Question

I’m trying to analyze the performance a recursive program I wrote.

The basic code is

Cost(x)
{
1 + MIN(Cost(x-1), Cost(x-2), Cost(x-3))
}

I want to write a recurrence relation for the number of calls made to Cost(). How would I start this?

Something like T(x) = T(x/2). But I don’t think that’s right

Edit: I can represent this as a tree with a branching factor of 3 for each of the 3 recursive calls to Cost(). So would it would more accurately be T(x) = T(x/3)?

Answer 1

The number of calls made to Cost() would be:

C(x) = 1 + C(x-1) + C(x-2) + C(x-3)

So, for an input x, Cost() was called once plus the amount of times it was called for x-1, x-2, and x-3. This is assuming that your solution does not use memoization. The recurrence relation is not pretty: http://www.wolframalpha.com/input/?i=T(x)+%3D+1+%2B+T(x-1)+%2B+T(x-2)+%2B+T(x-3)

Using memoization, however, your “number of calls” becomes C(x) = x because you only need to evaluate C(i) once for all i between 0 and x. (Might be C(x) = x + 1, depending on your initial conditions)