Fibonacci Using DP in DSA Rust - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Fibonacci DP with memoization

What is the output of this Rust code that calculates the 7th Fibonacci number using memoization?

DSA Rust

fn fib(n: usize, memo: &mut Vec<Option<u64>>) -> u64 {
    if let Some(val) = memo[n] {
        return val;
    }
    if n <= 1 {
        memo[n] = Some(n as u64);
        return n as u64;
    }
    let result = fib(n - 1, memo) + fib(n - 2, memo);
    memo[n] = Some(result);
    result
}

fn main() {
    let n = 7;
    let mut memo = vec![None; n + 1];
    let result = fib(n, &mut memo);
    println!("{}", result);
}

A 21

B 13

C 8

D 34

💡 Hint

Remember that Fibonacci sequence starts with 0, 1, 1, 2, 3, 5, 8, 13...

The 7th Fibonacci number (0-based index) is 13. The code uses memoization to store intermediate results and avoid repeated calculations.

❓ Predict Output

intermediate

Output of bottom-up Fibonacci DP

What is the output of this Rust code that calculates the 10th Fibonacci number using bottom-up dynamic programming?

DSA Rust

fn fib(n: usize) -> u64 {
    let mut dp = vec![0u64; n + 1];
    dp[0] = 0;
    if n > 0 {
        dp[1] = 1;
    }
    for i in 2..=n {
        dp[i] = dp[i - 1] + dp[i - 2];
    }
    dp[n]
}

fn main() {
    let result = fib(10);
    println!("{}", result);
}

A 55

B 89

C 34

D 144

💡 Hint

The 10th Fibonacci number (0-based) is 55.

The bottom-up approach fills the dp array from 0 to n. For n=10, the Fibonacci number is 55.

🔧 Debug

advanced

Identify the error in this Fibonacci DP code

What error does this Rust code produce when calculating the 5th Fibonacci number using memoization?

DSA Rust

fn fib(n: usize, memo: &mut Vec<Option<u64>>) -> u64 {
    if memo[n].is_some() {
        return memo[n].unwrap();
    }
    if n <= 1 {
        return n as u64;
    }
    let result = fib(n - 1, memo) + fib(n - 2, memo);
    memo[n] = Some(result);
    result
}

fn main() {
    let n = 5;
    let mut memo = vec![None; n + 1];
    let result = fib(n, &mut memo);
    println!("{}", result);
}

A Index out of bounds panic

B Compilation error due to missing return

C Infinite recursion causing stack overflow

D Correct output 5

💡 Hint

Check the size of the memo vector compared to the index n.

The memo vector is created with size n (5), but the code accesses memo[5], which is out of bounds (indices 0 to 4). This causes a runtime panic.

🧠 Conceptual

advanced

Why use dynamic programming for Fibonacci?

Which reason best explains why dynamic programming improves Fibonacci number calculation compared to naive recursion?

A It uses less memory by not storing any results

B It converts Fibonacci into a sorting problem

C It uses randomization to guess the next number

D It avoids repeated calculations by storing intermediate results

💡 Hint

Think about what happens when naive recursion calls the same function multiple times with the same input.

Dynamic programming stores results of subproblems to avoid recalculating them, making Fibonacci calculation efficient.

🚀 Application

expert

Calculate sum of first N Fibonacci numbers using DP

Given the Rust function below that calculates Fibonacci numbers using DP, which option correctly prints the sum of the first 8 Fibonacci numbers (0-based)?

DSA Rust

fn fib(n: usize) -> u64 {
    let mut dp = vec![0u64; n + 1];
    dp[0] = 0;
    if n > 0 {
        dp[1] = 1;
    }
    for i in 2..=n {
        dp[i] = dp[i - 1] + dp[i - 2];
    }
    dp[n]
}

fn main() {
    let n = 8;
    // Calculate sum of first n Fibonacci numbers here
    // Fill in the blank
}

A let sum: u64 = (0..n).map(|i| fib(i)).sum(); println!("{}", sum);

B let sum: u64 = (1..=n).map(|i| fib(i)).sum(); println!("{}", sum);

C let sum: u64 = (0..=n).map(|i| fib(i)).sum(); println!("{}", sum);

D let sum: u64 = (1..n).map(|i| fib(i)).sum(); println!("{}", sum);

💡 Hint

Sum includes Fibonacci numbers from index 0 to 8 inclusive.

The range 0..=n includes all indices from 0 to n. Using map and sum calculates the total sum correctly.

Practice

1. What is the main advantage of using dynamic programming (DP) to calculate Fibonacci numbers compared to a simple recursive approach?

easy

A. It avoids repeated calculations by storing intermediate results.

B. It uses less memory by not storing any results.

C. It calculates Fibonacci numbers using only recursion without loops.

D. It always runs in constant time regardless of input size.

Solution

1. Step 1: Understand recursive Fibonacci inefficiency

   Simple recursion recalculates the same Fibonacci values many times, causing exponential time complexity.

2. Step 2: How DP improves calculation

   DP stores intermediate Fibonacci values in a table (memoization or tabulation), so each value is computed once.

3. Final Answer:

   It avoids repeated calculations by storing intermediate results. -> Option A

4. Quick Check:

   DP stores results to avoid repeats [OK]

Hint: DP saves previous results to speed up Fibonacci [OK]

Common Mistakes:

• Thinking DP uses no extra memory
• Believing DP always runs in constant time
• Confusing recursion with DP

2. Which of the following Rust code snippets correctly initializes a vector to store Fibonacci numbers for dynamic programming?

easy

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

B. let mut fib = vec![0; n+1];

C. let mut fib = Vec::new();

D. let fib = [0; n+1];

Solution

1. Step 1: Check vector size for DP storage

   We need a vector of size n+1 to store Fibonacci numbers from 0 to n.

2. Step 2: Verify mutability and initialization

   Vector must be mutable to update values; initialized with zeros for all positions.

3. Final Answer:

   let mut fib = vec![0; n+1]; -> Option B

4. Quick Check:

   Mutable vector with n+1 zeros [OK]

Hint: Use mutable vector with size n+1 for DP array [OK]

Common Mistakes:

• Using immutable vector when updates needed
• Incorrect vector size causing index errors
• Using fixed-size array without mutability

3. What will be the output of the following Rust code that uses bottom-up DP to compute Fibonacci numbers?

fn fib(n: usize) -> usize {
    let mut dp = vec![0; n+1];
    dp[1] = 1;
    for i in 2..=n {
        dp[i] = dp[i-1] + dp[i-2];
    }
    dp[n]
}

fn main() {
    println!("{}", fib(5));
}

medium

A. 13

B. 8

C. 5

D. 3

Solution

1. Step 1: Trace DP array filling for n=5

   dp[0]=0, dp[1]=1; then dp[2]=1, dp[3]=2, dp[4]=3, dp[5]=5.

2. Step 2: Return dp[5]

   Function returns dp[5], which is 5.

3. Final Answer:

   5 -> Option C

4. Quick Check:

   Fibonacci(5) = 5 [OK]

Hint: Bottom-up DP sums previous two values [OK]

Common Mistakes:

• Confusing index starting points
• Off-by-one errors in loop range
• Misreading dp array values

4. Identify the error in this Rust code for Fibonacci using DP and select the correct fix:

fn fib(n: usize) -> usize {
    let mut dp = vec![0; n];
    dp[0] = 0;
    dp[1] = 1;
    for i in 2..=n {
        dp[i] = dp[i-1] + dp[i-2];
    }
    dp[n]
}

medium

A. Change vector size to n+1 and return dp[n]

B. Change vector size to n and return dp[n-1]

C. Initialize dp with zeros and ones alternatively

D. Remove dp[0] assignment since it's zero by default

Solution

1. Step 1: Check vector size and indexing

   Vector size is n (indices 0 to n-1), but loop 2..=n and dp[n] access index n, causing out of bounds panic.

2. Step 2: Fix vector size

   Change to vec![0; n+1] to store values up to index n.

3. Final Answer:

   Change vector size to n+1 and return dp[n] -> Option A

4. Quick Check:

   Size n+1 for indices 0 to n [OK]

Hint: Vector of size n+1 stores fib[0..n] [OK]

Common Mistakes:

• Accessing dp[n] causing panic
• Incorrect vector size for indexing
• Ignoring zero-based indexing

5. You want to compute the 10th Fibonacci number using top-down DP (memoization) in Rust. Which code snippet correctly implements memoization to avoid repeated calculations?

fn fib_memo(n: usize, memo: &mut Vec) -> usize {
    if n <= 1 {
        return n;
    }
    if let Some(val) = memo[n] {
        return val;
    }
    let result = fib_memo(n-1, memo) + fib_memo(n-2, memo);
    memo[n] = Some(result);
    result
}

fn main() {
    let n = 10;
    let mut memo = vec![None; n+1];
    println!("{}", fib_memo(n, &mut memo));
}

hard

A. Incorrect because base case should return 0 for n=1

B. Incorrect because memo vector should store usize, not Option<usize>

C. Incorrect because memo is not passed as mutable reference

D. Correct implementation of top-down DP with memoization

Solution

1. Step 1: Check memo vector type and initialization

   Memo uses Option<usize> to track computed values; initialized with None for all indices.

2. Step 2: Verify memoization logic

   Function returns stored value if present; otherwise computes recursively and stores result.

3. Step 3: Confirm base case correctness

   Returns n for n=0 or 1, which matches Fibonacci definition.

4. Final Answer:

   Correct implementation of top-down DP with memoization -> Option D

5. Quick Check:

   Memo stores Option, uses recursion with caching [OK]

Hint: Use Option to track computed Fibonacci values [OK]

Common Mistakes:

• Using usize vector without Option causes wrong caching
• Wrong base case returns
• Not passing memo as mutable reference