Computational complexity of Fibonacci Sequence

Question

I understand Big-O notation, but I don\'t know how to calculate it for many functions. In particular, I\'ve been trying to figure out the computational complexity of the nai

Answer 1

I agree with pgaur and rickerbh, recursive-fibonacci's complexity is O(2^n).

I came to the same conclusion by a rather simplistic but I believe still valid reasoning.

First, it's all about figuring out how many times recursive fibonacci function ( F() from now on ) gets called when calculating the Nth fibonacci number. If it gets called once per number in the sequence 0 to n, then we have O(n), if it gets called n times for each number, then we get O(n*n), or O(n^2), and so on.

So, when F() is called for a number n, the number of times F() is called for a given number between 0 and n-1 grows as we approach 0.

As a first impression, it seems to me that if we put it in a visual way, drawing a unit per time F() is called for a given number, wet get a sort of pyramid shape (that is, if we center units horizontally). Something like this:

n              *
n-1            **
n-2           ****  
...
2           ***********
1       ******************
0    ***************************

Now, the question is, how fast is the base of this pyramid enlarging as n grows?

Let's take a real case, for instance F(6)

F(6)                 *  <-- only once
F(5)                 *  <-- only once too
F(4)                 ** 
F(3)                ****
F(2)              ********
F(1)          ****************           <-- 16
F(0)  ********************************    <-- 32

We see F(0) gets called 32 times, which is 2^5, which for this sample case is 2^(n-1).

Now, we want to know how many times F(x) gets called at all, and we can see the number of times F(0) is called is only a part of that.

If we mentally move all the *'s from F(6) to F(2) lines into F(1) line, we see that F(1) and F(0) lines are now equal in length. Which means, total times F() gets called when n=6 is 2x32=64=2^6.

Now, in terms of complexity:

O( F(6) ) = O(2^6)
O( F(n) ) = O(2^n)