大O而不是小O意味着Theta？ 同样，大欧米茄和小欧米茄暗示Theta吗？

Question

If f(n) is O(g(n)) but not o(g(n)) , is it true that f(n) is theta(g(n)) ?

Similarly, f(n) is Omega(g(n)) but not omega(g(n)) implies f(n) is theta(g(n)) .

If not, can you provide an explanation/counter-example please?

Answer 1

*NOTE : think of O as <= and o as <.

If f(n) is O(g(n)) but not o(g(n)), is it true that f(n) is theta(g(n))?

Yes, f(n) ∈ Θ(g(n)).

f(n) = O(g(n)) means f(n) ≤ Cg(n). 

f(n) = o(g(n)) is possible if and only if f(n) = O(g(n)), but f(n) ≠ Θ(g(n)).

So, since f(n) is not o(g(n)), but it is O(g(n)), hence, f(n) ∈ Θ(g(n)).

---

*NOTE : think of Ω as >= and ω as >.

Similarly, f(n) is Omega(g(n)) but not omega(g(n)) implies f(n) is theta(g(n)).

Yes, f(n) ∈ Θ(g(n)). Following the similar logic :

f(n) = Ω(g(n)) means f(n) ≥ cg(n).

f(n) = ω(g(n)) is possible if and only if f(n) = Ω(g(n)), but f(n) ≠ Θ(g(n)).

So, since f(n) is not ω(g(n)), but it is Ω(g(n)), hence, f(n) ∈ Θ(g(n)).