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N-Queens Problem with Backtracking

Certification: Advanced Level Accuracy: 25% Submissions: 4 Points: 15

Write a C++ program to solve the N-Queens problem using backtracking. The N-Queens problem is to place N chess queens on an N×N chessboard so that no two queens attack each other. Queens can attack horizontally, vertically, and diagonally. The program should find and display one valid solution for placing N queens on the board.

Example 1
    
      
  • Input: n = 4
    • 
      
  • Output: 
    . Q . .
    
        . . . Q
    
        Q . . .
    
        . . Q .
    • 
      
  • Explanation: 
    • We need to place 4 queens on a 4×4 chessboard. 
    • Starting from the first row, we place the first queen at position (0,1). 
    • For the second row, we find a safe position at (1,3) that doesn't conflict with the first queen. 
    • For the third row, we place the queen at (2,0) which is safe from previous queens. 
    • Finally, we place the fourth queen at (3,2) completing a valid solution.
    •
Example 2
    
      
  • Input: n = 8
    • 
      
  • Output: 
    Q . . . . . . .
    
        . . . . Q . . .
    
        . . . . . . . Q
    
        . . . . . Q . .
    
        . . Q . . . . .
    
        . . . . . . Q .
    
        . Q . . . . . .
    
        . . . Q . . . .
    • 
      
  • Explanation: 
    • We need to place 8 queens on an 8×8 chessboard. 
    • Using backtracking, we systematically try placing queens in each row. 
    • For each position, we check if it's safe from all previously placed queens. 
    • If a position leads to no solution, we backtrack and try the next position. 
    • The algorithm continues until all 8 queens are successfully placed without conflicts.
    •
Constraints
    
      
  • 1 ≤ n ≤ 15
    • 
      
  • You must use backtracking algorithm
    • 
      
  • Display the board with 'Q' for queens and '.' for empty spaces
    • 
      
  • Find and display only one valid solution
    • 
      
  • Time Complexity: O(N!)
    • 
      
  • Space Complexity: O(N²)
    •
RecursionBacktracking AmazonHCL Technologies
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Hint
Solution

Solution Hints

    
      
  • Use a 2D vector or array to represent the chessboard
    • 
      
  • Create a recursive function that tries to place queens row by row
    • 
      
  • For each row, try placing a queen in each column and check if it's safe
    • 
      
  • A position is safe if no other queen can attack it (same row, column, or diagonal)
    • 
      
  • If placing a queen leads to no solution, backtrack by removing it and trying the next position
    • 
      
  • Use helper functions to check if a position is safe from attacks
    • 
      
  • Once all N queens are placed successfully, display the solution
    •

Steps to solve by this approach:

 Step 1: Initialize an n×n chessboard with all positions marked as empty (0).
 Step 2: Start with the first row (row = 0) and try to place a queen in each column systematically.
 Step 3: For each position (row, col), check if it's safe by verifying no other queen can attack this position from the same column, upper-left diagonal, and upper-right diagonal.
 Step 4: If the position is safe, place the queen (mark as 1) and recursively try to solve for the next row.
 Step 5: If the recursive call returns true, we have found a valid solution and can return true.
 Step 6: If the recursive call returns false or no safe position is found in the current row, backtrack by removing the queen (mark as 0) and try the next column.
 Step 7: If all rows are successfully filled with queens (base case: row >= n), return true indicating a complete solution is found.

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