Write the exact as well as asymptotic $($Big Oh$)$ for the time complexity of the following algorithms. Assume that all operations $($arithmetic$,$ logical$,$ read$/$write take constant time $$c$).$ For example, your answer should be as detailed as shown in the following sample example.

Sample algorithm

for $($int i $=1$; i $<=$ n; i$++)\{$

int a $=4$;

int b $=5$;

int c $=$ a $+$ b;

$\}$

Algorithms

Sample analysis

for $($int i $=1$ to n$)\{//$loop is running $$n$$ times

int a $=4$; $//$constant time c$.$

int b $=5$; $//$constant time c$.$

int c $=$ a $+$ b; $//$constant time c$.$

$\}$

Exact time $=$ n $($c$+$c$+$c$)=3$cn

Asymptotic time $=$ O$($n$)$

$-------$

$1.$A function $($or set of statements$)$ that doesn$$t contain loop, recursion and call to any other non$-$constant time function, such as follows:

Fun$($int array A of size n$)\{$

int x $=5$;

int y $=4$;

int z $=$ x $+$ y;

$\}$

Exact $=$ Asymptotic $=$

$2.$A loop or recursion that runs a constant number of times, such as follows:

Fun$($int array A of size n$)\{$

for $($int i $=1$ to k$)\{//$ Here k is a constant, it could be any constant number

for $($int j $=1$ to k$)$

$\{$

int a $=5$;

a$++$;

a$--$;

$\}$

$\}$

$\}$

Exact $=$ Asymptotic $=$

$3.$A loop where the loop variables is incremented $/$ decremented by a constant amount k$,$ as follows:

Fun$($int array A of size n$)\{$

$//$ Here k is a positive integer constant

for $($int i $=1$; i $<=$ n; i $+=$ k$)\{$

int a $=5$;

a$+=$i ;

$\}$

$\}$

Exact $=$ Asymptotic $=$

$4.$Time complexity of nested loops as follows:

Fun$($int array A of size n$)\{$

$//$ Here k is a positive integer constant

for $($int i $=1$; i $<=$n; i $+=$ k$)\{$

for $($int j $=$ n; j $>=1$; j $=$ k$)\{$

int a $=$ i $+$ j;

$\}$

$\}$

$\}$

Exact $=$ Asymptotic $=$

$5.$Time Complexity of a loop if the loop variables is divided $/$ multiplied by a constant amount k as follows:

Fun$($int array A of size n$)\{$

for $($int i $=1$; i $<=$n; i $*=$ k$)\{$

int a $=$ i $+$ i ;

$\}$ Exact $=$ Asymptotic $=$

$6.$Time Complexity of a loop if the loop variables is reduced $/$ increased exponentially by a constant amount as follows:

Fun$($int array A of size n$)\{$

for $($int i $=2$; i $<=$n; i $=$ i$2)\{$

int a $=$ i;

$\}$

$\}$

Exact $=$ Asymptotic $=$

$7.$Time Complexity of a loop if the loop variables is reduced $/$ increased by a sqrt amount as follows:

Fun$($int array A of size n$)\{$

for $($int i $=$ n; i $>2$; i $=$ sqrt$($i$))\{//$here sqrt means square root

int a $=$ i;

$\}$

$\}$

Exact $=$ Asymptotic $=$

$8.$Time Complexity of a function with a variety of loops:

Fun$($int array A of size n$)\{$

for $($int i $=1$; i $<=$n; i$*=2)\{$

for $($int j $=1$; j $<=$n; j$*=3)\{$

int a $=$ i$+$j;

$\}$

for $($int j $=2$; j $<=$n; j $=$ j$2)\{$

int a $=$ j;

$\}$

$\}$

for $($int i $=1$; i $<=$n; i$+=4)\{$

for $($int k $=2$; k $<=$n; k $=$ k$2)\{$

int a $=$ i$+$k;

$\}$

$\}$

$\}$

Exact $=$ Asymptotic $=$