Definition:Little-Omega Notation/Definition 2
Jump to navigation
Jump to search
Definition
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:
- $\ds \lim_{n \mathop \to \infty} {\frac {\map f n} {\map g n} } = \infty$
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 little-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 minuscule counterpart of the capital letter $\Omega$.
Hence the former is called little-omega and the latter big-omega.
Some sources, therefore, write $\omega$ notation as little-$\omega$ notation, despite the fact that $\omega$'s "little"-ness is intrinsic.
$\mathsf{Pr} \infty \mathsf{fWiki}$ may sometimes adopt this convention if clarity is improved.
Also see
- Results about little-$\omega$ notation can be found here.
Sources
- 1990: Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest: Introduction to Algorithms ... (previous): $2$: Growth of Functions: $2.1$ Asymptotic Notation: $\omega$-notation