Which Algorithm Should You Use to Create a Sudoku Solver?

Creating a Sudoku solver is an interesting challenge that involves various algorithms and techniques. If you are looking to build a Sudoku solver, several algorithms can be employed to achieve this. Here, we will explore one of the most common and effective methods, backtracking, along with other approaches such as constraint propagation, Dancing Links, and heuristic methods.

Backtracking Algorithm

Backtracking is a well-known and robust algorithm widely used for solving Sudoku puzzles. It is a depth-first search algorithm that incrementally builds candidates for solutions. When it determines that a candidate cannot lead to a valid solution, the algorithm abandons it and tries another. This process repeats until a valid solution is found or all possibilities are explored.

How Backtracking Works for Sudoku

Choose an empty cell: Start from the first empty cell in the Sudoku grid. Try numbers: For each number from 1 to 9, check if placing the number in the cell violates Sudoku rules by ensuring the number is not already present in the same row, column, or 3x3 subgrid. Place the number: If the number is valid, place it in the cell and move to the next empty cell. Backtrack if necessary: If placing a number leads to a dead end (no valid numbers for subsequent cells), remove the number, backtrack, and try the next number. Repeat: Continue this cycle until the Sudoku is solved or no numbers are left to try.

Backtracking Algorithm in Python

def is_valid(row, col, num):
    for i in range(9):
        if board[row][i]  num or board[i][col]  num:
            return False
    start_row, start_col  3 * (row // 3), 3 * (col // 3)
    for i in range(3):
        for j in range(3):
            if board[start_row   i][start_col   j]  num:
                return False
    return True
def solve_sudoku(board):
    for row in range(9):
        for col in range(9):
            if board[row][col]  0:  # 0 represents an empty cell
                for num in range(1, 10):
                    if is_valid(row, col, num):
                        board[row][col]  num
                        if solve_sudoku(board):
                            return True
                        board[row][col]  0  # backtrack
                return False  # trigger backtracking
    return True  # solved

Here’s an example board where 0 represents empty cells:

sudoku_board  [
    [5, 3, 0, 0, 7, 0, 0, 0, 0],
    [6, 0, 0, 1, 9, 5, 0, 0, 0],
    [0, 9, 8, 0, 0, 0, 0, 6, 0],
    [8, 0, 0, 0, 6, 0, 0, 0, 3],
    [4, 0, 0, 8, 0, 3, 0, 0, 1],
    [7, 0, 0, 0, 2, 0, 0, 0, 6],
    [0, 6, 0, 0, 0, 0, 2, 8, 0],
    [0, 0, 0, 4, 1, 9, 0, 0, 5],
    [0, 0, 0, 0, 8, 0, 0, 7, 9]
]

solve_sudoku(sudoku_board)
print(sudoku_board)

Constraint Propagation

In addition to backtracking, techniques such as constraint propagation can be employed to enhance the solver's efficiency. Constraint propagation involves reducing the number of possibilities for cells based on the current state of the board, using techniques like:

Naked pairs/triples: Identify pairs or triples of numbers that can only fit in certain cells. Hidden singles: If a number can only fit in one cell of a row, column, or box, it must go there.

Dancing Links (Dancing Links) Algorithm

Another advanced algorithm that can be used is the Dancing Links (Dancing Links) algorithm. This algorithm is more complex to implement but can significantly improve performance for larger and more complex Sudoku puzzles. It is specifically designed for solving exact cover problems, which can be reduced to Sudoku problems.

Heuristic Methods

Lastly, heuristic methods can also be used to optimize the backtracking algorithm. Some common heuristic methods include:

Minimum Remaining Values (MRV): Choose the empty cell with the fewest legal options first. Degree Heuristic: Select the cell that is involved in the largest number of constraints.

For the majority of Sudoku puzzles, a backtracking algorithm with some heuristic optimizations should work well. However, if you are seeking optimal performance, consider exploring constraint propagation or more advanced algorithms like Dancing Links.