Problem 6. The Fibonacci sequence is a seemingly simple series of numbers that holds remarkable significance across various fields. Fibonacci sequence is defined as: F(0) = 0 F(1) = 1 F(n) = F(n − 1) + F(n − 2), n > 1 Consider the following code for generating the nth Fibonacci number. def fibonacci_iter(n): """Calculates the nth Fibonacci number using an iterative approach.""" if n <= 1: return n fib_prev, fib_next = 0, 1 for _ in range(2, n + 1): fib_prev, fib_next = fib_next, fib_prev + fib_next return fib_next What is the time complexity of fibonacci_iter? Problem 7. The Fibonacci sequence is a seemingly simple series of numbers that holds remarkable significance across various fields. Fibonacci sequence is defined as: F(0) = 0 F(1) = 1 F(n) = F(n − 1) + F(n − 2), n > 1 Consider the following code that uses recursion for generating the nth Fibonacci number. def fibonacci_recur(n): """Recursive function to calculate the nth Fibonacci number.""" if n <= 1: return n else: return fibonacci_recur(n-1) + fibonacci_recur(n-2) What is the time complexity of fibonacci_recur?

Let's analyze the time complexity for both the iterative and recursive implementations of the Fibonacci sequence.

### Problem 6: Time Complexity of `fibonacci_iter`

**Function:**
```python
def fibonacci_iter(n):
    """Calculates the nth Fibonacci number using an iterative approach."""
    if n <= 1:
        return n
    fib_prev, fib_next = 0, 1
    for _ in range(2, n + 1):
        fib_prev, fib_next = fib_next, fib_prev + fib_next
    return fib_next
```

**Analysis:**

- The function iterates from `2` to `n` using a loop. Each iteration computes the next Fibonacci number by updating `fib_prev` and `fib_next`.
- There is a single loop that runs `n - 1` times (from `2` to `n`).

**Time Complexity:**

- Since the loop runs linearly with respect to `n`, the time complexity of this function is **Θ(n)**.

### Problem 7: Time Complexity of `fibonacci_recur`

**Function:**
```python
def fibonacci_recur(n):
    """Recursive function to calculate the nth Fibonacci number."""
    if n <= 1:
        return n
    else:
        return fibonacci_recur(n-1) + fibonacci_recur(n-2)
```

**Analysis:**

- This is a direct recursive implementation of the Fibonacci sequence.
- The function calls itself twice for each non-base case (`n > 1`), which leads to a binary tree of recursive calls.

**Time Complexity:**

- Let \( T(n) \) be the time complexity of calculating `fibonacci_recur(n)`.
- The recurrence relation for this algorithm is:
  \[
  T(n) = T(n-1) + T(n-2) + \Theta(1)
  \]
- This is a well-known recurrence relation that resolves to **Θ(2^n)**.

**Explanation:**

- Each Fibonacci number requires the computation of two previous Fibonacci numbers, leading to an exponential number of calls.
- The number of nodes (function calls) in the recursion tree grows exponentially, approximately doubling at each level of the tree.
- Therefore, the time complexity of this recursive approach is **Θ(2^n)**.

### Summary:

1. **`fibonacci_iter(n)` Time Complexity:** Θ(n)
2. **`fibonacci_recur(n)` Time Complexity:** Θ(2^n)

Do you have any questions or need further clarification?

---

**Follow-up Questions:**

1. How would the space complexity of `fibonacci_iter` compare to `fibonacci_recur`?
2. What optimizations could be made to improve the time complexity of the recursive Fibonacci function?
3. What would happen to the time complexity if `fibonacci_iter` was implemented using memoization?
4. How does the time complexity of `fibonacci_recur` affect its performance for large inputs?
5. Can dynamic programming be applied to improve the time complexity of the recursive approach?

**Tip:** For recursive algorithms, consider the number of recursive calls and how the input size is reduced in each call. This helps in determining whether the algorithm has an exponential, polynomial, or linear time complexity.

Explore the time complexity analysis of iterative and recursive methods for computing Fibonacci numbers. Understand the linear time complexity of fibonacci_iter and the exponential time complexity of fibonacci_recur. Learn about recurrence relations and dynamic programming in algorithmic analysis.

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