Question

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 $C=81$。

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

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;
  }
}

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;
  }
};

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
}

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;
}

/**
 * @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;
};