Sudoku Solver

Problem 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 $3 \times 3$ sub-boxes of the grid.

The '.' character indicates empty cells.

Examples

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 1-9 or '.'.
• It is guaranteed that the input board has only one solution.

Approach

👉 Iterate Through Each Cell: The algorithm iterates through each cell of the Sudoku board

👉 Empty Cell Check: If the current cell is empty (contains ‘.’), the algorithm proceeds to try different numbers (from ‘1’ to ‘9’) in that cell

👉 Number Placement: For each number, the algorithm checks if placing that number in the current cell is valid according to the Sudoku rules (no repetition in the same row, column, or $3 \times 3$ sub-box).

👉 Recursive Exploration: If placing a number is valid, the algorithm sets that number in the cell and recursively explores the next empty cell.

👉 Backtracking: If the recursive exploration leads to an invalid solution, the algorithm backtracks by undoing the choice (resetting the cell to ‘.’) and trying the next number.

👉 Solution Check: The algorithm continues exploring possibilities until a valid solution is found, or it exhaustively searches all possibilities.

Solution Code

Python

class Solution:
  def solveSudoku(self, board: List[List[str]]) -> None:
    def isValid(row: int, col: int, c: str) -> bool:
      for i in range(9):
        if board[i][col] == c or \
           board[row][i] == c or \
           board[3 * (row // 3) + i // 3][3 * (col // 3) + i % 3] == c:
          return False
      return True

    def solve(s: int) -> bool:
      if s == 81:
        return True

      i = s // 9
      j = s % 9

      if board[i][j] != '.':
        return solve(s + 1)

      for c in string.digits[1:]:
        if isValid(i, j, c):
          board[i][j] = c
          if solve(s + 1):
            return True
          board[i][j] = '.'

      return False

    solve(0)

Java

class Solution {
    public void solveSudoku(char[][] board) {
        int[][] f=new int[9][10];
        int[][] row=new int[9][10];
        int[][] col=new int[9][10];
        int cell=0;
        for(int i=0;i<9;i++){
            for(int j=0;j<9;j++){
             cell=((i/3)*3)+j/3;
             if(board[i][j]!='.'){
             f[cell][(int)(board[i][j]-'0')]=1;
             row[i][(int)(board[i][j]-'0')]=1;
             col[j][(int)(board[i][j]-'0')]=1;
            }}
        }
        get(f,board,0,0,row,col);
        return;

    }
    static boolean get(int[][] f,char[][] board, int i, int j, int[][] row, int[][] col ){
        if(i==9)return true;
        if(board[i][j]!='.'){
            if(j==8){
                return get(f,board,i+1,0,row,col);
            }else{
                return get(f,board,i,j+1,row,col);
            }
        }else{
            boolean t=false;
            int cell=((i/3)*3) + j/3;
            for(int k=1;k<=9;k++){
                if(f[cell][k]==0 && row[i][k]==0 && col[j][k]==0){
                    f[cell][k]=1;
                    row[i][k]=1;
                    col[j][k]=1;
                    board[i][j]=(char) (k + '0');
                    if(j==8){
                        t=get(f,board,i+1,0,row,col);
                    }else{
                        t=get(f,board,i,j+1,row,col);
                    }
                    if(t)return true;
                    f[cell][k]=0;
                    row[i][k]=0;
                    col[j][k]=0;
                    board[i][j]='.';
                }
            }
        }
       return false;
    }


}

C++

class Solution {
public:
    void solveSudoku(vector<vector<char>>& board) {
        solve(board);
    }
    bool solve(vector<vector<char>>& board){
        for(int i=0;i<board.size();i++)
        {
            for(int j=0;j<board[0].size();j++)
            {
                if(board[i][j]=='.')
                {
                    for(char c='1';c<='9';c++)
                    {
                        if(isValid(board,i,j,c))
                        {
                            board[i][j]=c;
                            if(solve(board)==true)
                                return true;
                            else
                                board[i][j]='.';
                        }
                    }
                    return false;
                }
            }
        }
        return true;
    }
    bool isValid(vector<vector<char>>& board,int row,int col, char c)
    {
        for(int i=0;i<9;i++)
        {
            if((board[row][i]==c) || (board[i][col]==c) || (board[3*(row/3) + i/3][3*(col/3) + i%3] ==c))
                return false;
        }
        return true;
    }
};

Conclusion

✅ Time Complexity:

✔ The solve method uses a backtracking approach to explore the solution space.

✔ In the worst case, the algorithm tries out all possibilities for each empty cell until a valid solution is found or all possibilities are exhausted.

✔ The time complexity of the backtracking algorithm is typically exponential, but in practice, it tends to be much less than the worst case.

✔ The algorithm explores 9 possibilities for each of the 81 cells, resulting in a time complexity of $O(9^m)$, where m represents the number of empty cells.

✅ Space Complexity:

✔ The algorithm uses a recursive approach to explore the solution space.

✔ The recursive stack depth is proportional to the number of empty cells in the Sudoku board.

✔ The space complexity of the backtracking algorithm is typically linear, but in practice, it tends to be much less.

✔ The algorithm explores 9 possibilities for each of the 81 cells, resulting in a space complexity of $O(m)$, where m represents the number of empty cells.

✅ The algorithm solves the Sudoku puzzle by backtracking through the solution space, exploring possibilities until a valid solution is found or all possibilities are exhausted.

✅ The algorithm uses a recursive approach to explore the solution space, setting numbers in empty cells and checking if the placement is valid according to the Sudoku rules.

✅ The algorithm backtracks when an invalid solution is found, undoing the choice and trying the next number.