$1)$ Supposetherun$-$timeofaserialprogramisgivenbyTserial $=$n$2,$wheretheunitsoftherun$-$ time are in microseconds. Suppose that a parallelization of this program has run$-$time Tparallel $=$ n$2/$p $+$ log$2$ p$.$ Write a program that finds the speedups and efficiencies of this program for variousvaluesofnandp.Runyourprogramwithn$=10,20,40,...,320,$andp$=1,2,4,...,128.$ What happens to the speedups and efficiencies as p is increased and n is held fixed? What happens when p is fixed and n is increased?

$2)$ SupposethatTparallel$=$Tserial$/$p$+$Toverhead.Alsosupposethatwefixpandincreasethe problem size.

a$)$ Show that if Toverhead grows more slowly than Tserial, the parallel efficiency will increase as we increase the problem size.

b$)$ Show that if$,$ on the other hand, Toverhead grows faster than Tserial, the parallel efficiency will decrease as we increase the problem size.