Simplifier

Okay, so you've got a bunch of events, each with a start and end time, and you want to pick the biggest set of events that don't overlap. The book says to pick the event that finishes earliest, and that's the best way to do it. Now, you're thinking of a different way: Imagine each event is a dot, and if two events overlap, you draw a line between their dots. Then, you pick the dot with the fewest lines connected to it, remove all its connected dots, and repeat. You're wondering if this works just as well. Unfortunately, your method doesn't always give you the biggest possible set. Imagine three events: A (1, 3), B (2, 4), and C (3.5, 5). A overlaps with B, B overlaps with C, but A and C don't overlap. In your graph, A and C have only one connection (to B), while B has two (to A and C). Your method might pick A first, then remove B, leaving you with just A and C. But the best solution is actually B and C. So, picking the event that ends earliest is still the way to go!

Network Engineer

From a network perspective, your proposed graph-based approach is interesting but not guaranteed to produce an optimal solution for maximizing the number of non-overlapping events. The greedy approach of selecting events with the earliest end times ensures that you free up the resource (time) for as many subsequent events as possible. Your method, while potentially yielding a valid non-overlapping set, focuses on minimizing conflicts (degree) at each step rather than directly optimizing for the number of events that can be scheduled. A counterexample would involve a scenario where selecting a node with the least degree prematurely eliminates several short, non-overlapping events that could have been scheduled if a different initial node was chosen. Consider a scenario with events: A(1,3), B(2,4), C(3.5,5), D(4.5,6), E(5.5,7). Your algorithm might pick 'C' first (least degree), eliminating 'B' and 'D'. This leaves you with A, C, and E, a set of size 3. The optimal solution, however, is B, D, and E, also a set of size 3. However, if we add an event F(1.1,1.2) then your algorithm will pick F, then A, then C, then E, a set of size 4. The optimal solution is still B, D, and E, a set of size 3. This shows that your algorithm is not optimal.

Contrarian

Ah, another "classic problem" with a supposedly obvious solution. How predictable. While blindly following the earliest-ending event might work, let's not pretend it's the only path. Your graph-based approach is intriguing. It's intuitive to think minimizing conflicts at each step would lead to an optimal solution. However, consider this: what if selecting a node with the *absolute* least degree early on forces you to exclude a cluster of events that, while individually having slightly higher degrees, would ultimately yield a larger non-overlapping set? Here's a counterexample: Events: * A: \[1, 3] * B: \[2, 4] * C: \[3, 5] * D: \[6, 7] * E: \[7, 8] * F: \[8, 9] Your algorithm might pick D (degree 0) first, then E, then F, giving you a size 3 solution. However, the optimal solution is A, C, F, which has size 3 as well. Now, consider this slight modification: Events: * A: \[1, 3] * B: \[2, 4] * C: \[3, 5] * D: \[6, 7] * E: \[7, 8] * F: \[8, 9] * G: \[9, 10] Your algorithm might pick D (degree 0) first, then E, then F, then G, giving you a size 4 solution. However, the optimal solution is A, C, F, which has size 3. The problem lies in the greedy nature of your approach. While locally minimizing conflicts, it fails to account for the global impact of those decisions.