So I am studying an introductory chapter to dynamic programming that suggests a general solution to an optimization problem that occurs straightforwardly from expressing the problem with a reccurence equation.
Compute-Opt(j){
If j =0 then
Return 0
Else
Return max(vj+Compute-Opt(p(j)), Compute-Opt(j −1))
Endif
}
Although, I do get the equation and the recursive tree I do not get how we are able to quickly determine that it takes exponential time.
I can see that computing the max of these two, means that we have to compute 2 subproblems
where $$T(n) = T(p(j)) + T(n-1) \leq 2T(n-1)$$ and I know how to solve that with master theorem but I want to grasp intuitively how to conclude that this is exponential.
To my mind: The worst case is always $vj + Compute -Opt(j) \leq Opt(j-1)$ so we take the long path to the base case (is that correct?)
The tree has then height at most $log(j)$ . How do I continue my thought from there (is it in the right direction)?
This is a common theme in many cs books : refer to fibbonacci simplest version of recursive algorithm (with no memoization) to introduce the reader to the idea that: bad recursive algorithms take exponential time (and I can see that we do in fact compute many times the same subproblems) but I never really grasped the mathematics behind it.
