Definition:Big-Omega Notation/Definition 2
Definition
 | The validity of the material on this page is questionable. In particular: This is the same definition as for Definition:Theta Notation -- need to review this You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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 $\Omega$ notation can be found here.