Backtracking algorithm

Backtracking algorithm is a recursive algorithm used to solve problems by incrementally building a solution and then backtracking and trying different choices if the solution fails to meet the problem’s constraints.

In general, the backtracking algorithm works as follows:

Start with an empty solution or a partial solution.
Examine the solution and test whether it satisfies the problem’s constraints.
If the solution is valid, we have found a solution and can stop.
If the solution is invalid, we modify the solution by adding more elements or changing previous choices, and we repeat step 2.
If there are no more choices, we backtrack to the previous decision point and try a different path.

Figure: queen problem (image by Mehmet Akif CIFCI)

One of the classic examples of backtracking is the N-Queens problem, which involves placing N chess queens on an N x N chessboard so that no two queens threaten each other. The backtracking algorithm can solve this problem by incrementally building a solution, testing if it satisfies the problem’s constraints, and if it doesn’t, backtracking and trying a different choice.

Another example of the backtracking algorithm is the Subset Sum problem, which involves finding whether a subset of a given set of integers has a sum equal to a given target value. The backtracking algorithm can solve this problem by incrementally building a solution, testing if it satisfies the problem’s constraints, and if it doesn’t, backtracking and trying a different choice.

In summary, the backtracking algorithm is a powerful technique for solving problems that involve searching through a large solution space. It works by incrementally building a solution and testing whether it satisfies the problem’s constraints. If it doesn’t, the algorithm backtracks and tries a different path.

Here’s an example implementation of the backtracking algorithm in Python for the N-Queens problem:

def solve_n_queens(n):
    def is_valid(board, row, col):
        # Check if the column is valid
        for i in range(row):
            if board[i] == col or abs(board[i] - col) == row - i:
                return False
        return True

    def backtrack(board, row):
        if row == n:
            # We have found a solution
            solutions.append(board[:])
            return

        for col in range(n):
            if is_valid(board, row, col):
                board[row] = col
                backtrack(board, row + 1)
                board[row] = -1

    solutions = []
    board = [-1] * n
    backtrack(board, 0)
    return solutions

In this implementation, the function solve_n_queens takes an integer n as input, which is the size of the chessboard and the number of queens to place. The function then defines two helper functions: is_valid and backtrack.

The is_valid function checks if a given placement of a queen is valid by checking if the column and diagonal are free from other queens.

The backtrack function recursively places queens on the board, starting from the first row. It tries each column in turn, and if a valid placement is found, it moves on to the next row. If no valid placement is found, the algorithm backtracks to the previous row and tries a different column.

The function then initializes an empty list to store the solutions, creates an empty board, and calls the backtrack function to find all possible solutions to the N-Queens problem. Finally, the function returns the list of solutions.

Note that this is just one example of how to implement the backtracking algorithm in Python, and some many variations and optimizations can be made depending on the specific problem being solved.