 遵循如下规则：

数字 1-9 在每一行只能出现一次。
数字 1-9 在每一列只能出现一次。
数字 1-9 在每一个以粗实线分隔的 3x3 宫内只能出现一次。（请参考示例图）

数独部分空格内已填入了数字，空白格用 '.' 表示。

示例：

输入：board =
[["5","3",".",".","7",".",".",".","."],
["6",".",".","1","9","5",".",".","."],
[".","9","8",".",".",".",".","6","."],
["8",".",".",".","6",".",".",".","3"],
["4",".",".","8",".","3",".",".","1"],
["7",".",".",".","2",".",".",".","6"],
[".","6",".",".",".",".","2","8","."],
[".",".",".","4","1","9",".",".","5"],
[".",".",".",".","8",".",".","7","9"]]
输出：
[["5","3","4","6","7","8","9","1","2"],
["6","7","2","1","9","5","3","4","8"],
["1","9","8","3","4","2","5","6","7"],
["8","5","9","7","6","1","4","2","3"],
["4","2","6","8","5","3","7","9","1"],
["7","1","3","9","2","4","8","5","6"],
["9","6","1","5","3","7","2","8","4"],
["2","8","7","4","1","9","6","3","5"],
["3","4","5","2","8","6","1","7","9"]]
解释：输入的数独如上图所示，唯一有效的解决方案如下所示：

提示：

board.length == 9
board[i].length == 9
board[i][j] 是一位数字或者 '.'
题目数据保证输入数独仅有一个解

以下程序实现了这一功能：

#include <bits/stdc++.h>
using namespace std;
class Solution
{
public:void solveSudoku(vector<vector<char>> &board){int size = board.size();vector<vector<bool>> rows(size, vector<bool>(10));vector<vector<bool>> cols(size, vector<bool>(10));vector<vector<bool>> boxes(size, vector<bool>(10));for (int i = 0; i < size; i++){for (int j = 0; j < size; j++){if (board[i][j] != '.'){int num = board[i][j] - '0';int idx = i / 3 * 3 + j / 3;rows[i][num] = true;cols[j][num] = true;boxes[idx][num] = true;}}}dfs(board, 0, rows, cols, boxes);}private:bool valid(int num, int row, int col, int idx, vector<vector<bool>> &rows,vector<vector<bool>> &cols, vector<vector<bool>> &boxes){return !rows[row][num] && !cols[col][num] && !boxes[idx][num];}bool dfs(vector<vector<char>> &board, int size, vector<vector<bool>> &rows,vector<vector<bool>> &cols, vector<vector<bool>> &boxes){if (size == 9 * 9){return true;}else{bool ok = false;int row = size / 9;int col = size % 9;int idx = row / 3 * 3 + col / 3;if (board[row][col] == '.'){for (int i = 1; i <= 9; i++){if (valid(i, row, col, idx, rows, cols, boxes)){board[row][col] = i + '0';rows[row][i] = true;cols[col][i] = true;boxes[idx][i] = true;ok = dfs(board, size + 1, rows, cols, boxes);if (!ok){rows[row][i] = false;cols[col][i] = false;boxes[idx][i] = false;board[row][col] = '.';}}}}else{ok = dfs(board, size + 1, rows, cols, boxes);}return ok;}}
};

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