Definition:Theta Notation/Definition 1
Definition
Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.
Then $\map \Theta g$ is defined as:
- $\map \Theta g = \map \OO g \cap \map \Omega g$
where:
- $\map \OO g$ is big-$\OO$ notation
- $\map \Omega g$ is big-$\Omega$ notation.
That is:
- $\map \Theta g = \set {f: \N \to \R: \exists c_1, c_2 \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c_1 \cdot \size {\map g n} \le \size {\map f n} \le c_2 \cdot \size {\map g n} }$
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 presented as
This definition can also be presented as:
- $\map f n \in \map \Theta {\map g n}$
- $\paren {\map f n \in \map \OO {\map g n} } \text { and } \paren {\map f n \in \map \Omega {\map g n} }$
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.
Sources
- 1990: Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest: Introduction to Algorithms ... (previous) ... (next): $2$: Growth of Functions: $2.1$ Asymptotic Notation: $\Theta$-notation