N Queens Problem in DSA C# - 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.

bool SolveNQueens(int row, int n, bool[][] board) {
  if (row == n) return true;
  for (int col = 0; col < n; col++) {
    if (IsSafe(row, col, board)) {
      board[row][col] = true;
      if (SolveNQueens(row + 1, n, board)) return true;
      board[row][col] = false;
    }
  }
  return false;
}
    

This code tries to place queens row by row, checking if each position is safe, and uses backtracking to find a solution.

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 the current row.
• How many times: For each row, it tries up to n columns, and for each valid placement, it recurses to the next row.

How Execution Grows With Input

As the board size n grows, the number of ways to place queens grows very fast because each row tries n columns, and this repeats for n rows.

Input Size (n)	Approx. Operations
4	Hundreds of checks
8	Thousands to tens of thousands of checks
12	Millions of checks

Pattern observation: The operations grow very quickly, much faster than just doubling or squaring; it grows exponentially with n.

Final Time Complexity

Time Complexity: O(n!)

This means the time needed grows roughly like the factorial of n, which is very fast growth making large boards much harder to solve.

Common Mistake

[X] Wrong: "The time grows like n squared because there are n rows and n columns."

[OK] Correct: The algorithm tries many combinations by placing queens row by row, and each placement leads to multiple recursive calls, causing factorial growth, not just n squared.

Interview Connect

Understanding this problem's time complexity helps you explain how backtracking explores many possibilities and why some problems become very slow as input grows.

Self-Check

"What if we added memoization to remember safe positions? How would the time complexity change?"

Practice

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

easy

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

B. Place N queens on an NxN board so that all queens attack each other

C. Place N queens on an NxN board in a single row

D. Place N queens on an NxN board in a single column

Solution

1. Step 1: Understand the problem statement

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

2. Step 2: Identify the correct goal

   This means no two queens share the same row, column, or diagonal.

3. Final Answer:

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

4. Quick Check:

   Goal = No two queens attack each other [OK]

Hint: Remember: Queens must not threaten each other [OK]

Common Mistakes:

• Thinking queens can share rows or columns
• Ignoring diagonal attacks
• Confusing with placing queens in one line

2. Which of the following C# code snippets correctly declares a 2D array to represent an 8x8 chessboard for the N Queens Problem?

easy

A. int[,] board = new int[8,8];

B. int[][] board = new int[8];

C. int board[8][8];

D. int board = new int[8,8];

Solution

1. Step 1: Identify correct 2D array declaration in C#

   In C#, a 2D array is declared as int[,] board = new int[8,8];.

2. Step 2: Check other options for syntax errors

   int[][] board = new int[8]; declares a jagged array but misses initialization of inner arrays. int board[8][8]; is invalid syntax in C#. int board = new int[8,8]; tries to assign an array to an int variable.

3. Final Answer:

   int[,] board = new int[8,8]; -> Option A

4. Quick Check:

   Correct 2D array syntax = int[,] board = new int[8,8]; [OK]

Hint: Use int[,] for 2D arrays in C# [OK]

Common Mistakes:

• Using jagged arrays without initializing inner arrays
• Using C++ style array declarations
• Assigning arrays to single int variables

3. Consider the following C# code snippet for checking if a queen can be placed at row 1, column 2 on a 4x4 board:

bool IsSafe(int[,] board, int row, int col) {
    for (int i = 0; i < col; i++) {
        if (board[row, i] == 1) return false;
    }
    for (int i = row, j = col; i >= 0 && j >= 0; i--, j--) {
        if (board[i, j] == 1) return false;
    }
    for (int i = row, j = col; i < 4 && j >= 0; i++, j--) {
        if (board[i, j] == 1) return false;
    }
    return true;
}

int[,] board = new int[4,4];
board[0, 1] = 1;
bool result = IsSafe(board, 1, 2);
Console.WriteLine(result);

What will be the output?

medium

A. true

B. Compile-time error

C. false

D. Runtime exception

Solution

1. Step 1: Analyze the board state and IsSafe checks

   The board has a queen at (0,1). We check if placing at (1,2) is safe. The function checks left row, upper-left diagonal, and lower-left diagonal.

2. Step 2: Check conflicts for position (1,2)

   Left row check: no queen at (1,0) or (1,1). Upper-left diagonal: (0,1) has a queen, so returns false.

3. Final Answer:

   false -> Option C

4. Quick Check:

   Queen at (0,1) blocks (1,2) = false [OK]

Hint: Check diagonals carefully for conflicts [OK]

Common Mistakes:

• Ignoring diagonal checks
• Assuming no conflict if row is clear
• Misreading array indices

4. The following C# code attempts to solve the N Queens Problem but has a bug. What is the bug?

bool SolveNQueens(int[,] board, int col) {
    if (col >= board.GetLength(0)) return true;
    for (int i = 0; i < board.GetLength(0); i++) {
        if (IsSafe(board, i, col)) {
            board[i, col] = 1;
            if (SolveNQueens(board, col + 1)) return true;
            board[i, col] = 0;
        }
    }
    return false;
}

// IsSafe method is correct

medium

A. The function does not backtrack properly

B. No bug; code works correctly

C. The board array is not initialized

D. The base case should check row instead of column

Solution

1. Step 1: Review the recursive backtracking logic

   The function tries to place a queen in each row of the current column, checks safety, places queen, and recurses to next column.

2. Step 2: Verify base case and backtracking

   Base case returns true when all columns are processed. Backtracking resets board[i, col] to 0 if recursion fails. This is correct.

3. Final Answer:

   No bug; code works correctly -> Option B

4. Quick Check:

   Backtracking logic and base case are correct [OK]

Hint: Check base case and backtracking steps carefully [OK]

Common Mistakes:

• Confusing row and column in base case
• Forgetting to reset board on backtrack
• Assuming uninitialized board causes error

5. You want to modify the N Queens solver to count all possible solutions for N=4 instead of stopping at the first. Which change should you make to the recursive method?

hard

A. Add a global boolean flag to stop after first solution

B. Remove the IsSafe check to speed up counting

C. Change the board size to 5x5

D. Replace the return type with int and sum all recursive calls instead of returning on first true

Solution

1. Step 1: Understand current behavior

   The current solver returns true on finding the first solution, stopping further search.

2. Step 2: Modify to count all solutions

   Change return type to int, accumulate counts from recursive calls, and do not return early. This counts all valid placements.

3. Final Answer:

   Replace the return type with int and sum all recursive calls instead of returning on first true -> Option D

4. Quick Check:

   Counting solutions = sum recursive counts [OK]

Hint: Sum all recursive results to count solutions [OK]

Common Mistakes:

• Stopping at first solution
• Changing board size unnecessarily
• Removing safety checks causing wrong counts