I have a question about big O notation. Let's say I have 3 algorithms which, for an input of size $n$, have time complexity $O(n)$, $O(n^2)$ and $O(n \log n)$, respectively. Assume that all 3 algorithms take $a$, $b$ and $c$ seconds for the worst, average and best cases given an input of size $n$. What could be the actual time of these 3 algorithms given an input of size $2n$? Based on my understanding it should be:
- $O(n)$: $2a$, $2b$, and $2c$.
- $O(n^2)$: $a^2$, $b^2$, and $c^2$.
- $O(n \log n)$: $a \log a$, $b \log b$, and $c \log c$.
My questions are:
- Is this correct?
- In case not, why?
- In case this is correct, would the same hold for $\Theta$ and $\Omega$?