N Queens Problem in DSA C - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the N Queens Problem?

To place N queens on an N×N chessboard so that no two queens attack each other. This means no two queens share the same row, column, or diagonal.

beginner

Which data structure is commonly used to represent the chessboard in the N Queens Problem?

A 2D array or matrix of size N×N, where each cell can be marked to indicate if a queen is placed there or not.

intermediate

How do we check if placing a queen at a given position is safe?

We check the column, the upper left diagonal, and the upper right diagonal for any existing queens. If none are found, the position is safe.

intermediate

What algorithmic technique is mainly used to solve the N Queens Problem?

Backtracking, which tries placing queens row by row and backtracks when a conflict is found.

intermediate

Why is backtracking efficient for the N Queens Problem compared to brute force?

Because it stops exploring a path as soon as it finds a conflict, avoiding unnecessary checks and reducing the search space.

What does a queen threaten on a chessboard?

A Rows, columns, and diagonals

B Only rows

C Only columns

D Only diagonals

A queen can move any number of squares along a row, column, or diagonal, threatening all those positions.

In the N Queens Problem, how many queens must be placed on the board?

A N

B N squared

C 2N

D N divided by 2

Exactly N queens must be placed on an N×N board.

Which of these is NOT a valid check when placing a queen?

A Checking the same row

B Checking the same column

C Checking diagonals

D Checking the same color squares

The color of squares is irrelevant; only rows, columns, and diagonals matter.

What happens when backtracking finds a conflict placing a queen?

A It stops the entire algorithm

B It removes the last placed queen and tries a new position

C It places another queen anyway

D It restarts from the first row

Backtracking removes the last queen and tries the next possible position.

What is the time complexity of the N Queens Problem using backtracking in the worst case?

A O(2^N)

B O(N^2)

C O(N!)

D O(N)

Backtracking explores permutations of queen placements, leading to factorial time complexity in the worst case.

Explain how backtracking solves the N Queens Problem step-by-step.

Think about trying positions row by row and undoing moves when stuck.

Describe how to check if a queen placement is safe on the board.

Remember queens attack vertically and diagonally upwards only because we place row by row from top.

Practice

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

easy

A. Place N queens on an NxN chessboard so that all queens attack each other

B. Find the shortest path for a queen to move across the board

C. Count the total number of queens on the board

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

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 chessboard so that no two queens attack each other -> Option D

4. Quick Check:

   Goal = Place queens safely [OK]

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

Common Mistakes:

• Thinking queens can attack each other
• Confusing the problem with counting queens
• Ignoring diagonal attacks

2. Which of the following is the correct way to declare a 2D array for the chessboard in C for the N Queens Problem with size N?

easy

A. int board[];

B. int board[N][N];

C. int board[N];

D. int board(N,N);

Solution

1. Step 1: Recall C syntax for 2D arrays

   In C, a 2D array is declared as type name[size1][size2]; where both sizes are specified.

2. Step 2: Match the correct declaration

   int board[N][N]; uses int board[N][N]; which correctly declares a 2D array for the chessboard.

3. Final Answer:

   int board[N][N]; -> Option B

4. Quick Check:

   2D array syntax = int board[N][N]; [OK]

Hint: Use int board[N][N] for 2D arrays in C [OK]

Common Mistakes:

• Using 1D array for 2D board
• Incorrect syntax like int board(N,N);
• Not specifying both dimensions

3. What will be the output of the following code snippet for N=4 when printing the board after placing queens at positions (0,1), (1,3), (2,0), (3,2)?

int board[4][4] = {0};
board[0][1] = 1;
board[1][3] = 1;
board[2][0] = 1;
board[3][2] = 1;
for(int i=0; i<4; i++) {
  for(int j=0; j<4; j++) {
    printf("%d ", board[i][j]);
  }
  printf("\n");
}

medium

A. 0 0 1 0
1 0 0 0
0 1 0 0
0 0 0 1

B. 1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

C. 0 1 0 0
0 0 0 1
1 0 0 0
0 0 1 0

D. 0 0 0 1
0 0 1 0
1 0 0 0
0 1 0 0

Solution

1. Step 1: Place queens on the board as per given positions

   Set board[0][1], board[1][3], board[2][0], board[3][2] to 1; others remain 0.

2. Step 2: Print the board row-wise

   Row 0: 0 1 0 0 Row 1: 0 0 0 1 Row 2: 1 0 0 0 Row 3: 0 0 1 0

3. Final Answer:

   0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 -> Option C

4. Quick Check:

   Positions match output [OK]

Hint: Map queen positions to 1s in 2D array [OK]

Common Mistakes:

• Mixing row and column indices
• Printing without spaces or newlines
• Incorrect initialization of board

4. Identify the error in this C function snippet for checking if a queen can be placed at board[row][col]:

int isSafe(int board[][N], int row, int col) {
  for(int i=0; i=0 && j>=0; i--, j--) {
    if(board[i][j] == 1) return 0;
  }
  for(int i=row, j=col; i>=0 && j

medium

A. The function assumes N is a global constant but it may not be

B. The function incorrectly checks columns beyond the current row

C. The function misses checking the lower diagonals

D. The function does not check the current row for queens

Solution

1. Step 1: Analyze the use of N in function parameter

   The function uses int board[][N] but N is not passed as a parameter or defined locally.

2. Step 2: Understand C array parameter rules

   In C, array sizes must be known or passed explicitly; assuming N as global can cause errors if not defined.

3. Final Answer:

   The function assumes N is a global constant but it may not be -> Option A

4. Quick Check:

   Array size N must be known [OK]

Hint: Always pass array size explicitly or define globally [OK]

Common Mistakes:

• Ignoring array size requirements in C
• Assuming N is always global
• Missing checks for diagonals

5. Given the backtracking approach to solve the N Queens Problem, which of the following best describes the role of recursion in the solution?

hard

A. Recursion tries to place queens row by row and backtracks when no safe position is found

B. Recursion counts the total number of queens placed on the board

C. Recursion moves queens randomly until all are placed

D. Recursion only checks if the board is full without placing queens

Solution

1. Step 1: Understand backtracking in N Queens

   Backtracking uses recursion to place queens one row at a time, checking safety.

2. Step 2: Identify recursion behavior

   If no safe position is found in a row, recursion backtracks to previous row to try other positions.

3. Final Answer:

   Recursion tries to place queens row by row and backtracks when no safe position is found -> Option A

4. Quick Check:

   Backtracking = recursive trial and error [OK]

Hint: Think recursion as trial and error with backtracking [OK]

Common Mistakes:

• Thinking recursion places queens randomly
• Confusing recursion with counting
• Ignoring backtracking step