有效的数独[中等]

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一、题目

请你判断一个9 x 9的数独是否有效。只需要根据以下规则，验证已经填入的数字是否有效即可。

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）  

注意： 一个有效的数独（部分已被填充）不一定是可解的。只需要根据以上规则，验证已经填入的数字是否有效即可。空白格用’.'表示。

示例 1：
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：true

示例 2：
输入：board =

[["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"]]

输出：false
解释：除了第一行的第一个数字从 5 改为 8 以外，空格内其他数字均与 示例1 相同。 但由于位于左上角的 3x3 宫内有两个 8 存在, 因此这个数独是无效的。

:::warning
board.length == 9
board[i].length == 9
board[i][j] 是一位数字（1-9）或者 ‘.’
:::

二、代码

思路：

可以使用哈希表记录每一行、每一列和每一个小九宫格中，每个数字出现的次数。只需要遍历数独一次，在遍历的过程中更新哈希表中的计数，并判断是否满足有效的数独的条件即可。

对于数独的第i行第j列的单元格，其中0≤i,j<9该单元格所在的行下标和列下标分别为i和j，该单元格所在的小九宫格的行数和列数分别为⌊i/3​⌋和⌊j/3⌋，其中0≤⌊i/3⌋,⌊j/3⌋<3。

由于数独中的数字范围是1到9，因此可以使用数组代替哈希表进行计数。

具体做法是，创建二维数组rows和columns分别记录数独的每一行和每一列中的每个数字的出现次数，创建三维数组subboxes记录数独的每一个小九宫格中的每个数字的出现次数，其中rows[i][index]、columns[j]和subboxes[⌊i/3⌋][⌊j/3⌋][index]分别表示数独的第i行第j列的单元格所在的行、列和小九宫格中，数字index+1出现的次数，其中0≤index<9对应的数字index+1满足1≤index+1≤9。

如果board[i][j]填入了数字n则将rows[i][n−1]、columns[j][n−1]和subboxes[⌊i/3⌋][⌊j/3⌋][n−1]各加1。如果更新后的计数大于1，则不符合有效的数独的条件，返回false。

如果遍历结束之后没有出现计数大于1的情况，则符合有效的数独的条件，返回true。

class Solution {
    public boolean isValidSudoku(char[][] board) {
        // X轴数字出现的次数
        int[][] rows = new int[9][9];
        // y轴数字出现的次数
        int[][] columns = new int[9][9];
        // 九宫格内出现的次数
        int[][][] subboxes = new int[3][3][9];
        // 遍历 9*9 表格
        for (int i = 0; i < 9; i++ ) {
            for (int j = 0; j < 9; j++) {
                char element = board[i][j];
                if (element != '.') {
                    int index = element - '1';
                    // x轴中的元素+1
                    rows[i][index]++;
                    // y轴中的元素+1
                    columns[j][index]++;
                    // 九宫格中的元素+1
                    subboxes[i/3][j/3][index]++;
                    //次数大于1直接退出
                    if (rows[i][index] > 1 || columns[j][index] > 1 || subboxes[i/3][j/3][index] > 1) return false;
                }
            }
        }
        return true;
    }
}