Definition:Big-Omega Notation

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

Let $f, g$ be two functions.

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


 * $\exists c > 0, k \ge 0: \forall n > k: \map f n \ge c \map g n$
 * $\exists c > 0, k \ge 0: \forall n > k: \map f n \ge c \map g n$

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

Another method of determining the condition is the following limit:


 * $\ds \lim_{n \mathop \to \infty} {\frac {\map f n} {\map g n} } = c > 0$

where $0 < c \le \infty$.

If such a $c$ does exist, then:
 * $\map f n \in \map \Omega {\map g n}$

To say that $\map f n \in \map \Omega {\map g n}$ is equivalent to:
 * $\map g n \in \map \OO {\map f n}$

where $\OO$ is big-$\OO$ notation.

Also see

 * Definition:O Notation
 * Definition:Little-Omega Notation