Fibonacci Using DP in DSA Rust - Time & Space Complexity

Time Complexity: Fibonacci Using DP

O(n)

Understanding Time Complexity

We want to understand how fast the Fibonacci number calculation using dynamic programming runs.

How does the time needed change as the input number grows?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

This code calculates the nth Fibonacci number by storing previous results to avoid repeated work.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The for-loop that runs from 2 to n, calculating each Fibonacci number once.
• How many times: Exactly n - 1 times, once for each number from 2 up to n.

How Execution Grows With Input

Each new Fibonacci number requires one addition operation, so the total work grows steadily with n.

Input Size (n)	Approx. Operations
10	About 9 additions
100	About 99 additions
1000	About 999 additions

Pattern observation: The number of operations grows roughly in a straight line as n increases.

Final Time Complexity

Time Complexity: O(n)

This means the time to compute the nth Fibonacci number grows directly in proportion to n.

Common Mistake

[X] Wrong: "Calculating Fibonacci with recursion is always fast because it just calls itself."

[OK] Correct: Simple recursion repeats many calculations, making it much slower than this dynamic programming approach.

Interview Connect

Understanding how dynamic programming improves speed by remembering past results is a key skill that shows you can optimize solutions thoughtfully.

Self-Check

"What if we changed the code to use recursion with memoization instead of a loop? How would the time complexity change?"

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