Fibonacci Using DP in DSA Rust

Mental Model

We remember past Fibonacci numbers so we don't repeat work. This saves time by building answers step-by-step.

Analogy: Like climbing stairs and writing down how many ways to reach each step so you don't count again.

Fib array: [0, 1, _, _, _, ...]
Index:     0  1  2  3  4  ...

Dry Run Walkthrough

Input: Calculate Fibonacci number for n=5 using DP

Goal: Find the 5th Fibonacci number efficiently by storing previous results

Step 1: Initialize array with base cases fib[0]=0, fib[1]=1

fib: [0, 1, _, _, _, _]

Why: We need starting points to build next numbers

Step 2: Calculate fib[2] = fib[1] + fib[0] = 1 + 0

fib: [0, 1, 1, _, _, _]

Why: Each number is sum of two before it

Step 3: Calculate fib[3] = fib[2] + fib[1] = 1 + 1

fib: [0, 1, 1, 2, _, _]

Why: Build next Fibonacci number using stored values

Step 4: Calculate fib[4] = fib[3] + fib[2] = 2 + 1

fib: [0, 1, 1, 2, 3, _]

Why: Continue building sequence step-by-step

Step 5: Calculate fib[5] = fib[4] + fib[3] = 3 + 2

fib: [0, 1, 1, 2, 3, 5]

Why: Final Fibonacci number for n=5 found

Result:

fib: [0, 1, 1, 2, 3, 5]
Answer: 5

Annotated Code

DSA Rust

fn fibonacci_dp(n: usize) -> usize {
    if n == 0 {
        return 0;
    }
    let mut fib = vec![0; n + 1];
    fib[0] = 0; // base case
    fib[1] = 1; // base case
    for i in 2..=n {
        fib[i] = fib[i - 1] + fib[i - 2]; // build fib[i] from previous two
    }
    fib[n] // return nth Fibonacci number
}

fn main() {
    let n = 5;
    let result = fibonacci_dp(n);
    println!("Fibonacci number for n={} is {}", n, result);
}

if n == 0 { return 0; }

handle base case n=0 immediately

fib[0] = 0; // base case

initialize first Fibonacci number

fib[1] = 1; // base case

initialize second Fibonacci number

for i in 2..=n { fib[i] = fib[i - 1] + fib[i - 2]; }

build each Fibonacci number from the two before it

fib[n]

return the final computed Fibonacci number

Output

Fibonacci number for n=5 is 5

Complexity Analysis

Time: O(n) because we compute each Fibonacci number once from the two before it

Space: O(n) because we store all Fibonacci numbers up to n in an array

vs Alternative: Compared to naive recursion with exponential time, DP is much faster by avoiding repeated calculations

Edge Cases

n = 0

Returns 0 immediately without looping

DSA Rust

if n == 0 { return 0; }

n = 1

Returns 1 from base case fib[1]

DSA Rust

fib[1] = 1; // base case

When to Use This Pattern

When you see a problem asking for Fibonacci or similar sequences, use DP to store previous results and avoid repeated work.

Common Mistakes

Mistake: Using recursion without memoization causing slow exponential time
Fix: Use an array to store computed Fibonacci numbers and build up iteratively

Mistake: Not handling n=0 base case causing index errors
Fix: Add explicit check for n=0 and return 0 immediately

Summary

Calculates Fibonacci numbers efficiently by storing previous results in an array.

Use when you want fast Fibonacci calculation without repeated work.

Remember: each Fibonacci number depends only on the two before it, so store and reuse them.

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