Finding the constants in Landau notation

Question

I am trying to find the constants $n_0$ and $c$ to show that some given functions belong to the $O(\cdot)$ equivalence class. But, while it seems easy, I am not sure whether I am allowed to do what I will showcase below, or rather, what decides which constants I should take into consideration. For example:

$$n^{\frac 2 3} \in \Omega(\log^8n).$$

The definition is: $g(n) \in \Omega(f(n))$ if there exist $c,n_0$ such that for all $n > n_0$, we have $cf(n) \leq g(n)$.

So, if I am asked to find the constants $n_0$ and $c$, with nothing more asked (no extra conditions etc), how can I decide which values to consider?

Answer 1

The assertion can be proved without finding a particular value of $n_0$, but by just proving there exists an $n_0$ satisfying the desired conditions. More specifically, to show $n^{2/3}$ grows asymptotically faster than $(\log n)^8$, it suffices to show that $\lim_{n \rightarrow \infty} \frac{n^{2/3}}{(\log n)^8} \ =\infty$, which can be shown by repeated application of L'Hospital's rule.

Suppose that $\lim_{n \rightarrow \infty} \frac{f(n)}{g(n)} = \infty$. By definition, this means the ratio $\frac{f(n)}{g(n)}$ grows without bound, i.e. for all $B > 0$, $\frac{f(n)}{g(n)} \ge B$ for all sufficiently large $n$. In other words, for all $B > 0$, there exists $n_0 > 0$ such that $\frac{f(n)}{g(n)} \ge B$ for all $n \ge n_0$. In particular, taking $B=1$, we get that there exists $n_0$ such that $f(n) \ge g(n)$ for all $n \ge n_0$. This implies $f(n) = \Omega(g(n))$.

Answer 2

By solving for $n_0$ and $c$ the system of inequations

$$\begin{cases}n>n_0,\\f(n)>c\,g(n).\end{cases}$$

Concretely, you can study the function $\dfrac{f(n)}{g(n)}$ and find a lower bound (f.i. the global minimum, or any gross approximation).