Greedy Algorithms Questions Long
The activity selection problem is a classic problem in computer science that involves selecting a maximum number of non-overlapping activities from a given set of activities, each with their own start and finish times. The goal is to find the optimal solution that maximizes the number of activities selected.
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 activity with the earliest finish time first.
2. Initialize an empty list to store the selected activities.
3. Select the first activity from the sorted list and add it to the selected activities list.
4. Iterate through the remaining activities in the sorted list. For each activity, check if its start time is greater than or equal to the finish time of the previously selected activity. If it is, add the current activity to the selected activities list.
5. Repeat step 4 until all activities have been considered.
6. Return the selected activities list as the optimal solution.
The greedy algorithm works for this problem because at each step, we choose the locally optimal choice (i.e., the activity with the earliest finish time) without considering the future consequences. By doing so, we ensure that we always select the maximum number of non-overlapping activities.
The time complexity of this greedy algorithm is dominated by the sorting step, which takes O(n log n) time, where n is the number of activities. The remaining steps take O(n) time, resulting in an overall time complexity of O(n log n).
In conclusion, the activity selection problem with time intervals can be efficiently solved using a greedy algorithm that selects activities based on their finish times. This approach guarantees an optimal solution by always choosing the locally optimal choice at each step.