Rec_2
What is the time complexity of the recursive fibonacci function below,

Choose the correct time complexity:

The Master Theorem is provided below. Use it as you see fit:

Question

Rec_2
What is the time complexity of the recursive  fibonacci function below,

Choose the correct time complexity:

The Master Theorem is provided below. Use it as you see fit:

BlackTom AI · Accepted Answer

Reframing the problem, we’re asked to determine the time complexity of the classic naive recursive Fibonacci function. This function computes fib(n) by making two recursive calls to fib(n-1) and fib(n-2), plus constant work per call, so the overall running time grows with the total number of nodes in the recursion tree.

Option: O(n)
- This would imply linear growth. However, the naive Fibonacci recursion does not progress in a strictly linear fashion; each call expands into two subcalls, producing a binary tree of calls rather than a single linear chain. Therefore linear time is not accurate for this recurrence.

Option: O(n2)
- This suggests quadratic time. The actual growth is far faster than quadratic due to the branching factor in the recursion tree; the number of calls increases exponentially with n, not merely as a polynomial in n. Thus O(n^2) underestimates the work by a large margin.

Option: O(log n)
- Logarithmic time would be expected in algorithms that halve the problem size at each step (e.g., binary search). The Fibonacci recurrence does not reduce n by a constant factor in a way that would yield log n behavior; it expands rather than compresses, so this is not a plausible bound.

Option: O(2n)
- This option is commonly read in practice as representing exponential growth with respect to n (often intended to denote something like O(2^n) in the context of recursive Fibonacci). The naive Fibonacci recursion indeed exhibits exponential time due to the doubling effect of the recursive expansion at each level of the recursion tree, which roughly follows a base-2 exponential growth pattern. While the notation O(2n) is nonstandard for exponential growth (the standard is O(2^n)), in many teaching and testing contexts this is treated as referring to exponential time caused by the two recursive calls per step. Consequently, this option captures the intended exponential behavior, even though the exact mathematical notation for exponential growth would be 2^n.

In summary, the incorrect choices mischaracterize the growth by forcing linear, quadratic, or logarithmic interpretations, while the remaining option aligns with the exponential explosion of recursive calls in the naive Fibonacci implementation. The core idea is that the number of calls grows exponentially with n because each level roughly doubles the number of subproblems to solve, leading to an overall exponential time complexity.

Rec_2 What is the time complexity of the recursive fibonacci function below, Choose the correct time complexity: The Master Theorem is provided below. Use it as you see fit:

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