# Understanding the relations between O(g(n)), Θ(g(n)) and Ω(g(n)) [duplicate]

I was reading the Cormen, Leiserson, Rivest and Stein textbook, Introduction to Algorithms.

The book explained the three asymptotic notations literally very well. However, there was this paragraph:

Technically, it is an abuse to say that the running time of insertion sort is $$O(n^2)$$, since for a given $$n$$, the actual running time varies, depending on the particular input of size $$n$$.

and this one:

It is not contradictory, however, to say that the worst-case running time of insertion sort is $$Ω(n^2)$$, since there exists an input that causes the algorithm to take $$Ω(n^2)$$ time.

1. I understand why did the author said that "it is an abuse to say $$O(n^2)$$ is the running time", as there are the inputs with best cases causing linear time and there are also inputs with worst case causing quadratic time.

2. I don't understand why it is not contradictory to say that the worst-case running time of insertion sort is $$Ω(n^2)$$. Isn't $$\Omega(g(n))$$ supposed to be the best running time? So should it not be $$Ω(n)$$?

It confuses me. Can you please explain why $$Ω(n^2)$$ is possible for insertion sort when it should be $$Ω(n)$$?

• $O$, $\Omega$ and $\Theta$ are just ways of describing how fast functions grow. It doesn't matter what a function is used to measure. Just like $>$ and $<$ are ways of describing relationships between numbers, regardless of what those numbers measure. You could say "The tallest person on Earth is at most 3m tall" and "The tallest person on Earth is at least 1m tall" and nobody would say "But you're talking about the tallest -- you're only allowed to use $>$." – David Richerby Mar 21 '19 at 9:50

Aren't $$\Omega(g(n))$$ supposed to tell best running time?
Not at all. It just means that for all large enough $$n$$, there exist inputs of size $$n$$ where the algorithm takes time at least proportional to $$g(n)$$.
Can you please explain why $$\Omega(n^2)$$ is possible for insertion sort when it should be $$\Omega(n)$$?
The worst-case running time of insertion sort is $$\Omega(n^2)$$ because for all $$n$$, there exist inputs of size $$n$$ where it takes time proportional to $$n^2$$.
The running time of insertion sort in general is not always $$\Omega(n^2)$$, as you observed; it depends on the input. But the worst-case running time (which does not depend on the input, only on $$n$$) is $$\Omega(n^2)$$.