# Definition:Big-Omega

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## 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:

- $\map f n \in \map \Omega {\map g n}$

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

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 the big-O notation.