Write an MPI program tasked with executing the multiplication of a vector

by a scalar and computing the dot product. Process $0$ will solicit input

from the user, distributing two vectors and a scalar among all processes.

Following computation, the results are consolidated onto process $0$ for

display. It's assumed that $\text{'}n\text{'},$ representing the vectors' order, is evenly

divisible by 'comm$\_$sz$\text{'},$ the size of the communicator.

Assuming comm$\_$sz$=8$ and the vector $x=(0,1,2,$dots,$15)$ has been

partitioned among processes using block distribution, create a diagram

outlining the stages involved in executing a butterfly implementation of

the all$-$gather operation for $\text{'}x\text{'}.$

Alter the parallel odd$-$even transposition sort by adjusting the

Merge functions to interchange array pointers after identifying the

smallest or largest elements. How does this modification impact the

total runtime of the sorting process?