37. Sudoku Solver

Problem:

Write a program to solve a Sudoku puzzle by filling the empty cells.

A sudoku solution must satisfy all of the following rules:

Each of the digits 1-9 must occur exactly once in each row.
Each of the digits 1-9 must occur exactly once in each column.
Each of the the digits 1-9 must occur exactly once in each of the 9 3x3 sub-boxes of the grid.

Empty cells are indicated by the character '.'.

A sudoku puzzle...

...and its solution numbers marked in red.

Note:

The given board contain only digits 1-9 and the character '.'.
You may assume that the given Sudoku puzzle will have a single unique solution.
The given board size is always 9x9.

思路：

采用递归的思想。基本的思路是按照从上到下，从左到右的顺序，在所有的空格处依次填入1-9的数字，然后判断填进去之后是否满足数独的要求，如果满足则继续填下一个空格，不满足则在当前空格处填入下一个数字。在填的时候利用了递归的思想，如果checkSudoku(board, i, j, c)，则说明数独有效，再进行solveSudoku(board, i, j+1)操作。

值得注意的是，虽然题目的函数返回值为空，但事实上的输出为board，所以我们在传递参数的时候传递的是board的引用&board，如果不传递引用则board不会被改变。

Solution (C++):

public:
    void solveSudoku(vector<vector<char>>& board) {
        solveSudoku(board, 0, 0);
    }
private:
    bool checkSudoku(vector<vector<char>> &board, int i, int j, char val) {            //检查填入val后数独是否有效
        int row = i - i%3, column = j - j%3;
        for (int x = 0; x < 9; x++) {
            if (board[i][x] == val)  return false;
            if (board[x][j] == val)  return false;
            if (board[row+x/3][column+x%3] == val)  return false;
        }
        return true;
    }
    bool solveSudoku(vector<vector<char>> &board, int i, int j) {
        if (i == 9)  return true;                                //说明数独填完
        if (j == 9)  return solveSudoku(board, i+1, 0);          //说明第i行填完，开始填第i+1行
        if (board[i][j] != '.')  return solveSudoku(board, i, j+1);
        
        for (char c = '1'; c <= '9'; c++) {
            if (checkSudoku(board, i, j, c)) {
                board[i][j] = c;
                if (solveSudoku(board, i, j+1))  return true;
                board[i][j] = '.';
            }
        }
        return false;
    }

性能：

Runtime: 16 ms Memory Usage: 8.9 MB