Greedy Algorithms Questions Long
To calculate the maximum profit that can be earned by completing jobs within their deadlines, considering penalties for missing deadlines, we can use a greedy algorithm known as the "Job Scheduling Problem with Deadlines and Penalties".
The problem statement is as follows: Given a set of jobs, each with a deadline and a penalty for missing the deadline, we need to schedule the jobs in such a way that the total penalty is minimized and the maximum profit is achieved.
Here is the step-by-step approach to solving this problem using a greedy algorithm:
1. Sort the jobs in descending order of their profits. This ensures that we prioritize jobs with higher profits.
2. Initialize an array of size equal to the maximum deadline among all jobs, and initialize all elements to -1. This array will be used to keep track of the scheduled jobs.
3. Iterate through the sorted jobs list. For each job, starting from the highest profit job:
a. Find the latest available slot in the array that is before or at the job's deadline. This can be done by starting from the job's deadline and moving left until an empty slot is found.
b. If a slot is found, assign the job to that slot in the array.
c. If no slot is found, skip the job as it cannot be scheduled within its deadline.
4. Calculate the total profit and penalty by iterating through the array and summing up the profits and penalties of the scheduled jobs.
5. Return the maximum profit and the scheduled jobs.
The time complexity of this greedy algorithm is O(n^2), where n is the number of jobs. This is because for each job, we may need to iterate through the array to find an available slot.
Note: It is important to note that this greedy algorithm may not always provide an optimal solution. There can be cases where a different scheduling order could yield a higher profit. However, this algorithm provides a good approximation and is efficient in terms of time complexity.