Calculating Fibonacci Numbers
In this lesson, we'll look at the classic method to find the nth Fibonacci number and its time complexity using recurrence relations.
Classic recursive implementation of the fibonacci series
Before we dive into what dynamic programming is, let’s have a look at a classic programming problem, the Fibonacci series. You have probably already seen it, but let’s start with a quick refresher. The Fibonacci series is a series of numbers, in which each number is the sum of the preceding two numbers. The first two numbers are 0 and 1. So, it looks like:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
Here is a Python function that returns the Fibonacci number.
This is the Fibonacci golden spiral. The width of each square represents a Fibonacci number.
def fib(num):"""Finds nth fibonacci number:param num: An integer number:return: nth fibonacci number"""if num <= 1:return numreturn fib(num - 1) + fib(num - 2)# Driver code to test above functionif __name__ == '__main__':num = 6print(fib(6))
Time complexity with recurrence relations
To calculate the time complexity of the code, we can solve a recurrence relation,
represents the fib function. and represents the recursive calls to fib(n-1) fib(n-2), whereas the ‘4’ represents the basic operations in the code. Namely, the comparison on line 7 and the two subtractions and one addition on line 9. Lastly, and ...