Complexity Estimator | Python

Fibonacci Time and Space Complexity Analysis

Analysis of the time and space complexity of a recursive Fibonacci function, exploring insights on inefficiencies and potential optimizations using memoization and iterative approaches.

Prompt

def fibonacci(n):
    if n <= 1:
        return n
    else:
        return fibonacci(n-1) + fibonacci(n-2)

print(fibonacci(5))

Answer

Time Complexity

Worst-case: O(2^n) due to the recursive calls for each Fibonacci number.
Best-case: O(1) when n <= 1.
Average-case: O(2^n) as each function call branches into two additional calls.

Space Complexity

Worst-case: O(n) due to the recursive calls stored in the call stack.
Best-case: O(1) when n <= 1.
Average-case: O(n) because the call stack will grow linearly with the input size.

Insights

The implementation has exponential time complexity, making it inefficient for large inputs.
Recursive calls result in redundant calculations for the same Fibonacci numbers.
Memoization technique can be utilized to optimize by storing previously computed Fibonacci values.
Converting the recursive function into an iterative approach can improve performance.

Potential Optimizations

Implement memoization to store intermediate results and avoid redundant calculations.
Refactor the code to an iterative solution to reduce the overhead associated with recursive calls.

Description

Analysis of the time and space complexity of a recursive Fibonacci function, exploring insights on inefficiencies and potential optimizations using memoization and iterative approaches.