What is the maximum number of activities that can be selected without overlapping and how can it be calculated using a greedy algorithm?

The maximum number of activities that can be selected without overlapping can be calculated using a greedy algorithm known as the Activity Selection Problem. This problem involves selecting a maximum number of activities from a given set of activities, where each activity has a start time and an end time, and the goal is to select the activities in such a way that no two activities overlap.

To solve this problem using a greedy algorithm, we can follow the following steps:

1. Sort the activities based on their end times in ascending order. This step ensures that the activity with the earliest end time comes first.

2. Initialize a variable, let's say "count," to keep track of the number of activities selected. Set count to 1, as we will always select the first activity.

3. Iterate through the sorted activities starting from the second activity. For each activity, check if its start time is greater than or equal to the end time of the previously selected activity. If it is, select the current activity and increment the count by 1.

4. Repeat step 3 until all activities have been considered.

5. The final value of count will represent the maximum number of activities that can be selected without overlapping.

The greedy algorithm works because by selecting the activity with the earliest end time, we create room for more activities to be selected in the remaining time slots. By always choosing the activity that starts after the previous activity ends, we ensure that no two activities overlap.

Here is an example to illustrate the algorithm:

Consider the following activities with their start and end times:
Activity 1: (1, 4)
Activity 2: (3, 5)
Activity 3: (0, 6)
Activity 4: (5, 7)
Activity 5: (3, 9)
Activity 6: (5, 9)
Activity 7: (6, 10)
Activity 8: (8, 11)
Activity 9: (8, 12)
Activity 10: (2, 14)

Step 1: Sorting the activities based on their end times:
Activity 3: (0, 6)
Activity 1: (1, 4)
Activity 10: (2, 14)
Activity 2: (3, 5)
Activity 5: (3, 9)
Activity 4: (5, 7)
Activity 6: (5, 9)
Activity 7: (6, 10)
Activity 8: (8, 11)
Activity 9: (8, 12)

Step 2: Initialize count = 1 (selecting the first activity)

Step 3: Iterate through the sorted activities:
- Activity 3: (0, 6) - Select this activity as it is the first one.
- Activity 1: (1, 4) - Cannot select this activity as it overlaps with the previous one.
- Activity 10: (2, 14) - Select this activity as it does not overlap with the previous one.
- Activity 2: (3, 5) - Cannot select this activity as it overlaps with the previous one.
- Activity 5: (3, 9) - Cannot select this activity as it overlaps with the previous one.
- Activity 4: (5, 7) - Cannot select this activity as it overlaps with the previous one.
- Activity 6: (5, 9) - Cannot select this activity as it overlaps with the previous one.
- Activity 7: (6, 10) - Cannot select this activity as it overlaps with the previous one.
- Activity 8: (8, 11) - Select this activity as it does not overlap with the previous one.
- Activity 9: (8, 12) - Cannot select this activity as it overlaps with the previous one.

Step 4: The final value of count is 3, which represents the maximum number of activities that can be selected without overlapping.

Therefore, the maximum number of activities that can be selected without overlapping in this example is 3, and it is calculated using the greedy algorithm described above.