Fibonacci Using DP in DSA Rust - Build from Scratch

Fibonacci Using DP

📖 Scenario: You are building a simple program to find the Fibonacci number at a given position. Fibonacci numbers start with 0 and 1, and each next number is the sum of the two before it. To make this fast, you will use a method called Dynamic Programming (DP) which remembers past results to avoid repeating work.

🎯 Goal: Build a Rust program that calculates the Fibonacci number at position n using dynamic programming with a vector to store intermediate results.

📋 What You'll Learn

Create a vector called fib with the first two Fibonacci numbers: 0 and 1

Create a variable called n and set it to 10

Use a for loop with variable i from 2 to n (inclusive) to fill the fib vector with Fibonacci numbers

Print the Fibonacci number at position n from the fib vector

💡 Why This Matters

🌍 Real World

Calculating Fibonacci numbers efficiently is useful in computer science problems like algorithm design, financial modeling, and nature simulations.

💼 Career

Understanding dynamic programming and efficient computation is important for software development, especially in roles involving algorithms and performance optimization.

1

Create the initial Fibonacci vector

Create a vector called fib with the first two Fibonacci numbers: 0 and 1.

DSA Rust

# Create a vector called fib with values 0 and 1
# Your code here

Hint

Use vec![0, 1] to create the vector with first two Fibonacci numbers.

2

Set the position variable n

Create a variable called n and set it to 10.

DSA Rust

fn main() {
    let mut fib = vec![0, 1];
    // Create a variable n and set it to 10
    # Your code here
}

Hint

Use let n = 10; to create the variable.

3

Fill the vector with Fibonacci numbers using a loop

Use a for loop with variable i from 2 to n (inclusive) to calculate Fibonacci numbers and push them into the fib vector. Calculate each new number as the sum of fib[i - 1] and fib[i - 2].

DSA Rust

fn main() {
    let mut fib = vec![0, 1];
    let n = 10;
    
    // Use a for loop from 2 to n inclusive
    // Calculate fib[i] = fib[i - 1] + fib[i - 2] and push to fib
    # Your code here
}

Hint

Use for i in 2..=n and inside the loop calculate next = fib[i - 1] + fib[i - 2] then push next to fib.

4

Print the Fibonacci number at position n

Print the Fibonacci number at position n from the fib vector using println!.

DSA Rust

fn main() {
    let mut fib = vec![0, 1];
    let n = 10;
    
    for i in 2..=n {
        let next = fib[i - 1] + fib[i - 2];
        fib.push(next);
    }
    
    // Print the Fibonacci number at position n
    # Your code here
}

Hint

Use println!("{}", fib[n]); to print the Fibonacci number at position n.

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