Question

There are n customers who are waiting for a service. Each customer's service time is known in advance: it is t; minutes for ith customer. So if, for example, the customers are ti served in order of increasing i, then the ith customer has to wait for ti minutes.

We want to minimize the total waiting time L

T= (time spent by customer i).

i=1

Design an algorithm that works in polynomial of n time that returns the optimal order in which to process the customers.

Answer 1

To find the optimal order to process customers, sort their service times in ascending order and serve them in this sequence; it's an efficient O(n log n) greedy algorithm.

Step-by-step explanation:

To determine the optimal order in which to process customers based on their service times, we can apply a greedy algorithm that schedules the customers by ascending service times (ti). This approach ensures each customer spends the least amount of time waiting. The algorithm follows these steps:

• Create a list of customers and their associated service times.
  
• Sort the list in non-decreasing order based on service times.
• Serve the customers in the sorted order.

This algorithm is efficient, operating with a time complexity of O(n log n), due to the sorting step. The rest of the operation—creating the list and serving the customers—is performed in O(n), making the overall algorithm polynomial-time in terms of the number of customers.

Example: If customers' service times are [15, 5, 10], sorting them gives [5, 10, 15]. Serving them in this order minimizes the cumulative waiting time, leading to an optimal solution.