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Complexity Analysis

Title text: PERPETUALLY OPTIMISTIC CASE: Early in the execution, our research group makes a breakthrough on proving P=NP.

Explanation

This explanation may be incomplete or incorrect: Created by a PERPETUALLY OPTIMISTIC BOT - Please change this comment when editing this page. Do NOT delete this tag too soon.

Cueball is teaching about an algorithm's complexity. The average-case runtime of the algorithm is written as O(n log n), in Big O notation, expressing the asymptotic runtime of the algorithm as the number of inputs to it grows larger and larger. This is an error by Randall (or Cueball), as Big-O notation represents only the worst-case, not average-case time complexity.

The "best case" for an algorithm is typically its runtime when its inputs have optimal values and it runs in as little time as possible. The joke here is that not only does it run this quickly, but that its runtime is actually an hour shorter because of an act of Congress changing daylight saving time, potentially giving it negative 'runtime'. Daylight saving time is a recurrent theme on xkcd, and it is clear that Randall is not a fan. so that congress just gives us an hour in a surprise DST is another way for Randall to mock the concept.

The "worst case" refers to the movie Groundhog Day, in which the same events occur over and over in a sort of time loop. And thus if the hardware running the algorithm is stuck in this kind of loop that resets to a previous time before it ever gets finished then the algorithm would never terminate. In the movie we must expect that the whole world is reset after every day, and not just the city where the movie takes place. But if it is just a town outside the rest of the world, where you can still connect to their hardware from outside, then this situation could arise. For those in the town, the algorithm never uses much time, since they do not experience the reset of the day. That was only the main character who lived the same day over and over. For all other in the movie it was their first time living that day, every day. And if the main character did nothing everyone else would have the same day over and over.

This may be an indirect reference to the halting problem, a famous problem in computer science of determining whether a given algorithm will ever halt. The halting problem is undecidable, meaning that there is no general algorithm that can tell whether a given algorithm will halt. The movie has been referenced before in 1076: Groundhog Day.

The title text refers to perhaps an even more famous problem in computer science, P versus NP. This asks whether every problem whose solution can be quickly verified (in nondeterministic polynomial time, NP) can also be quickly solved (in polynomial time, P). The P-versus-NP problem is one of the seven Millennium Prize Problems, and as such has a $1 million prize for its solution. Presumably, the problem discussed here is in NP, so if P=NP, its worst-case runtime would be some polynomial O(n^k). However, P vs. NP is a Millennium Prize Problem for a reason, and most computer scientists expect that P != NP, so hoping for a breakthrough in proving P=NP is "perpetually optimistic".

Transcript

[Cueball is holding a presentation pointer stick, pointing to a table behind him that towers above him. The table has a heading above it and then two columns and three rows. the first column is slim and the second much broader.]

Results of algorithm complexity analysis:

Average case	O(n log n)
Best case	Algorithm turns out to be unnecessary and is halted, then Congress enacts surprise daylight saving time and we gain an hour
Worst case	Town in which hardware is located enters a Groundhog Day scenario, algorithm never terminates

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