$(10$ points$)$ Minimum job completion time.

Suppose there are $n$ distinct and independent jobs, labeled $J_1,J_2,$dots,$J_n.$ Also, suppose there

is one supercomputer and sufficient number of PCs $($which is more than $n).$ Each job consists

of two stages: first it needs to be preprocessed on the supercomputer, and then it needs to be

finished on one of the PCs$.$ Job $J_i$ needs $p_i$ seconds of time on the supercomputer, followed

by $f_i$ seconds of time on a PC$.$ Since we have sufficient PCs$,$ the finishing of the jobs can be

performed fully in parallel $-$ all the jobs can be processed at the same time on different PCs$.$

However, the supercomputer can only work on a single job at a time.

A schedule is an ordering of the jobs for the supercomputer, and the completion time of the

schedule is the earliest time at which all jobs will have finished processing on the PCs$.$ Design a

polynomial$-$time greedy algorithm that finds a schedule with the minimum job completion

time.