Big-$\Omega$ Asymptotic Notation: Lower Bound vs Upper Bound

In asymptotic notation, Big-$\Omega$ (Omega) is used to represent a lower bound on the growth rate of a function—meaning the function grows at least as fast as another function beyond some input size. Therefore, among the choices, the correct statement is (ii) The lower bound (Best-case).

To reason precisely: in the definition of $f(n) \in \Omega(g(n))$, there exist constants $c>0$ and $n_0$ such that for all $n \ge n_0$,
$f(n) \ge c\cdot g(n).$
This inequality is inherently about guaranteeing a minimum growth rate (lower bound), not an upper bound.

A helpful visual intuition is:

• Big-$O$ ensures $f(n)$ does not grow faster than $g(n)$ (upper bound),
• Big-$\Omega$ ensures $f(n)$ grows no slower than $g(n)$ (lower bound),
• Big-$\Theta$ ensures $f(n)$ grows at the same order as $g(n)$ (tight bound).

We’ll use these ideas to answer the multiple-choice question and clarify what “tight” and “average-case” correspond to in asymptotic analysis.

Key definitions and how they map to answer choices

We use these standard asymptotic sets:

• Big-O
• [Big-$\Omega$]{def="Lower bound: f(n) grows at least as fast as g(n) up to constants"}
• Big-Theta
• Asymptotic bound

Big-$\Omega$ is a lower bound, which directly corresponds to choice (ii).
It is not an upper bound (choice i), and not a tight bound on its own (choice iii), because tightness requires Big-$\Theta$.

“Average-case” is typically analyzed using probabilistic models (expected values), not captured by the classic Big-$\Omega$ / Big-$O$ / Big-$\Theta$ definitions alone.

Visual: Big-$O$, Big-$\Omega$, and Big-$\Theta$

This reinforces the mapping:

• Big-$\Omega$ ⇢ lower bound
• Big-$\Theta$ ⇢ tight bound
• Big-$O$ ⇢ upper bound

So the correct option is (ii).