What is the difference between Θ(n) and O(n)?

Question

Sometimes I see Θ(n) with the strange Θ symbol with something in the middle of it, and sometimes just O(n). Is it just laziness of typing because nobody knows how to type th

Answer 1

Conclusion: we regard big O, big θ and big Ω as the same thing.

Why? I will tell the reason below:

Firstly, I will clarify one wrong statement, some people think that we just care the worst time complexity, so we always use big O instead of big θ. I will say this man is bullshitting. Upper and lower bound are used to describe one function, not used to describe the time complexity. The worst time function has its upper and lower bound; the best time function has its upper and lower bound too.

In order to explain clearly the relation between big O and big θ, I will explain the relation between big O and small o first. From the definition, we can easily know that small o is a subset of big O. For example：

T(n)= n^2 + n, we can say T(n)=O(n^2), T(n)=O(n^3), T(n)=O(n^4). But for small o, T(n)=o(n^2) does not meet the definition of small o. So just T(n)=o(n^3), T(n)=o(n^4) are correct for small o. The redundant T(n)=O(n^2) is what? It's big θ!

Generally, we say big O is O(n^2), hardly to say T(n)=O(n^3), T(n)=O(n^4). Why? Because we regard big O as big θ subconsciously.

Similarly, we also regard big Ω as big θ subconsciously.

In one word, big O, big θ and big Ω are not the same thing from the definitions, but they are the same thing in our mouth and brain.