N Queens Problem in DSA Rust - 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

How do we check if a queen placement is safe in the N Queens Problem?

We check three conditions: no other queen is in the same column, no other queen is in the same major diagonal, and no other queen is in the same minor diagonal.

intermediate

What technique is commonly used to solve the N Queens Problem?

Backtracking is used. It tries placing queens row by row and backtracks if a conflict is found, exploring all possible safe placements.

intermediate

In Rust, which data structure can efficiently track columns and diagonals for queen placement?

Boolean vectors (Vec<bool>) can track columns and diagonals. For example, one vector for columns, one for major diagonals, and one for minor diagonals.

beginner

Why is the N Queens Problem considered a classic example of backtracking?

Because it systematically explores all possible queen placements, abandoning paths that lead to conflicts, which is the essence of backtracking.

What does a solution to the N Queens Problem represent?

A Positions of N queens on the main diagonal only

B Positions of N queens with no attacks

C Positions of N queens all in one column

D Positions of N queens all in one row

A solution places N queens so none attack each other, meaning no shared rows, columns, or diagonals.

Which of these is NOT a condition to check when placing a queen?

A Same color square

B Same column

C Same row

D Same diagonal

Queens can be on any color square; color does not affect attacks.

What is the time complexity of the naive backtracking solution for N Queens?

A O(N!)

B O(N^2)

C O(2^N)

D O(N)

Backtracking tries permutations of queen placements, roughly O(N!) in worst case.

In Rust, which type is best to represent the chessboard for N Queens?

A Vec<bool>

B String

C HashMap<usize, usize>

D Vec<Vec<char>>

A 2D vector of chars can represent the board visually with 'Q' and '.' characters.

What does backtracking do when it finds a conflict placing a queen?

A Stops the entire algorithm

B Moves to the next row without placing

C Removes the last placed queen and tries a new position

D Places the queen anyway

Backtracking removes the last queen and tries other positions to find a safe placement.

Explain how backtracking helps solve the N Queens Problem step-by-step.

Think of exploring a decision tree and undoing choices when stuck.

Describe how you would track columns and diagonals to quickly check queen safety in Rust.

Imagine marking attacked lines on the board to avoid conflicts.

Practice

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

easy

A. Place queens only on the diagonal cells

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

C. Move queens to the center of the board

D. Count the total number of queens on a 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: Identify the correct goal

   Only Place N queens on an NxN chessboard so that no two queens attack each other correctly states this goal, while others describe unrelated tasks.

3. Final Answer:

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

4. Quick Check:

   N Queens Problem = Place queens safely [OK]

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

Common Mistakes:

• Thinking queens can attack if placed on same diagonal only
• Confusing the problem with counting queens
• Assuming queens can share rows or columns

2. Which Rust syntax correctly declares a 2D vector to represent an NxN chessboard initialized with zeros?

easy

A. let board = vec![vec![0; n]; n];

B. let board = vec![0; n][vec![0; n]];

C. let board = vec![vec![n; 0]; n];

D. let board = vec![vec![0, n]; n];

Solution

1. Step 1: Understand Rust vector initialization

   To create a 2D vector with n rows and n columns filled with 0, use vec![vec![0; n]; n].

2. Step 2: Check each option

   let board = vec![vec![0; n]; n]; matches correct syntax; others have syntax errors or wrong order.

3. Final Answer:

   let board = vec![vec![0; n]; n]; -> Option A

4. Quick Check:

   2D vector init = vec![vec![0; n]; n] [OK]

Hint: Use vec![vec![value; cols]; rows] for 2D vectors [OK]

Common Mistakes:

• Using incorrect order of brackets
• Mixing values and sizes in vec! macro
• Trying to initialize with vec![0; n][vec![0; n]] which is invalid

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

fn is_safe(board: &Vec, row: usize, col: usize, n: usize) -> bool {
    for i in 0..col {
        if board[row][i] == 1 {
            return false;
        }
    }
    let mut i = row as i32;
    let mut j = col as i32;
    while i >= 0 && j >= 0 {
        if board[i as usize][j as usize] == 1 {
            return false;
        }
        i -= 1;
        j -= 1;
    }
    let mut i = row as i32;
    let mut j = col as i32;
    while i < n as i32 && j >= 0 {
        if board[i as usize][j as usize] == 1 {
            return false;
        }
        i += 1;
        j -= 1;
    }
    true
}

fn main() {
    let n = 4;
    let mut board = vec![vec![0; n]; n];
    board[1][0] = 1;
    println!("{}", is_safe(&board, 2, 1, n));
}

medium

A. true

B. Compilation error

C. Runtime panic

D. false

Solution

1. Step 1: Analyze the board and is_safe call

   Board has a queen at (1,0). We check if placing at (2,1) is safe.

2. Step 2: Check row, upper diagonal, and lower diagonal

   Row check: no queen in row 2 before col 1. Upper diagonal: (1,0) has queen, so is_safe returns false.

3. Final Answer:

   false -> Option D

4. Quick Check:

   Queen attacks on upper diagonal = false [OK]

Hint: Check all three directions: row, upper and lower diagonals [OK]

Common Mistakes:

• Ignoring diagonal checks
• Confusing row and column indices
• Assuming safe when queen is on diagonal

4. Identify the bug in this Rust function snippet for placing queens and suggest the fix:

fn solve_n_queens(board: &mut Vec, col: usize, n: usize) -> bool {
    if col >= n {
        return true;
    }
    for i in 0..n {
        if is_safe(board, i, col, n) {
            board[i][col] = 1;
            if solve_n_queens(board, col + 1, n) {
                return true;
            }
            board[i][col] = 0;
        }
    }
    false
}

medium

A. No bug; function is correct

B. Backtracking step resets board[i][col] incorrectly

C. Function should return false instead of true when col >= n

D. Missing base case for col == n

Solution

1. Step 1: Check base case correctness

   Base case returns true when col >= n, meaning all queens placed successfully.

2. Step 2: Verify backtracking logic

   Backtracking resets board[i][col] to 0 if deeper call fails, which is correct.

3. Final Answer:

   No bug; function is correct -> Option A

4. Quick Check:

   Backtracking logic and base case are correct [OK]

Hint: Backtracking resets state after failed recursive call [OK]

Common Mistakes:

• Removing or changing base case incorrectly
• Not resetting board cell during backtracking
• Returning wrong boolean values

5. Given the partial Rust code below, which modification will correctly print all distinct solutions for the N Queens Problem for n = 4?

fn solve_all(board: &mut Vec, col: usize, n: usize, solutions: &mut Vec>) {
    if col == n {
        solutions.push(board.clone());
        return;
    }
    for i in 0..n {
        if is_safe(board, i, col, n) {
            board[i][col] = 1;
            solve_all(board, col + 1, n, solutions);
            board[i][col] = 0;
        }
    }
}

fn main() {
    let n = 4;
    let mut board = vec![vec![0; n]; n];
    let mut solutions = Vec::new();
    solve_all(&mut board, 0, n, &mut solutions);
    // Which code snippet below correctly prints all solutions?
}

hard

A. for sol in solutions { println!("{}", sol); }

B. println!("{:?}", solutions);

C. for sol in &solutions { for row in sol { println!("{:?}", row); } println!("---"); }

D. for sol in &solutions { println!("{}", sol); }

Solution

1. Step 1: Understand solutions data structure

   solutions is a vector of boards, each board is Vec representing queen positions.

2. Step 2: Choose correct printing method

   for sol in &solutions { for row in sol { println!("{:?}", row); } println!("---"); } iterates over each solution and each row, printing rows clearly. Others print entire vectors directly or incorrectly.

3. Final Answer:

   for sol in &solutions { for row in sol { println!("{:?}", row); } println!("---"); } -> Option C

4. Quick Check:

   Iterate solutions and rows for clear output [OK]

Hint: Print each row of each solution separately for clarity [OK]

Common Mistakes:

• Printing entire vector at once causing unreadable output
• Not iterating by reference causing ownership errors
• Using {} instead of {:?} for vectors