I knew the recursive and iterative solutions cold, but O(log n) is a different beast.
Start by clarifying the problem constraints (e.g., n can be large, so O(n) is too slow). Then explain that matrix exponentiation or fast doubling can compute F(n) in O(log n) by using the identities F(2k) = F(k)*(2*F(k+1) - F(k)) and F(2k+1) = F(k+1)^2 + F(k)^2. Finally, discuss implementation details like handling large numbers and recursion depth.
Pro tip: Mention that fast doubling is generally more efficient than matrix exponentiation because it uses fewer multiplications and avoids 2x2 matrix operations, and be prepared to discuss how to handle very large n (e.g., using memoization or iterative implementation to avoid stack overflow).
Ask about the expected range of n, whether the result should be modulo something, and if recursion is acceptable. This shows you think about edge cases and practical limits.
Describe either matrix exponentiation or fast doubling, highlighting the logarithmic time complexity. Derive the key identities or matrix power to demonstrate understanding.
Outline the steps of the chosen method, e.g., for fast doubling: recursively compute F(k) and F(k+1) for k = n/2, then use formulas to get F(n). Mention base cases.
State that time complexity is O(log n) and space complexity is O(log n) for recursion or O(1) for iterative. Compare with naive O(n) and discuss when each method is preferable.
Mention handling large numbers (e.g., using arbitrary precision or modulo), avoiding recursion depth issues, and potential optimizations like iterative fast doubling.
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