Consider a problem in which you have a list J of jobs, each of which takes the same amount of time to complete. You need to schedule these jobs in array S, indexed from 0 to n. Each job occupies a single index in the array, because they each take the same amount of time to complete. Each job j e J has: •deadline j.deadline from 0 to n, denoting the latest index in S at which it can be scheduled. •profit j.profit, denoting the profit or gain from completing job j. The goal is to maximize the sum of profits of the jobs scheduled in S. |J| can be greater than |S|, so it might not be possible to schedule all jobs. There is no penalty for leaving a job undone. Consider the following algorithm: Job Scheduling IN : List J of jobs, max index n 1 S+ an array indexed 0 to n, with null at each index 2 Sort J in non-increasing order of profits 3 for i from 0 to n Find the largest t such that S[t] = null and t < J[i].deadline (if one exists) if an index t was found | S[t] + J[i] OUT: S maximizes the profit of scheduled jobs Note that – is used to denote assignment; x + y means that x is assigned the value of y. Prove that the algorithm is correct. HINT: A schedule S is promising if it can be extended to an optimal schedule by scheduling jobs that have not yet been considered. If S is promising after all jobs have been considered, then S is an optimal schedule.

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