Definition:Big-Omega Notation
Definition
Definition 1
Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.
Then $\map \Omega g$ is defined as:
- $\map \Omega g = \set {f: \N \to \R: \exists c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c \cdot \size {\map g n} \le \size {\map f n} }$
Definition 2
Let $f: \N \to \R, g: \N \to \R$ be two real sequences, expressed here as real-valued functions on the set of natural numbers $\N$.
Let there exist $c \in \R_{>0}$ such that:
- $\ds \lim_{n \mathop \to \infty} {\frac {\map f n} {\map g n} } = c > 0$
Then:
- $\map f n \in \map \Omega {\map g n}$
Notation
The expression $\map f n \in \map \Omega {\map g n}$ is read as:
- $\map f n$ is big-omega of $\map g n$
While it is correct and accurate to write:
- $\map f n \in \map \Omega {\map g n}$
it is a common abuse of notation to write:
- $\map f n = \map \Omega {\map g n}$
This notation offers some advantages.
Also known as
Note that in the Greek alphabet, $\Omega$ is the capital letter counterpart of the minuscule $\omega$.
Hence the former is called big-omega and the latter little-omega.
Some sources, therefore, write $\Omega$ notation as big-$\Omega$ notation, despite the fact that $\Omega$'s "big"-ness is intrinsic.
$\mathsf{Pr} \infty \mathsf{fWiki}$ may sometimes adopt this convention if clarity is improved.
Motivation
$\Omega$ notation is a type of order notation for typically comparing run-times or growth rates between two growth functions.
Also see
- Results about big-$\Omega$ notation can be found here.