Consider the following algorithm.

ALGORITHM: Mystery$($A$[0..$n$-1],$ B$[0..$n$-1],$n$)$

$//$Input: Two same$-$sized arrays A$[0..$n$-1],$ B$[0..$n$-1]$ of n real numbers, size of A$,$ B

a$$ A$[$n$-1]$

b $$ B$[$n$-1]$

for i $$ n$-2$ downto $0$ do

if A$[$i$]<$ a then

a$$ A$[$i$]$

if B$[$i$]>$b then

b$$ B$[$i$]$

return b$$a

a$)$ What does this algorithm compute?

b$)$ Which of the following statements about the output are true and which are false? Explain your answers by providing an argument for true statements and a counter example for false statements.

i$.$ It will be equal to a value in one of the arrays.

ii$.$ It can be negative.

iii. It will be between $0$ and n$-1.$

c$)$ What is its basic operation?

d$)$ Show and simplify a summation for the number of times the basic operation is executed.

e$)$ What is the $\backslash$Theta time$-$efficiency class of this algorithm? Prove your result.

f$)$ Suggest an improvement, or a better algorithm altogether, and indicate its efficiency class, or make a convincing argument that no improvement is possible.