AI Big-O Analyzer | Python

Recursive Fibonacci Complexity Analysis

Detailed analysis of time and space complexities for the recursive Fibonacci algorithm with insights on different cases and non-obvious parts. Example illustrated for an average-case scenario with n=10 showing exponential time complexity


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Prompt

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

print(recursive_fibonacci(10))

Answer

Complexity Analysis of Recursive Fibonacci Algorithm

Code Inspection

The provided code snippet implements the recursive Fibonacci algorithm to calculate the nth Fibonacci number. It recursively calls the function with parameters n-1 and n-2 until it reaches the base case where n is less than or equal to 1.

Time Complexity

  • Worst Case: The worst-case time complexity of the recursive Fibonacci algorithm is O(2^n). This is because each call branches into two more recursive calls, leading to an exponential increase in function calls.
  • Best Case: The best-case time complexity occurs when n <= 1, resulting in O(1) since a constant number of operations are required to return the result.
  • Average Case: The average-case time complexity is also O(2^n) as it considers scenarios where the function completes after multiple recursive calls.

Space Complexity

  • Worst Case: The worst-case space complexity of the recursive Fibonacci algorithm is O(n) due to the recursive calls stack. The function requires memory space proportional to the input value n for each recursive call.
  • Best Case: The best-case space complexity is O(1) when n <= 1, as only a constant amount of space is needed throughout the execution.
  • Average Case: The average-case space complexity remains O(n) as the number of recursive calls grows linearly with the input size.

Non-obvious Parts

  • In each recursive call, the function computes the Fibonacci number by recursively summing the results of the two preceding Fibonacci numbers.
  • The repetitive calculations in this algorithm lead to a high time complexity, especially for larger values of n.

Average-Case Scenario Illustration

For an input value of n=10 in the average-case scenario:

  • The algorithm will make multiple recursive calls to calculate the 10th Fibonacci number.
  • The expected runtime will be around O(2^10) in terms of time complexity due to the recursive branching.
  • The memory usage will increase linearly with the input size, approximately requiring O(10) stack space in this scenario.

Overall, the recursive Fibonacci algorithm exhibits exponential time complexity and linear space complexity, making it inefficient for large values of n.

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Description

Detailed analysis of time and space complexities for the recursive Fibonacci algorithm with insights on different cases and non-obvious parts. Example illustrated for an average-case scenario with n=10 showing exponential time complexity and linear space complexity.

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