Let us assume that each invocation of a collective communication across the P processes takes $\backslash$alpha $$logP microseconds, where $\backslash$alpha is a constant in microseconds. Let us also assume that each invocation of the numerical differentiation function, Diff$($x$),$ takes fixed $\backslash$beta microseconds to return.

Please derive the efficiency of your parallel MPI program for the array size, N$.$ Please just consider the computational cost in the code snippets.

In order for this program to be strongly scalable with an efficiency approaching $1,$ what value should the ratio of $\backslash$beta to $\backslash$alpha approach? In other words, how should computation cost determined by $\backslash$beta compare relative to the communication cost determined by $\backslash$alpha in order for this program to be strongly scalable?