19.4.2. Example Execution of the Algorithm

So, now let us look at a completely different problem, a problem called interval scheduling. So, suppose we have a special video class room, where we can deliver online lectures. Now, different teachers want to book the class room to deliver classes and each instructor has a slot that he would like to deliver this lecture in. So, instructor i has a slot, let us starts at a time s i and finishes it at f i. So, you have a slot which starts at s i and finishes at f i, now two instructors may have overlapping slot. So, the maybe somebody who wants the slot like this, so the blue slot starts before the red slot ends, so obviously both these slots cannot be in given bookings, because there were interfere with each other. So, our task is to look at the set of bookings and chooses subset which is feasible that is no two bookings that we choose interfere with each other.

The problem at hand involves scheduling classes for multiple instructors in a classroom that can only accommodate one instructor at a time. Each instructor provides their specific time slot defined by a start time (sᵢ) and end time (fᵢ). Given that overlapping time slots can't be scheduled together, the goal is to maximize the number of teachers who can deliver their lectures in the available time slots without conflicts.

Imagine a meeting room in an office that can be booked by different employees for meetings. If Employee A wants to meet from 10 AM to 11 AM and Employee B also wants the same room from 10:30 AM to 11:30 AM, both cannot meet simultaneously. Thus, the office must find a way to allow as many employees as possible to use the meeting room without scheduling conflicts.

So, broadly if we follow a greedy approach, this is what we would do: Among all the bookings that are not yet allocated and which are still available to allocate, we will pick one based on some local strategy, then we would remove all conflicting bookings, bookings that overlapped with this booking that we just allocated and somehow we have to argue that this sequence of bookings that we are choosing maximizes the number of teachers who get to use the room.

The greedy approach consists of making a series of local choices that seem best at the moment. In this scheduling problem, this means selecting the next available booking that doesn't conflict with previously allocated bookings. After choosing one booking, the algorithm then removes all other bookings that overlap with the chosen one from further consideration. The challenge is to ensure that this method actually maximizes the number of accepted bookings.

Think of it like packing a suitcase for a trip. Each clothing item has a specific size (represented by its time slot). You pick the first item that fits nicely into your suitcase (your first booking) and then discard all items that won't fit anymore (the overlapping bookings). By doing this sequentially, you want not only to fill your suitcase but to pack the maximum number of items possible, ensuring you have what you need for the trip.

So, let us look at some typical greedy strategies that one might wanted. So, one strategy might be to choose the booking whose start time is earliest, but it is not difficult to come up with the counter example.

In algorithm design, it's important to evaluate different greedy strategies. One naive strategy might involve always choosing the booking that starts the earliest. However, this can lead to suboptimal solutions because picking a booking too early may occupy the time slot for too long, preventing the accommodation of additional bookings. Counterexamples can demonstrate that this strategy fails in practice.

Imagine you're at a buffet with limited space to hold your plate. If you grab the first dish you see (the earliest start time), which is very large, you may miss an opportunity to fill your plate with a more varied selection of smaller dishes that come later. Thus, starting with the earliest may not always lead to the most delicious meal!

So, here is a fourth strategy, instead of choosing the one like we begin with whose start time is earliest, let us choose the one whose finished time is earliest. So, can we come up with a counter example or should we instead try to prove this is correct. So, in fact this strategy does work and let us see how we can prove it.

After testing multiple strategies, one effective approach emerges: always selecting the booking that finishes the earliest. This strategy helps maximize the remaining time for additional bookings that can follow. By focusing on ending times rather than starting times, we can create an optimal schedule. The effectiveness of this strategy can be proven mathematically, demonstrating that it never misses opportunities compared to other choices.

Consider a concert hall with multiple performances throughout a day. If you choose to attend the concert that finishes the earliest, you leave the concert hall early enough to catch another show, maximizing your enjoyment throughout the day. By prioritizing the earliest finale, you ensure that you can fit more performances into your schedule.

So, having shown that it is correct, let us just quickly look at how we would implement this and estimate the upper bound of the complexity. So, initially we sort the m bookings by finishing time, this takes time n log n for n bookings and now let us assume if the bookings are renumbered 1, 2 up to n in this sorted order.

To implement the greedy strategy for interval scheduling, the first step is to sort all bookings based on their finishing times. This sorting operation has a complexity of O(n log n). After sorting, the algorithm proceeds to select the earliest finishing booking that is still feasible and continues this process until no further bookings can be accommodated. This systematic approach ensures an optimal solution while maintaining an acceptable computational time.

Think of the algorithm like organizing a schedule for a series of events. First, you lay out all the events in order of when they end (like sorting), and then you carefully pick and finalize which events you can attend without overlapping timings. This structured method helps you experience the most events possible!

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