Problem $3.$ Each of the algorithms below takes as input a positive integer n and then performs some steps.

For each algorithm, state the running time of the algorithm as a function of n$,$ using $\backslash$Theta $-$notation.

$(15$ points$)$

a$)$ For i $=1$ to n$/3$

$$ j $10$

$$ While $($j $<=$ i$/20)$

$$ Do placeholder stuff that takes O$(1)$ time

$$ j $$ j $+10$

b$)$ While $($n $>=1)$

$$ n $$ n$/3$

$$ do placeholder stuff that takes O$(1)$ time.

c$)$ While $($n $>=2)$

$$ n $$

$$

n

$$ Do placeholder stuff that takes O$(1)$ time.

d$)$ For i $=1$ to n

$$ For j $=1$ to i

$2$

$$ Do placeholder stuff that takes O$(1)$ time

e$)$ For i $=1$ to n

$$ For j $=1$ to log$($i$)$

$$ For k $=1$ to n

$$ Do placeholder stuff that takes O$(1)$ time