Can you explain the N-Queens algorithm with a code example?

Can you explain the N-Queens algorithm with a code example?

Sure! The N-Queens problem is a classic algorithmic problem that involves placing N queens on an N×N chessboard so that no two queens threaten each other. Here's a simple backtracking solution in Python:

def is_safe(board, row, col):
    for i in range(col):
        if board[row][i] == 1:
            return False
    for i, j in zip(range(row, -1, -1), range(col, -1, -1)):
        if board[i][j] == 1:
            return False
    for i, j in zip(range(row, len(board)), range(col, -1, -1)):
        if board[i][j] == 1:
            return False
    return True


def solve_n_queens_util(board, col):
    if col >= len(board):
        return True
    for i in range(len(board)):
        if is_safe(board, i, col):
            board[i][col] = 1
            if solve_n_queens_util(board, col + 1):
                return True
            board[i][col] = 0
    return False


def solve_n_queens(n):
    board = [[0] * n for _ in range(n)]
    if not solve_n_queens_util(board, 0):
        return "No solution exists"
    return board

# Example usage:
print(solve_n_queens(4))

What does the code do step by step?

The code defines a function to solve the N-Queens problem using backtracking. Here's a step-by-step breakdown:

1. is_safe: This function checks if placing a queen at a given position is safe by checking the row and both diagonals.
2. solve_n_queens_util: This recursive function attempts to place queens column by column. If it successfully places all queens, it returns true.
3. solve_n_queens: This function initializes the board and starts the solving process. If a solution is found, it returns the board configuration; otherwise, it indicates no solution exists.

Can you show me the output for N=4?

Certainly! For N=4, the output of the solve_n_queens(4) function will be a 4x4 board configuration where the queens are placed safely. The output will look like this:

[[0, 1, 0, 0],
 [0, 0, 0, 1],
 [1, 0, 0, 0],
 [0, 0, 1, 0]]

In this representation, '1' indicates a queen's position, and '0' indicates an empty space.