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36. 有效的数独

题目描述

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。

  1. 数字 1-9 在每一行只能出现一次。
  2. 数字 1-9 在每一列只能出现一次。
  3. 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。

 

提示：

  • board.length == 9
  • board[i].length == 9
  • board[i][j] 是一位数字（1-9）或者 '.'

解法

方法一：一次遍历

有效的数独满足以下三个条件：

  • 每一行中的数字都不重复；
  • 每一列中的数字都不重复；
  • 每一个 \(3 \times 3\) 的宫格中的数字都不重复。

遍历数独，对于每个数字，判断其所在的行、列 以及 \(3 \times 3\) 的宫格是否已经出现过该数字，如果是，则返回 false。遍历结束，返回 true。

时间复杂度 \(O(C)\)，空间复杂度 \(O(C)\)，其中 \(C\) 是数独中的空格数。本题中 \(C=81\)。

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class Solution:
    def isValidSudoku(self, board: List[List[str]]) -> bool:
        row = [[False] * 9 for _ in range(9)]
        col = [[False] * 9 for _ in range(9)]
        sub = [[False] * 9 for _ in range(9)]
        for i in range(9):
            for j in range(9):
                c = board[i][j]
                if c == '.':
                    continue
                num = int(c) - 1
                k = i // 3 * 3 + j // 3
                if row[i][num] or col[j][num] or sub[k][num]:
                    return False
                row[i][num] = True
                col[j][num] = True
                sub[k][num] = True
        return True

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class Solution {
    public boolean isValidSudoku(char[][] board) {
        boolean[][] row = new boolean[9][9];
        boolean[][] col = new boolean[9][9];
        boolean[][] sub = new boolean[9][9];
        for (int i = 0; i < 9; ++i) {
            for (int j = 0; j < 9; ++j) {
                char c = board[i][j];
                if (c == '.') {
                    continue;
                }
                int num = c - '0' - 1;
                int k = i / 3 * 3 + j / 3;
                if (row[i][num] || col[j][num] || sub[k][num]) {
                    return false;
                }
                row[i][num] = true;
                col[j][num] = true;
                sub[k][num] = true;
            }
        }
        return true;
    }
}

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class Solution {
public:
    bool isValidSudoku(vector<vector<char>>& board) {
        vector<vector<bool>> row(9, vector<bool>(9, false));
        vector<vector<bool>> col(9, vector<bool>(9, false));
        vector<vector<bool>> sub(9, vector<bool>(9, false));
        for (int i = 0; i < 9; ++i) {
            for (int j = 0; j < 9; ++j) {
                char c = board[i][j];
                if (c == '.') continue;
                int num = c - '0' - 1;
                int k = i / 3 * 3 + j / 3;
                if (row[i][num] || col[j][num] || sub[k][num]) {
                    return false;
                }
                row[i][num] = true;
                col[j][num] = true;
                sub[k][num] = true;
            }
        }
        return true;
    }
};

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func isValidSudoku(board [][]byte) bool {
    row, col, sub := [9][9]bool{}, [9][9]bool{}, [9][9]bool{}
    for i := 0; i < 9; i++ {
        for j := 0; j < 9; j++ {
            num := board[i][j] - byte('1')
            if num < 0 || num > 9 {
                continue
            }
            k := i/3*3 + j/3
            if row[i][num] || col[j][num] || sub[k][num] {
                return false
            }
            row[i][num] = true
            col[j][num] = true
            sub[k][num] = true
        }
    }
    return true
}

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function isValidSudoku(board: string[][]): boolean {
    const row: boolean[][] = Array.from({ length: 9 }, () =>
        Array.from({ length: 9 }, () => false),
    );
    const col: boolean[][] = Array.from({ length: 9 }, () =>
        Array.from({ length: 9 }, () => false),
    );
    const sub: boolean[][] = Array.from({ length: 9 }, () =>
        Array.from({ length: 9 }, () => false),
    );
    for (let i = 0; i < 9; ++i) {
        for (let j = 0; j < 9; ++j) {
            const num = board[i][j].charCodeAt(0) - '1'.charCodeAt(0);
            if (num < 0 || num > 8) {
                continue;
            }
            const k = Math.floor(i / 3) * 3 + Math.floor(j / 3);
            if (row[i][num] || col[j][num] || sub[k][num]) {
                return false;
            }
            row[i][num] = true;
            col[j][num] = true;
            sub[k][num] = true;
        }
    }
    return true;
}

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impl Solution {
    pub fn is_valid_sudoku(board: Vec<Vec<char>>) -> bool {
        let mut row = vec![vec![false; 9]; 9];
        let mut col = vec![vec![false; 9]; 9];
        let mut sub = vec![vec![false; 9]; 9];

        for i in 0..9 {
            for j in 0..9 {
                let c = board[i][j];
                if c == '.' {
                    continue;
                }
                let num = (c as u8 - b'0' - 1) as usize;
                let k = i / 3 * 3 + j / 3;
                if row[i][num] || col[j][num] || sub[k][num] {
                    return false;
                }
                row[i][num] = true;
                col[j][num] = true;
                sub[k][num] = true;
            }
        }
        true
    }
}

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/**
 * @param {character[][]} board
 * @return {boolean}
 */
var isValidSudoku = function (board) {
    const row = [...Array(9)].map(() => Array(9).fill(false));
    const col = [...Array(9)].map(() => Array(9).fill(false));
    const sub = [...Array(9)].map(() => Array(9).fill(false));
    for (let i = 0; i < 9; ++i) {
        for (let j = 0; j < 9; ++j) {
            const num = board[i][j].charCodeAt() - '1'.charCodeAt();
            if (num < 0 || num > 8) {
                continue;
            }
            const k = Math.floor(i / 3) * 3 + Math.floor(j / 3);
            if (row[i][num] || col[j][num] || sub[k][num]) {
                return false;
            }
            row[i][num] = true;
            col[j][num] = true;
            sub[k][num] = true;
        }
    }
    return true;
};

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public class Solution {
    public bool IsValidSudoku(char[][] board) {
        bool[,] row = new bool[9, 9];
        bool[,] col = new bool[9, 9];
        bool[,] sub = new bool[9, 9];

        for (int i = 0; i < 9; i++) {
            for (int j = 0; j < 9; j++) {
                char c = board[i][j];
                if (c == '.') {
                    continue;
                }
                int num = c - '0' - 1;
                int k = (i / 3) * 3 + (j / 3);
                if (row[i, num] || col[j, num] || sub[k, num]) {
                    return false;
                }
                row[i, num] = true;
                col[j, num] = true;
                sub[k, num] = true;
            }
        }
        return true;
    }
}

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class Solution {
    /**
     * @param String[][] $board
     * @return Boolean
     */
    function isValidSudoku($board) {
        $row = array_fill(0, 9, array_fill(0, 9, false));
        $col = array_fill(0, 9, array_fill(0, 9, false));
        $sub = array_fill(0, 9, array_fill(0, 9, false));

        for ($i = 0; $i < 9; $i++) {
            for ($j = 0; $j < 9; $j++) {
                $c = $board[$i][$j];
                if ($c === '.') {
                    continue;
                }
                $num = intval($c) - 1;
                $k = intdiv($i, 3) * 3 + intdiv($j, 3);
                if ($row[$i][$num] || $col[$j][$num] || $sub[$k][$num]) {
                    return false;
                }
                $row[$i][$num] = true;
                $col[$j][$num] = true;
                $sub[$k][$num] = true;
            }
        }
        return true;
    }
}

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