Algorithm to Solve Sudoku | Sudoku Solver

Sudoku is a logic-based number puzzle that requires filling a 9×9 grid with numbers 1 through 9 such that each row, column, and 3×3 subgrid contains no repeated numbers. Below is a common algorithm to solve a Sudoku puzzle:

Step-by-Step Algorithm

1. Input:
   • A partially filled 9×9 Sudoku grid.
2. Check Validity:
   • Ensure that the initial grid follows Sudoku rules: no duplicate numbers in any row, column, or 3×3 subgrid.
3. Backtracking Algorithm:
   • Use recursion and trial-and-error to solve the puzzle.
   • Here’s the breakdown:

Sudoku Solver Algorithm

1. Find an Empty Cell:
   • Traverse the grid to find an empty cell (denoted by 0 or another placeholder).
2. Try Numbers 1 to 9:
   • For the empty cell, try placing numbers 1 through 9.
3. Check Validity:
   • After placing a number, check if it is valid:
     • It must not repeat in the same row.
     • It must not repeat in the same column.
     • It must not repeat in the same 3×3 subgrid.
4. Recursive Call:
   • If the number is valid, place it in the cell and recursively try to solve the rest of the grid.
5. Backtrack:
   • If placing a number leads to an invalid state later, remove it (backtrack) and try the next number.
6. Repeat:
   • Continue until the entire grid is filled.
7. Solution Found:
   • If all cells are filled without conflicts, the puzzle is solved.

Pseudocode

def solve_sudoku(grid):
    # Find an empty cell
    row, col = find_empty_cell(grid)
    if row is None:  # No empty cells left
        return True

    for num in range(1, 10):  # Try numbers 1 to 9
        if is_valid(grid, num, row, col):
            grid[row][col] = num  # Place the number

            if solve_sudoku(grid):  # Recursive call
                return True

            grid[row][col] = 0  # Backtrack

    return False

def is_valid(grid, num, row, col):
    # Check row, column, and 3x3 subgrid for validity
    # Implementation details here...
    pass

def find_empty_cell(grid):
    # Find the first empty cell (0)
    for r in range(9):
        for c in range(9):
            if grid[r][c] == 0:
                return r, c
    return None, None

Complexity

• Time Complexity: Depends on the number of empty cells and the constraints. Worst-case: O(9N)O(9^{N}), where NN is the number of empty cells.
• Space Complexity: O(N)O(N) due to recursive calls.

Applications

• Writing Sudoku solvers in Python, Java, or C++.
• Developing mobile apps for Sudoku games.
• Automating Sudoku-solving processes.