For each of the following three program fragments, give an analysis

of the running time. First, give the total number of times of each

statement being executed. Then give the summed$/$total time in Big$-$O

notation. Note that you must give the most accurate bigO notation.

For example, if the bigO notation time complexity is $O(N^2),$ don't

write $O(N^9).$

$($a$)$ i $=1$;

while i N do

i $=2\backslash$times i;

$($b$)$ Sum $=0$;

for i $=1$ to N$3$ do

for j $=1$ to N do

Sum $=$ Sum $+1$;

$($c$)$ Sum $=0$;

for i $=1$ to N do

for j $=1$ to i

$2$ do

Sum $=$ Sum $+1$;

$($d$)$ x $=100$;

y $=0$;

for i $=1$ to N step $2\backslash$times i do

x $=$ x $+$ i;

for j $=1$ to sqrt$($N$)$ do

if x $>$y$/$j :

y $=$ y $+$ i$/$j;

else

y $=$ y $-1$;

$($e$)$ x $=0$;

for i $=1$ to N do

y $=1$;

for j $=1$ to i do

y $=$ y $*$ j;

x $=$ x $+$ y;