Definition:Theta Notation/Definition 1

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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:

$\map f n \in \map \Theta {\map g n}$

if and only if:

$\paren {\map f n \in \map \OO {\map g n} } \text { and } \paren {\map f n \in \map \Omega {\map g n} }$


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:

There is no Little-Theta Notation

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

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