What is the maximum profit that can be earned by completing jobs within their deadlines and how can it be calculated using a greedy algorithm?

To calculate the maximum profit that can be earned by completing jobs within their deadlines using a greedy algorithm, we can follow the steps outlined below:

1. Sort the jobs in descending order based on their profits. This step ensures that we prioritize jobs with higher profits.

2. Initialize an array or list to keep track of the time slots for each job. Initially, all time slots are set to 0, indicating that no job has been scheduled yet.

3. Iterate through the sorted jobs list. For each job, find the latest available time slot before its deadline. If a time slot is available, assign the job to that time slot and update the time slot as occupied. If no time slot is available, skip the job.

4. Calculate the total profit earned by summing up the profits of all scheduled jobs.

5. Return the maximum profit earned.

By following this greedy algorithm, we ensure that we always select the job with the highest profit available at each step, while also considering the deadlines to ensure that jobs are completed within their specified time frames.

Here is an example to illustrate the algorithm:

Consider the following jobs with their respective profits and deadlines:
Job 1: Profit = 100, Deadline = 2
Job 2: Profit = 50, Deadline = 1
Job 3: Profit = 70, Deadline = 2
Job 4: Profit = 60, Deadline = 1

Step 1: Sort the jobs in descending order based on their profits:
Job 1: Profit = 100, Deadline = 2
Job 3: Profit = 70, Deadline = 2
Job 4: Profit = 60, Deadline = 1
Job 2: Profit = 50, Deadline = 1

Step 2: Initialize the time slots array: [0, 0, 0, 0]

Step 3: Iterate through the sorted jobs list:
- Job 1: The latest available time slot before its deadline is 2. Assign it to time slot 2. Update the time slots array: [0, 100, 0, 0]
- Job 3: The latest available time slot before its deadline is 1. Assign it to time slot 1. Update the time slots array: [70, 100, 0, 0]
- Job 4: The latest available time slot before its deadline is 1. However, time slot 1 is already occupied. Skip this job.
- Job 2: The latest available time slot before its deadline is 1. However, time slot 1 is already occupied. Skip this job.

Step 4: Calculate the total profit earned: 70 + 100 = 170

Step 5: Return the maximum profit earned: 170

Therefore, the maximum profit that can be earned by completing jobs within their deadlines using the greedy algorithm for this example is 170.