Computational Complexity Of Fibonacci Sequence

Answer :

You model the time function to calculate Fib(n) as sum of time to calculate Fib(n-1) plus the time to calculate Fib(n-2) plus the time to add them together (O(1)). This is assuming that repeated evaluations of the same Fib(n) take the same time - i.e. no memoization is use.

T(n<=1) = O(1)

T(n) = T(n-1) + T(n-2) + O(1)

You solve this recurrence relation (using generating functions, for instance) and you'll end up with the answer.

Alternatively, you can draw the recursion tree, which will have depth n and intuitively figure out that this function is asymptotically O(2n). You can then prove your conjecture by induction.

Base: n = 1 is obvious

Assume T(n-1) = O(2n-1), therefore

T(n) = T(n-1) + T(n-2) + O(1) which is equal to

T(n) = O(2n-1) + O(2n-2) + O(1) = O(2n)

However, as noted in a comment, this is not the tight bound. An interesting fact about this function is that the T(n) is asymptotically the same as the value of Fib(n) since both are defined as

f(n) = f(n-1) + f(n-2).

The leaves of the recursion tree will always return 1. The value of Fib(n) is sum of all values returned by the leaves in the recursion tree which is equal to the count of leaves. Since each leaf will take O(1) to compute, T(n) is equal to Fib(n) x O(1). Consequently, the tight bound for this function is the Fibonacci sequence itself (~θ(1.6n)). You can find out this tight bound by using generating functions as I'd mentioned above.

Just ask yourself how many statements need to execute for F(n) to complete.

For F(1), the answer is 1 (the first part of the conditional).

For F(n), the answer is F(n-1) + F(n-2).

So what function satisfies these rules? Try aⁿ (a > 1):

aⁿ == a^(n-1) + a^(n-2)

Divide through by a^(n-2):

a² == a + 1

Solve for a and you get (1+sqrt(5))/2 = 1.6180339887, otherwise known as the golden ratio.

So it takes exponential time.

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)