Definition:Theta Notation/Definition 2
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 there exist $c \in \R_{>0}$ such that:
- $\ds \lim_{n \mathop \to \infty} \frac {\map f n} {\map g n} = c$
Then:
- $\map f n \in \map \Theta {\map g n}$
Notation
The expression $\map f n \in \map \Theta {\map g n}$ is read as:
- $\map f n$ is theta of $\map g n$
While it is correct and accurate to write:
- $\map f n \in \map \Theta {\map g n}$
it is a common abuse of notation to write:
- $\map f n = \map \Theta {\map g n}$
This notation offers some advantages.
Also known as
Some sources refer to $\Theta$ notation as big-$\Theta$ notation, in parallel with big-$\OO$ and big-$\Omega$.
However, it is worth bearing in mind that:
and so there is no need to distinguish between big-$\Theta$ and little-$\theta$.
Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ consistently use the term $\Theta$ notation, voicing it as theta notation.
Motivation
$\Theta$ notation is a type of order notation for typically comparing run-times or growth rates between two growth functions.
$\Theta$ is a stronger statement than big-$\OO$ and big-$\Omega$.
Also see
- Results about $\Theta$ notation can be found here.