GATE-CS-2001 Problem 9

Introduction

This is a problem from the Graduate Aptitude Test in Engineering (GATE) for Computer Science and Information Technology (CS) conducted in the year 2001. The problem tests the understanding of recursion and computational complexity.

Problem Statement

Write a recursive function to compute the Fibonacci sequence. How many additions/subtractions and multiplications/divisions are performed when computing the nth Fibonacci number?

Solution

The Fibonacci sequence is defined as follows:

fib(0) = 0
fib(1) = 1
fib(n) = fib(n-1) + fib(n-2) for n > 1

Here is the recursive function for computing the nth Fibonacci number:

def fib(n):
    if n == 0:
        return 0
    elif n == 1:
        return 1
    else:
        return fib(n-1) + fib(n-2)

To compute the nth Fibonacci number using this function, we need to perform the addition or subtraction of the two previous numbers in the sequence. Each recursive call to the function will result in two additional recursive calls until the base case is reached.

The number of additions/subtractions and multiplications/divisions performed for computing the nth Fibonacci number can be computed as follows:

• The number of additions/subtractions is equal to the nth Fibonacci number minus one.
• The number of multiplications/divisions is equal to the golden ratio to the nth power, where the golden ratio is defined as (1 + sqrt(5)) / 2.

Therefore, the time complexity of this function is O(2^n) and the space complexity is O(n) due to the recursive calls.

Conclusion

This problem tested the understanding of recursion and computational complexity by asking to write a recursive function for computing the Fibonacci sequence and to determine the number of additions/subtractions and multiplications/divisions performed. The recursive function and the analysis of the time and space complexity were presented in this solution.