Fibonacci Using DP in DSA Rust - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main idea behind using Dynamic Programming (DP) to calculate Fibonacci numbers?

DP stores the results of smaller subproblems (previous Fibonacci numbers) to avoid repeated calculations, making the process faster and efficient.

beginner

In Fibonacci DP, what does memoization mean?

Memoization means saving the results of Fibonacci calculations in a data structure (like an array) so that when the same Fibonacci number is needed again, it can be retrieved instantly without recalculating.

intermediate

Why is the DP approach better than the simple recursive approach for Fibonacci?

The simple recursive approach recalculates the same Fibonacci numbers many times, causing exponential time complexity. DP calculates each Fibonacci number once and reuses it, resulting in linear time complexity.

beginner

What is the time complexity of Fibonacci calculation using DP?

The time complexity is O(n) because each Fibonacci number from 0 to n is calculated once and stored.

beginner

Show the base cases used in Fibonacci DP and explain why they are important.

Base cases are fib(0) = 0 and fib(1) = 1. They are important because they stop the recursion or iteration and provide starting values to build the rest of the Fibonacci sequence.

What does DP store to optimize Fibonacci calculation?

A Previously calculated Fibonacci numbers

B Random numbers

C Only the last Fibonacci number

D Prime numbers

DP stores previously calculated Fibonacci numbers to avoid repeated calculations.

What is the time complexity of Fibonacci using DP?

A O(n)

B O(2^n)

C O(log n)

D O(n^2)

DP calculates each Fibonacci number once, resulting in O(n) time complexity.

Which of these is a base case for Fibonacci DP?

A fib(3) = 5

B fib(2) = 3

C fib(0) = 0

D fib(1) = 2

fib(0) = 0 is a base case; fib(1) = 1 is another base case.

What problem does DP solve in Fibonacci calculation?

A Sorting numbers

B Searching in array

C Finding maximum

D Repeated calculations

DP avoids repeated calculations by storing results.

Memoization in DP means:

A Calculating twice

B Saving results for reuse

C Deleting old results

D Ignoring base cases

Memoization saves results to reuse them later.

Explain how Dynamic Programming improves the calculation of Fibonacci numbers compared to simple recursion.

Think about how storing answers helps.

Describe the role of base cases in the Fibonacci DP approach and why they are necessary.

Base cases stop infinite loops.

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