# Is $T(n^2) = Ω(n)$?

We need the notation for the lower bound. A capital omega $$\Omega$$ notation is used in this case.

We say that $$f(n) = \Omega(g(n))$$ when there exist constant $$c$$ that $$f(n) \geq c \cdot g(n)$$ for for all sufficiently large $$n$$. Examples:

1. $$n = \Omega(1)$$
2. $$n^2 = \Omega(n)$$
3. $$n^2 = \Omega(n \cdot \log n)$$
4. $$2n + 1 = O(n)$$

I need to verify whether $$T(n^2) = Ω(n)$$. Because I can't seem to think that a quadratic run-time can have a linear lower bound run-time.

I am confused so I might be wrong though.

• What is the function $T$? – Yuval Filmus Feb 9 at 4:38

You haven't said what the function $$T$$ is, so let me guess that what you need to verify is that $$n^2 = \Omega(n)$$. Note that this statement has nothing to do with the running time of any algorithm. It is purely a statement about two functions, $$n^2$$ and $$n$$.
When trying to prove something, it's always a good idea to use the definition. In order to show that $$n^2 = \Omega(n)$$, we need to find $$c > 0$$ so that $$n^2 \geq c \cdot n$$ holds for all large enough $$n$$. I'm sure you can think of such a $$c$$ that works for all $$n \geq 1$$.
If you have an algorithm running in time $$\Theta(n^2)$$, then its running time is also $$\Omega(n)$$. This just means that the running time is linear or worse. In particular, a quadratic running time always has a linear time lower bound, just as $$2 \geq 1$$ holds although $$2 \neq 1$$.