Question

Given a set S = {a1,.., an} of proposed activities (e.g., classes) that wish to use a resource (e.g., classroom) which can serve one activity at a time. Each a, has a start times, and a finish time f; with 0s; < fi < [infinity], and takes place in time interval [Si, fi). Activities a; and a, are compatible if [a;, fi) [a,j, fj) = 0. The scheduling all intervals problem is to find a minimum number of resources to serve all activities, each resource serves a subset of mutually compatible activities from S. Give an efficient greedy algorithm to solve the scheduling all intervals problem; prove the correctness and analyze the running time of the algorithm.

Answer 1

To solve the scheduling all intervals problem, use a greedy algorithm. Sort the activities by their finish times and assign them to resources based on compatibility. The time complexity of the algorithm is O(n log n).

Step-by-step explanation:

The scheduling all intervals problem can be solved using a greedy algorithm. Here is the step-by-step explanation:

1. Sort the activities in ascending order of their finish times.
2. Initialize an empty list of resources.
3. For each activity in the sorted list:

This algorithm works because by prioritizing activities with earlier finish times, it ensures that resources are used efficiently. The correctness can be proved by showing that the final assignment of activities to resources is a valid solution. The running time of the algorithm is O(n log n) due to the sorting step.