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中等
数组
哈希表
矩阵

36. 有效的数独

English Version

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 $C=81$。

Python3

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

Java

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;
    }
}

C++

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;
    }
};

Go

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
}

TypeScript

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;
}

JavaScript

/**
 * @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;
};

PHP

class Solution {
    /**
     * @param string[][] $board
     * @return boolean
     */

    function isValidSudoku($board) {
        $rows = [];
        $columns = [];
        $boxes = [];

        for ($i = 0; $i < 9; $i++) {
            $rows[$i] = [];
            $columns[$i] = [];
            $boxes[$i] = [];
        }

        for ($row = 0; $row < 9; $row++) {
            for ($column = 0; $column < 9; $column++) {
                $cell = $board[$row][$column];

                if ($cell != '.') {
                    if (
                        in_array($cell, $rows[$row]) ||
                        in_array($cell, $columns[$column]) ||
                        in_array($cell, $boxes[floor($row / 3) * 3 + floor($column / 3)])
                    ) {
                        return false;
                    }

                    $rows[$row][] = $cell;
                    $columns[$column][] = $cell;
                    $boxes[floor($row / 3) * 3 + floor($column / 3)][] = $cell;
                }
            }
        }
        return true;
    }
}