Time complexity for recursion in for loop

Question

func(n){
 if(n == 0){
   //print sth
   return;
 } 
 for(i=1;i<=6;i++){
   func(n-1)
 }
}

Please help me understand time complexity of above pseudo code.

templatetypedef · Accepted Answer

Your recursive code has the following properties:

When you call the function with the argument 0, it does a constant amount of work and then returns.
When you call the function with an argument n > 0, it does a constant amount of work and makes six recursive calls on problems of size n - 1.

Mathematically, we can model the total work done as a recurrence relation, letting T(n) denote the amount of work done when you call func(n):

Let's play around with this and see what we find.

T(0) = 1.
T(1) = 6T(0) + 1 = 6 + 1 = 7.
T(2) = 6T(1) + 1 = 6(7) + 1 = 43.
T(3) = 6T(2) + 1 = 6(43) + 1 = 259.
T(4) = 6T(3) + 1 = 6(259) + 1 = 1555.

It might not be obvious just by looking at this, the numbers here are actually given by the formula

T(n) = (6ⁿ⁺¹ - 1) / 5

We can check this briefly as follows:

Asymptotically, this means that T(n) = Θ(6ⁿ), so the overall runtime is Θ(6ⁿ).

So... where did (6ⁿ⁺¹ - 1) / 5 come from? Notice that

More generally, T(n) seems to have the form

6ⁿ + 6^n-1 + ... + 6¹ + 6⁰.

This is the sum of a geometric series, which simplifies to (6ⁿ⁺¹ - 1) / 5.

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