Master theorem where f(n) = cn^k

Question

I'm wondering about a specific case for the master theorem that arises when f(n) = xn ^k and n^(log _b a) = n ^k where x is an integer greater than 1. For this case f(n) is larger than n^(log _b a) however it is not polynomially larger than it so case 3 can not arise.

For a case like this I would assume you use case 2 as the big-O of them both are the same but that doesn't seem to fit with the equation that I can find. It seems possible that I'm making a mistake in taking f(n) directly out of the original recursive relation rather than it's big-O as that seems to make sense to me yet I can't find any clarification on this or any examples where the space of f(n) in the equation is not already it's own big-O.

Edit: When I say "the equation that I can find" what I mean is that assumption doesn't fit with the master theorem as I can work it out. As I have it the master theorem for case 2 which I am talking about looks like f(n) = Θ(n^(log _b a)) . I think the important bit really is whether out of an equation ending in + xn ^k I pull out f(n) = xn ^k or f(n) = n ^k . Apologies for the poor wording.

Answer 1

I think the important bit really is whether out of an equation ending in + xnk I pull out f(n) = xnk or f(n) = nk.

Normally you should take f(n) = x * n ^k . Because master theorem defines T(n) to be in the form aT(n/b) + f(n) . But in your example, it doesn't really matter.

Growth of f(n) and x * f(n) are the same, if x is a positive constant. In the case of f(n) = xn ^k , They are both Θ(n ^k ). (Or you could say they are both Θ(x * n ^k ). This is the same set as Θ(n ^k ).)

Since f(n) = Θ(n ^{log _b a} ), case 2 of master theorem should be used here. The theorem says T(n) = Θ(n ^{log _b a} * lgn) in this case.

Again, it doesn't matter here if you write Θ(n ^{log _b a} * lgn) or Θ(5 * n ^{log _b a} * lgn) or Θ(x * n ^{log _b a} * lgn) . Multiplying a function with a positive constant doesn't change its asymptotic bounds. Master theorem gives you just the asymptotical bounds of the function, not its exact value.