The complexity of an ineffective division and conquest algorithm

Question

A size instance nis divided into p≥2instances of each size n-a, where it ais small integerand pis constant. The estimated cost of this transaction (i.e. division by instances) is unit, sC(0)=1.

I am trying to find the complexity of this design. It's hard for me to put words in the equation, this is what I think recursion should look like this:

C(n) = (n-a)*C(n/p) + 1

Is it correct?

Answer 1

I think it will be something like this:

C(n) = (p)*C(n-a) + 1

, "p≥2 n-a" . , C(n-a) p . , - p*C(n-a). . C(0) = 1, .

Answer 2

, , - " ", .

a=1 p=2. n-1, 1 . 1 n=1, .. C(1)=1,

C(1)=1
C(2) = 2*C(1) + 1 = 3
C(3) = 2*C(2) + 1 = 7
C(4) = 2*C(3) + 1 = 15

... , C(n) =2^n - 1. a 1, : n/a n.

, a " ", p ""? , . "", .

Answer 3

, C (n) = p⋅C (n-a) + 1 .

,

C (n) = Σ _{i = 0, ..., n / a} p ⁱ = p ^{n / a + 1} - 1

So, C (n) is in O (p ^{n / a} ), which is exponential in n.