Growth - Tag - rbloghttps://korhal.h4ck.me/tags/growth/Growth - Tag - rblogHugo -- gohugo.ioen© [rblog contributors](https://github.com/rltyty/rblog/graphs/contributors)Sun, 26 Apr 2026 16:54:24 +0800Growth of Functionshttps://korhal.h4ck.me/notes/algorithm/growth-of-functions/Sun, 26 Apr 2026 16:54:24 +0800rltytyhttps://korhal.h4ck.me/notes/algorithm/growth-of-functions/Asymptotic Notations

Assume all functions are asymptotically nonnegative unless otherwise stated.

$\Theta$-notation $\underline{渐近紧界}$

$$ \begin{aligned} \Theta(g(n)) = \{f(n): &\text{ there exist positive constants } c_1, c_2 \text{ and } n_0 \\ &\text{ such that } c_1 g(n) \le f(n) \le c_2 g(n) \text{ for all } n \ge n_0\} \end{aligned} $$

We say that $f(n)$ is $\Theta(g(n))$, i.e., $g(n)$ is a tight asymptotic bound for $f(n)$.

$O$-notation $\underline{渐近上界}$

$$ \begin{aligned} O(g(n)) = \{f(n): &\text{ there exist positive constants } c \text{ and } n_0 \\ &\text{ such that } f(n) \le c g(n) \text{ for all } n \ge n_0\} \end{aligned} $$

We say that $f(n)$ is $O(g(n))$, i.e., $g(n)$ is an asymptotic upper bound for $f(n)$.

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