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Is there any pattern that is a combinatorial analogue to the Big-O notation?
Is there any pattern that is a combinatorial analogue to the Big-O notation?
Big-O notation is for computational complexity (i.e., the run time) of a function, and has the following general form (Wikipedia):
$f(n) = c \cdot n^d$, where $d \in \mathbb{N}$, and $n$ is the size of an input.
The big-O notation gives a measure of the complexity of a function by simply comparing their growth rates.
An alternative way of expressing the Big-O notation is using the following asymptotic form:
$f(n) = \Theta (g(n))$, where $g$ is a bounded function.
A:
Yes, there is. The asymptotic speed is called big-Theta notation, defined in terms of a function $g$ as follows:
$ f(n) = \Theta(g(n)) = \Bigl(\exists c\in\mathbb{R}\Bigr) \Bigl(\exists n_0\in\mathbb{N}\Bigr) \Bigl( c\cdot n^d = g(n) \geq 0 \Bigr)$
For functions $f,g$ such that $f(n) = \Theta(g(n))$ (which include many common functions), the following simple inequalities hold:
$ f(n) \leq \Theta(g(n)) \leq g(n) $
$ f(n) \geq \Theta(g(n)) \geq \Omega(g(n)) $
Note that the first inequality is just the definition, and the second is a consequence of the above definition, if $n_0$ 0b46394aab
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