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37. Sudoku Solver

Description

Write a program to solve a Sudoku puzzle by filling the empty cells.

A sudoku solution must satisfy all of the following rules:

  1. Each of the digits 1-9 must occur exactly once in each row.
  2. Each of the digits 1-9 must occur exactly once in each column.
  3. Each of the digits 1-9 must occur exactly once in each of the 9 3x3 sub-boxes of the grid.

The '.' character indicates empty cells.

 

Example 1:

Input: 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"]]
Output: [["5","3","4","6","7","8","9","1","2"],["6","7","2","1","9","5","3","4","8"],["1","9","8","3","4","2","5","6","7"],["8","5","9","7","6","1","4","2","3"],["4","2","6","8","5","3","7","9","1"],["7","1","3","9","2","4","8","5","6"],["9","6","1","5","3","7","2","8","4"],["2","8","7","4","1","9","6","3","5"],["3","4","5","2","8","6","1","7","9"]]
Explanation: The input board is shown above and the only valid solution is shown below:

 

Constraints:

  • board.length == 9
  • board[i].length == 9
  • board[i][j] is a digit or '.'.
  • It is guaranteed that the input board has only one solution.

Solutions

Solution 1: Backtracking

We use arrays row, col, and box to record whether a number has appeared in each row, each column, and each 3x3 grid respectively. If the number i has appeared in the rth row, the cth column, and the bth 3x3 grid, then row[r][i], col[c][i], and box[b][i] are all true.

We traverse each empty space in board, enumerate the numbers v that it can fill in. If v has not appeared in the current row, the current column, and the current 3x3 grid, then we can try to fill in the number v and continue to search for the next empty space. If we search to the end and all spaces are filled, it means that a feasible solution has been found.

The time complexity is $O(9^{81})$, and the space complexity is $O(9^2)$.

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class Solution:
    def solveSudoku(self, board: List[List[str]]) -> None:
        def dfs(k):
            nonlocal ok
            if k == len(t):
                ok = True
                return
            i, j = t[k]
            for v in range(9):
                if row[i][v] == col[j][v] == block[i // 3][j // 3][v] == False:
                    row[i][v] = col[j][v] = block[i // 3][j // 3][v] = True
                    board[i][j] = str(v + 1)
                    dfs(k + 1)
                    row[i][v] = col[j][v] = block[i // 3][j // 3][v] = False
                if ok:
                    return

        row = [[False] * 9 for _ in range(9)]
        col = [[False] * 9 for _ in range(9)]
        block = [[[False] * 9 for _ in range(3)] for _ in range(3)]
        t = []
        ok = False
        for i in range(9):
            for j in range(9):
                if board[i][j] == '.':
                    t.append((i, j))
                else:
                    v = int(board[i][j]) - 1
                    row[i][v] = col[j][v] = block[i // 3][j // 3][v] = True
        dfs(0)

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class Solution {
    private boolean ok;
    private char[][] board;
    private List<Integer> t = new ArrayList<>();
    private boolean[][] row = new boolean[9][9];
    private boolean[][] col = new boolean[9][9];
    private boolean[][][] block = new boolean[3][3][9];

    public void solveSudoku(char[][] board) {
        this.board = board;
        for (int i = 0; i < 9; ++i) {
            for (int j = 0; j < 9; ++j) {
                if (board[i][j] == '.') {
                    t.add(i * 9 + j);
                } else {
                    int v = board[i][j] - '1';
                    row[i][v] = col[j][v] = block[i / 3][j / 3][v] = true;