Question

Good evening people,

I would like some help to compare a big-O and a Θ algorithm.
I can understand how to compare two big-O’s but something troubles
my understanding on how to compare big-O with Θ or big-O with Ω etc etc.

I will post some examples below :

Θ(2ⁿ)     vs      Ο(2ⁿ)
Θ(n^0.6)  vs      Θ(n^logn)
O(n)      vs      Ω(n⋅logn)

Answer 1

Θ(2^n) vs Ο(2^n)

I have one thing that’s the same size as an elephant[*], and another thing that’s no bigger than an elephant. Compare their sizes.

Θ(n^0.6) vs Θ(n^logn)

n^log n is bigger than n^0.6, because log n is bigger than a constant. But I can’t be bothered thinking of animals for them.

O(n) vs Ω(nlogn)

I have one thing that’s no bigger than a mouse, and another thing that’s no smaller than a cat. Compare their sizes.

[*] erm… as the thing and the elephant tend to infinity they’re the same size, anyway. The analogy isn’t perfect, but the point is that big-O means “is no bigger than”, big-Omega means “is no smaller than”, and big-Theta means, “is both no bigger than and no smaller than”. “Bigger” and “smaller” are both judged by the same standard, actually meaning “f(n) no bigger/smaller in magnitude than a constant multiple times g(n), for sufficiently large n“