What is the activity selection problem and how can it be solved using a greedy algorithm?

The activity selection problem is a classic optimization problem in computer science that involves selecting a maximum number of compatible activities from a given set of activities, where each activity has a start time and an end time. The goal is to find the largest possible subset of non-overlapping activities.

A greedy algorithm can be used to solve the activity selection problem efficiently. The algorithm works by iteratively selecting the activity with the earliest finish time and discarding any activities that overlap with it. This greedy approach ensures that the selected activities do not conflict with each other and maximizes the number of activities that can be performed.

Here is the step-by-step process for solving the activity selection problem using a greedy algorithm:

1. Sort the activities based on their finish times in ascending order.
2. Initialize an empty set to store the selected activities.
3. Select the first activity from the sorted list and add it to the selected set.
4. Iterate through the remaining activities:
- If the start time of the current activity is greater than or equal to the finish time of the last selected activity, add the current activity to the selected set.
- Otherwise, discard the current activity as it overlaps with the last selected activity.
5. Return the selected set of activities as the solution.

By always choosing the activity with the earliest finish time, the greedy algorithm ensures that the selected activities do not conflict with each other and maximizes the number of activities that can be performed. The time complexity of this algorithm is O(n log n), where n is the number of activities, due to the initial sorting step.