Definition:Theta Notation

Definition
Big-$\Theta$ notation is a type of order notation for typically comparing run-times or growth rates between two growth functions.

Big-$\Theta$ is a stronger statement than big-$\OO$ and big-$\Omega$.

Suppose $f: \N \to \R, g: \N \to \R$ are two real functions.

Then:
 * $\map f n \in \map \Theta {\map g n}$


 * $\paren {\map f n \in \map \OO {\map g n} } \land \paren {\map f n \in \map \Omega {\map g n} }$
 * $\paren {\map f n \in \map \OO {\map g n} } \land \paren {\map f n \in \map \Omega {\map g n} }$

where $\map \OO {\map g n}$ is big-$\OO$ and $\map \Omega {\map g n}$ is big-$\Omega$.

This is read as:
 * $\map f n$ is big-theta of $\map g n$.

Another method of determining the condition is the following limit:
 * $\ds \lim_{n \mathop \to \infty} \frac {\map f n} {\map g n} = c$, where $0 < c < \infty$

If such a $c$ does exist, then $\map f n \in \map \Theta {\map g n}$.

Also see

 * Definition:Little-Theta Notation