Description:

Determine if a 9x9 Sudoku board is valid. Only the filled cells need to be validated according to the following rules:

Each row must contain the digits 1-9 without repetition.
Each column must contain the digits 1-9 without repetition.
Each of the 9 3x3 sub-boxes of the grid must contain the digits 1-9 without repetition.

A partially filled sudoku which is valid. The Sudoku board could be partially filled, where empty cells are filled with the character '.'. Example 1:

Input:
[
[“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”]
]
Output: true

Example 2:

Input:
[
[“8”,”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”]
]
Output: false
Explanation: Same as Example 1, except with the 5 in the top left corner being
modified to 8. Since there are two 8’s in the top left 3x3 sub-box, it is invalid.

Note:

A Sudoku board (partially filled) could be valid but is not necessarily solvable.
Only the filled cells need to be validated according to the mentioned rules.
The given board contain only digits 1-9 and the character '.'.
The given board size is always 9x9.

解法：

此题无需判断数独是否有解，只需要保证3个基本条件，即每一行的元素不重复，每一列的元素不重复，每一个子格的元素不重复。对于重复的判断，借用数据结构Map（因为Map的Key不允许重复）。

整体上来看，判断是否重复的对象为9行，9列，9个子格，故外层需循环9次。而内层循环针对外层循环的内部元素进行循环。对于行，列，子格，其内部都有9个最小单位元素。

对于行而言，第i行对应的元素为

1	board[i][j]

对于列而言，第i列对应的元素为

1	board[j][i]

对于子格元素的对应

观察行号规律：

第0个九宫格：000111222; 第1个九宫格：000111222; 第2个九宫格：000111222;

第3个九宫格：333444555; 第4个九宫格：333444555; 第5个九宫格：333444555;

第6个九宫格：666777888; 第7个九宫格：666777888; 第8个九宫格：666777888;

可见对于每三个九宫格行号增3；对于单个九宫格，每三个格点行号增1。

因此第i个九宫格的第j个格点的行号可表示为i/3*3+j/3

观察列号规律：

第0个九宫格：012012012; 第1个九宫格：345345345; 第2个九宫格：678678678;

第3个九宫格：012012012; 第4个九宫格：345345345; 第5个九宫格：678678678;

第6个九宫格：012012012; 第7个九宫格：345345345; 第8个九宫格：678678678;

可见对于下个九宫格列号增3，循环周期为3；对于单个九宫格，每个格点行号增1，周期也为3。

周期的数学表示就是取模运算mod。

因此第i个九宫格的第j个格点的列号可表示为i%3*3+j%3

故为：

1	board[i/33 + j/3][i%33 + j%3]

具体代码如下：

class Solution {
    public boolean isValidSudoku(char[][] board) {
        for (int i = 0; i < 9; i++) {
            Map<Character, Boolean> judge_row = new HashMap<Character, Boolean>();
            Map<Character, Boolean> judge_col = new HashMap<Character, Boolean>();
            Map<Character, Boolean> judge_sub = new HashMap<Character, Boolean>();
            for (int j = 0; j < 9; j++) {
                if (board[i][j] != '.') {
                    if (judge_row.containsKey(board[i][j])) {
                        return false;
                    }
                    judge_row.put(board[i][j], true);
                }
                if (board[j][i] != '.') {
                    if (judge_col.containsKey(board[j][i])) {
                        return false;
                    }
                    judge_col.put(board[j][i], true);
                }
                if (board[i / 3 * 3 + j / 3][i % 3 * 3 + j % 3] != '.') {
                    if (judge_sub.containsKey(board[i / 3 * 3 + j / 3][i % 3 * 3 + j % 3])) {
                        return false;
                    }
                    judge_sub.put(board[i / 3 * 3 + j / 3][i % 3 * 3 + j % 3], true);
                }
            }
        }
        return true;
    }
}