N Queens Problem in DSA Python - Time & Space Complexity

Time Complexity: N Queens Problem

O(n!)

Understanding Time Complexity

We want to understand how the time needed to solve the N Queens problem grows as the board size increases.

Specifically, how does the number of steps change when we try to place queens on larger boards?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

def solve_n_queens(board, row, n):
    if row == n:
        return True
    for col in range(n):
        if is_safe(board, row, col, n):
            board[row][col] = 1
            if solve_n_queens(board, row + 1, n):
                return True
            board[row][col] = 0
    return False

# is_safe checks if placing a queen is safe

This code tries to place queens row by row, checking safe positions and backtracking if needed.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The recursive calls that try placing a queen in each column of a row.
• How many times: For each row, it tries up to n columns, and for each placement, it recurses to the next row.

How Execution Grows With Input

As the board size n grows, the number of ways to try placing queens grows very fast.

Input Size (n)	Approx. Operations
4	~256 tries
8	~16,777,216 tries
12	~8.9 trillion tries

Pattern observation: The tries grow roughly like n to the power of n, which means the work grows very fast as n increases.

Final Time Complexity

Time Complexity: O(n!)

This means the time needed grows very fast, roughly like n factorial, making it very costly for large boards.

Common Mistake

[X] Wrong: "The algorithm only tries n times because it places one queen per row."

[OK] Correct: Actually, it tries many combinations because it explores all columns for each row and backtracks, leading to an exponential number of tries.

Interview Connect

Understanding this complexity helps you explain why backtracking can be slow and when to expect such growth in recursive search problems.

Self-Check

"What if we used a heuristic to skip unsafe columns early? How would the time complexity change?"

Practice

1. What is the main goal of the N Queens Problem?

easy

A. Place N queens on an NxN chessboard so that no two queens attack each other

B. Count the total number of queens on a chessboard

C. Move queens to capture all opponent pieces

D. Arrange queens in a single row on the chessboard

Solution

1. Step 1: Understand the problem statement

   The N Queens Problem requires placing N queens on an NxN board so that no two queens threaten each other.

2. Step 2: Analyze the options

   Only Place N queens on an NxN chessboard so that no two queens attack each other correctly describes placing queens so none attack each other. Other options describe unrelated tasks.

3. Final Answer:

   Place N queens on an NxN chessboard so that no two queens attack each other -> Option A

4. Quick Check:

   Goal = Place queens safely [OK]

Hint: Remember: no two queens can share row, column, or diagonal [OK]

Common Mistakes:

• Thinking queens can be in the same row or column
• Confusing the problem with counting queens
• Assuming queens can attack each other

2. Which of the following Python code snippets correctly initializes an empty chessboard for the N Queens Problem with size n?

easy

A. board = [[0 for _ in range(n)] for _ in range(n)]

B. board = [[0]*n]

C. board = [0]*n

D. board = [[0]*n]*n

Solution

1. Step 1: Understand board initialization

   We need a 2D list with independent rows to represent the chessboard.

2. Step 2: Analyze each option

   board = [[0]*n]*n creates rows referencing the same list, causing bugs. board = [[0 for _ in range(n)] for _ in range(n)] creates independent rows correctly. board = [0]*n creates a 1D list, not 2D. board = [[0]*n] creates a single row only.

3. Final Answer:

   board = [[0 for _ in range(n)] for _ in range(n)] -> Option A

4. Quick Check:

   Independent rows = board = [[0 for _ in range(n)] for _ in range(n)] [OK]

Hint: Use nested list comprehension for independent rows [OK]

Common Mistakes:

• Using [[0]*n]*n causing shared references
• Creating 1D list instead of 2D
• Forgetting to repeat for each row

3. What will be the output of the following Python code snippet for n = 4?

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), 1), range(col, -1, -1)):
        if board[i][j] == 1:
            return False
    return True

board = [[0]*4 for _ in range(4)]
board[1][0] = 1
print(is_safe(board, 1, 1))
print(is_safe(board, 0, 1))

medium

A. True\nFalse

B. False\nFalse

C. False\nTrue

D. True\nTrue

Solution

1. Step 1: Check is_safe for position (1,1)

   Column check: board[1][0] == 1, so returns False.

2. Step 2: Check is_safe for position (0,1)

   Row left: board[0][0] == 0. Upper diagonal: board[0][1] == 0. Lower diagonal: board[1][0] == 1, so returns False.

3. Final Answer:

   False\nFalse -> Option B

4. Quick Check:

   Check columns and diagonals = False, False [OK]

Hint: Check columns and diagonals left of position [OK]

Common Mistakes:

• Ignoring diagonal checks
• Confusing row and column indices
• Assuming safe if no queen in same row only

4. Identify the error in the following code snippet for solving the N Queens Problem:

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

board = [[0]*4 for _ in range(4)]
solve_n_queens(board, 0)
print(board)

medium

A. The base case condition is incorrect

B. The board is not reset properly after backtracking

C. The function returns True too early, missing all solutions

D. The is_safe function is not called

Solution

1. Step 1: Analyze the recursion and return statements

   The function returns True immediately after finding the first solution, so it stops searching further.

2. Step 2: Understand impact on solution finding

   This means it finds only one solution and does not explore all possible solutions.

3. Final Answer:

   The function returns True too early, missing all solutions -> Option C

4. Quick Check:

   Early return stops full search [OK]

Hint: Return True only after exploring all columns [OK]

Common Mistakes:

• Stopping recursion after first solution
• Not resetting board on backtrack
• Incorrect base case check

5. Given the N Queens Problem, which approach best improves efficiency by reducing unnecessary checks during queen placement?

hard

A. Place queens randomly and retry if conflicts occur

B. Check the entire board for conflicts every time before placing a queen

C. Use nested loops to try all possible board configurations without pruning

D. Use backtracking with arrays to track columns and diagonals already occupied

Solution

1. Step 1: Understand efficiency in N Queens

   Checking the entire board repeatedly wastes time; pruning reduces checks.

2. Step 2: Analyze options for pruning

   Using arrays to track columns and diagonals occupied avoids repeated scanning, improving speed.

3. Final Answer:

   Use backtracking with arrays to track columns and diagonals already occupied -> Option D

4. Quick Check:

   Pruning with tracking arrays = Use backtracking with arrays to track columns and diagonals already occupied [OK]

Hint: Track columns and diagonals to skip invalid spots fast [OK]

Common Mistakes:

• Checking full board every time
• Trying random placements without pruning
• Using brute force without backtracking