Implement nth Fibonacci number

Problem

Write a function that returns the n-th Fibonacci number.

The Fibonacci sequence is defined as:

• $F(0)=0$
• $F(1)=1$
• $F(n)=F(n-1)+F(n-2)$ for $n \ge 2$

Requirements

• Input: integer n (assume n >= 0).
• Output: integer F(n).
• Discuss time/space complexity and how you would handle large n.

Follow-ups (if asked)

• Avoid recursion stack overflow.
• Optimize for time (e.g., better than $O(n)$) or for very large values.

What a Strong Answer Covers

• A clear algorithm with the right data structures and enough pseudocode or code-level detail to implement it.
• A correctness argument that explains why the algorithm covers all required cases.
• Time and space complexity, plus at least one alternative approach when relevant.
• Focused tests for normal cases, edge cases, and failure modes.

Follow-up Questions

• How would the approach change if the input were streaming or too large for memory?
• What invariants would you assert in production code?
• Which tests would catch off-by-one, duplicate, or tie-breaking bugs?

nth Fibonacci Number

Return F(n) with F(0)=0, F(1)=1 using fast doubling.

Constraints

• n >= 0

Examples

Example 1

Input: (0,)
Output: 0

Explanation: Base case.

Example 2

Input: (1,)
Output: 1

Explanation: Base case.

Hints

1. Fast doubling computes F(2k) and F(2k+1) from F(k), F(k+1).

Solution

Solution Alignment

The prompt asks for an implementation-level answer. The safest way to present it is to define the state, maintain clear invariants, then walk through complexity and tests.

Problem Write a function that returns the n-th Fibonacci number. The Fibonacci sequence is defined as: - $F(0)=0$ - $F(1)=1$ - $F(n)=F(n-1)+F(n-2)$ for $n \ge 2$ ### Requirements - Input: integer n (assume n >= 0). - Output: integer F(n). - Discuss time/space complexity and how you would handle large n. ### Follow-ups (if asked) - Avoid recursion stack overflow. - Optimize for time (e.g., better than $O(n)$) or for very large values.

Recommended Approach

Start with a brute-force baseline to confirm correctness, then identify the repeated work or ordering property that enables a better data structure such as a hash map, heap, stack, queue, two pointers, prefix sums, BFS/DFS, or dynamic programming. Write the implementation around a small invariant and test that invariant directly.

Correctness

The implementation should maintain an invariant after each loop or operation that directly matches the problem statement. At termination, that invariant implies the returned value has considered every valid candidate exactly once, or has preserved the required data-structure state after every API call.

Complexity

State the baseline complexity and the optimized complexity. For most interview constraints, justify why the optimized approach meets the expected input size.

Edge Cases and Tests

Empty and singleton inputs, duplicates, ties, invalid inputs, boundary values, and tests that exercise the main invariant.