Definition:Theta Notation

Definition

Informal Definition

Let $f: \N \to \R$ and $g: \N \to \R$ be real sequences, expressed as real-valued functions on the set of natural numbers $\N$.

$f$ is $\Theta$ of $g$
there exist positive constants $c_1$ and $c_2$ such that $\map f n$ can be "sandwiched" between $c_1 \map g n$ and $c_2 \map g n$ for sufficiently large $n \ge n_0$.

It is not as important to determine the values of $c_1$, $c_2$ as it is to establish that such constants exist.

Definition 1

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:

$\map \Theta g = \set {f: \N \to \R: \exists c_1, c_2 \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c_1 \cdot \size {\map g n} \le \size {\map f n} \le c_2 \cdot \size {\map g n} }$

Definition 2

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:

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

Also defined as

Sources which utilise order notation so as to explore the behaviour of algorithms are concerned only with algorithm run times, necessarily positive.

Hence they may define the $\Theta$ notation on positive real sequences only, as follows:

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:

$\map \Theta g = \set {f: \N \to \R: \exists c_1, c_2 \in \R_{>0}: \exists n_0 \in \N: \forall n \ge n_0: 0 \le c_1 \cdot \map g n \le \map f n \le c_2 \cdot \map g n}$

Some sources define some or all of the inequalities in this expression to be strict, that is:

$\map \Theta g = \set {f: \N \to \R: \exists c_1, c_2 \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: 0 \le c_1 \cdot \map g n < \map f n < c_2 \cdot \map g n}$

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.