Question

Job Selection Problem. As a software engineer, you are tasked with designing a real- time job scheduler for an operating system running on a single-threaded CPU. Each job consists of a duration (in milliseconds) and the profit it generates once scheduled. The goal of this problem is to select a subset of all given jobs such that the profit is maximized without exceeding the given input time limit $t$. You may choose to run a fraction of the job to get a proportionate fraction of the profit. However, you are only allowed to run the same job at most once. Let $X = \{x_1, x_2, ..., x_n\}$ be the set of jobs where each $x_i = (i, d_i, p_i)$ represents the du- ration $d_i \ge 0$ and profit $p_i \ge 0$ of job $i$. To illustrate this task, consider for example the following set of jobs to be scheduled within a time period of $t = 8$: \begin{tabular}{|c|c|c|c|c|} \hline Job No.& 1 & 2 & 3 & 4 \\ \hline Duration & 3 & 6 & 4 & 2 \\ \hline Profit & 15 & 24 & 12 & 16 \\ \hline \end{tabular} Then an optimal selection of these jobs is 1, 4, and 1/2 of job 2 which will yield a maximum of 43 within the time limit of 8 ms. 1. (15 points) Greedy Approach. Devise a greedy algorithm to solve this problem: (a) (3 points) Argue why the problem has optimal substructure. (b) (2 points) State your greedy choice. (c) (10 points) Write a program that implements the greedy algorithm to solve the job selection problem. Record the average running time for a problem size of $n = 1000$ (i.e. there are 1000 jobs with random durations/profits to choose from) within a time period of $t = 500$ms.

          Job Selection Problem. As a software engineer, you are tasked with designing a real-
time job scheduler for an operating system running on a single-threaded CPU. Each job
consists of a duration (in milliseconds) and the profit it generates once scheduled. The goal
of this problem is to select a subset of all given jobs such that the profit is maximized
without exceeding the given input time limit $t$. You may choose to run a fraction of the
job to get a proportionate fraction of the profit. However, you are only allowed to run the
same job at most once.
Let $X = \{x_1, x_2, ..., x_n\}$ be the set of jobs where each $x_i = (i, d_i, p_i)$ represents the du-
ration $d_i \ge 0$ and profit $p_i \ge 0$ of job $i$. To illustrate this task, consider for example the
following set of jobs to be scheduled within a time period of $t = 8$:
\begin{tabular}{|c|c|c|c|c|}
\hline
Job No.& 1 & 2 & 3 & 4 \\
\hline
Duration & 3 & 6 & 4 & 2 \\
\hline
Profit & 15 & 24 & 12 & 16 \\
\hline
\end{tabular}
Then an optimal selection of these jobs is 1, 4, and 1/2 of job 2 which will yield a maximum
of 43 within the time limit of 8 ms.
1. (15 points) Greedy Approach. Devise a greedy algorithm to solve this problem:
(a) (3 points) Argue why the problem has optimal substructure.
(b) (2 points) State your greedy choice.
(c) (10 points) Write a program that implements the greedy algorithm to solve the job
selection problem. Record the average running time for a problem size of $n = 1000$ (i.e.
there are 1000 jobs with random durations/profits to choose from) within a time period
of $t = 500$ms.

$Job Selection Problem. As a software engineer, you are tasked with designing a real- time job scheduler for an operating system running on a single-threaded CPU. Each job consists of a duration (in milliseconds) and the profit it generates once scheduled. The goal of this problem is to select a subset of all given jobs such that the profit is maximized without exceeding the given input time limit t. You may choose to run a fraction of the job to get a proportionate fraction of the profit. However, you are only allowed to run the same job at most once. Let X = {x1, x2, ..., xn} be the set of jobs where each xi = (i, di, pi) represents the du- ration di ≥ 0 and profit pi ≥ 0 of job i. To illustrate this task, consider for example the following set of jobs to be scheduled within a time period of t = 8: Job No. 1 2 3 4 Duration 3 6 4 2 Profit 15 24 12 16 Then an optimal selection of these jobs is 1, 4, and 1/2 of job 2 which will yield a maximum of 43 within the time limit of 8 ms. 1. (15 points) Greedy Approach. Devise a greedy algorithm to solve this problem: (a) (3 points) Argue why the problem has optimal substructure. (b) (2 points) State your greedy choice. (c) (10 points) Write a program that implements the greedy algorithm to solve the job selection problem. Record the average running time for a problem size of n = 1000 (i.e. there are 1000 jobs with random durations/profits to choose from) within a time period of t = 500ms.$

Added by Bradley B.

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