This is a Python program to minimize maximum lateness using greedy algorithm.

We are given n requests numbered 0 to n – 1. Each request i has a time that it takes to complete t(i) and a deadline d(i). If a request i starts at time s(i), then its finish time is f(i) = s(i) + t(i). The lateness of request i is L(i) = f(i) – d(i) if f(i) > d(i). The maximum value of L(i) over all i is called the maximum lateness. That is, maximum lateness = max(L(0), L(1), …, L(n – 1)). The problem is to schedule all the requests such that the value of maximum lateness is minimized.

1. The function minimize_lateness defined.

2. It takes two lists ttimes and dtimes as arguments.

3. ttimes[i] is the time taken to complete request i.

4. dtimes[i] is the deadline of request i.

5. The algorithm works by sorting the requests in order of earliest deadline.

6. This ordering is an optimal way to schedule the requests to minimize maximum lateness.

7. The minimum maximum lateness is then calculated.

8. The minimum maximum lateness and the optimal schedule ordering is returned.

Here is the source code of a Python program to minimize maximum lateness using greedy algorithm. The program output is shown below.

def minimize_lateness(ttimes, dtimes): """Return minimum max lateness and the schedule to obtain it. (min_lateness, schedule) is returned. Lateness of a request i is L(i) = finish time of i - deadline of if request i finishes after its deadline. The maximum lateness is the maximum value of L(i) over all i. min_lateness is the minimum value of the maximum lateness that can be achieved by optimally scheduling the requests. schedule is a list that contains the indexes of the requests ordered such that minimum maximum lateness is achieved. ttime[i] is the time taken to complete request i. dtime[i] is the deadline of request i. """ # index = [0, 1, 2, ..., n - 1] for n requests index = list(range(len(dtimes))) # sort according to deadlines index.sort(key=lambda i: dtimes[i]) min_lateness = 0 start_time = 0 for i in index: min_lateness = max(min_lateness, (ttimes[i] + start_time) - dtimes[i]) start_time += ttimes[i] return min_lateness, index n = int(input('Enter number of requests: ')) ttimes = input('Enter the time taken to complete the {} request(s) in order: ' .format(n)).split() ttimes = [int(tt) for tt in ttimes] dtimes = input('Enter the deadlines of the {} request(s) in order: ' .format(n)).split() dtimes = [int(dt) for dt in dtimes] min_lateness, schedule = minimize_lateness(ttimes, dtimes) print('The minimum maximum lateness:', min_lateness) print('The order in which the requests should be scheduled:', schedule)

1. The user is prompted to enter the number of requests n.

2. The user is then asked to enter the time taken and the deadline of all n requests.

3. The function minimize_lateness is called to get the minimum maximum lateness and the order in which the requests should be scheduled.

4. This result is then displayed.

Case 1: Enter number of requests: 5 Enter the time taken to complete the 5 request(s) in order: 2 4 1 3 5 Enter the deadlines of the 5 request(s) in order: 3 1 5 6 3 The minimum maximum lateness: 9 The order in which the requests should be scheduled: [1, 0, 4, 2, 3] Case 2: Enter number of requests: 3 Enter the time taken to complete the 3 request(s) in order: 3 4 5 Enter the deadlines of the 3 request(s) in order: 4 3 2 The minimum maximum lateness: 8 The order in which the requests should be scheduled: [2, 1, 0] Case 3: Enter number of requests: 1 Enter the time taken to complete the 1 request(s) in order: 3 Enter the deadlines of the 1 request(s) in order: 2 The minimum maximum lateness: 1 The order in which the requests should be scheduled: [0]

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