Greedy Algorithms Questions Long
The activity selection problem is a classic optimization problem that involves selecting a maximum-weight subset of activities from a given set of activities, each with its own time interval, weight, and value. The goal is to maximize the total value of the selected activities while ensuring that no two activities overlap in their time intervals.
To solve this problem using a greedy algorithm, we can follow the following steps:
1. Sort the activities based on their finish times in ascending order. This step ensures that we always consider the activities that end earliest.
2. Initialize an empty set, let's call it "selected_activities," to store the selected activities.
3. Iterate through the sorted activities. For each activity, check if it overlaps with any of the previously selected activities. If it does not overlap, add it to the "selected_activities" set and update the total value.
- To check for overlap, compare the start time of the current activity with the finish time of the last selected activity. If the start time is greater than the finish time, there is no overlap.
4. Return the "selected_activities" set and the total value as the solution to the problem.
The greedy algorithm works for this problem because it always selects the activity that finishes earliest among the remaining activities. By doing so, it maximizes the available time for selecting other activities and increases the chances of selecting activities with higher values.
The time complexity of this greedy algorithm is O(n log n), where n is the number of activities. This complexity arises from the initial sorting step. The subsequent iteration through the sorted activities takes linear time.
In conclusion, the activity selection problem with time intervals, weights, and values can be efficiently solved using a greedy algorithm. The algorithm selects activities based on their finish times, ensuring that no two activities overlap, and maximizes the total value of the selected activities.