Definition:Theta Notation/Definition 2

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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:

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.