How can I solve the N-Queens problem using backtracking?

How can I solve the N-Queens problem using backtracking?

To solve the N-Queens problem using backtracking, you can follow these steps: 1. Place a queen in a column of the first row. 2. Move to the next row and try placing a queen in a column that is not attacked by any previously placed queens. 3. If you can place a queen, move to the next row. If not, backtrack to the previous row and move the queen to the next column. 4. Repeat until all queens are placed or all possibilities are exhausted.

What are the constraints I need to consider?

You need to ensure that no two queens threaten each other. This means checking that no two queens share the same row, column, or diagonal.

Can you provide a code example?

Sure! Here's a simple Python implementation:

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

This code defines a function to solve the N-Queens problem using backtracking.

What is the time complexity of this algorithm?

The time complexity of the backtracking algorithm for the N-Queens problem is O(N!), where N is the number of queens. This is because, in the worst case, you may need to explore all possible arrangements of queens.