* Comparing Growth rate of funtions
** Applying limit
To compare two funtions $f(n)$ and $g(n)$. We can use limit
\[ \lim_{n\to\infty} \frac{f(n)}{g(n)} \]
+ If result is 0 then growth of $g(n)$ > growth of $f(n)$
+ If result is $\infty$ then growth of $g(n)$ < growth of $f(n)$
+ If result is any finite number (constant), then growth of $g(n)$ = growth of $f(n)$
*Note* : L'Hôpital's rule can be used in this limit.

** Using logarithm
Using logarithm can be useful to compare exponential functions. When comaparing functions $f(n)$ and $g(n)$, 
+ If growth of $\log(f(n))$ is greater than growth of $\log(g(n))$, then growth of $f(n)$ is greater than growth of $g(n)$
+ If growth of $\log(f(n))$ is less than growth of $\log(g(n))$, then growth of $f(n)$ is less than growth of $g(n)$
+ When using log for comparing growth, comaparing constants after applying log is also required. For example, if functions are $2^n$ and $3^n$, then their logs are $n.log(2)$ and $n.log(3)$. Since $log(2) < log(3)$, the growth rate of $3^n$ will be higher.
+ On equal growth after applying log, we can't decide which function grows faster.

** Common funtions
Commonly, growth rate in increasing order is
\[  c < c.log(log(n)) < c.log(n) < c.n < n.log(n) < c.n^2 < c.n^3 < c.n^4 ...  \]
\[ n^c < c^n < n! < n^n  \]
Where $c$ is any constant.

* Properties of Asymptotic Notations

** Big-Oh
+ *Product* :  \[ Given\ f_1 = O(g_1)\ \ and\ f_2 = O(g_2) \implies f_1 f_2 = O(g_1 g_2) \] \[ Also\  f.O(g) = O(f g) \]

+ *Sum* : For a sum of two functions, the big-oh can be represented with only with funcion having higer growth rate. \[ O(f_1 + f_2 + ... + f_i) = O(max\ growth\ rate(f_1, f_2, .... , f_i )) \]

+ *Constants* : For a constant $c$ \[ O(c.g(n)) = O(g(n)) \], this is because the constants don't effect the growth rate.

** Properties
# Taken from https://www.youtube.com/watch?v=pmGau4xHjFM&ab_channel=UnacademyComputerScience (Analysis of an Algorithm - 2 | L 2 | Algorithms | Infinity Batch | GATE 2022 CS/IT | Ankush Sir)
[[file:./imgs/asymptotic-notations-properties.png]]

+ *Reflexive* :  $f(n) = O(f(n)$ and $f(n) = \Omega (f(n))$ and $f(n) = \theta (f(n))$
+ *Symmetric* : If $f(n) = \theta (g(n))$ then $g(n) = \theta (f(n))$
+ *Transitive* : If $f(n) = O(g(n))$ and $g(n) = O(h(n))$ then $f(n) = O(h(n))$
+ *Transpose* : If $f(n) = O(g(n))$ then we can also conclude that $g(n) = \Omega (f(n))$ so we say Big-Oh is transpose of Big-Omega and vice-versa.
+ *Antisymmetric* : If $f(n) = O(g(n))$ and $g(n) = O(f(n))$ then we conclude that $f(n) = g(n)$
+ *Asymmetric* : If $f(n) = \omega (g(n))$ then we can conclude that $g(n) \ne \omega (f(n))$