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| Complexity Analysis |
 Title text: PERPETUALLY OPTIMISTIC CASE: Early in the execution, our research group makes a breakthrough in proving P=NP. |
Explanation
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This explanation may be incomplete or incorrect: Created by a PROBABLY DETERMINISTIC 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 complexity of the algorithm is written in Big O notation as O(n log n), expressing the asymptotic runtime of the algorithm as the number of inputs to it grows larger and larger.
The comic's joke involves taking the terms "best case" and "worst case" far more broadly and literally than intended. Cueball presents not just the best/worst cases for the data input into the function, but also the global environment as a whole, taking in factors such as the United States Congress which should fall far outside the algorithm's scope.
In particular, the joke regards the analysis of a closed system, which is common in engineering. An algorithm's "best case" is typically its runtime when its inputs have optimal values and it runs in as little time as possible. One example would be a sorting algorithm that is called with an already-sorted list of numbers; an algorithm may only need to check each item in the list, in one pass, to confirm this, compared with having to compare an arbitrary number of items against an arbitrary number of others across a number of cycles. The worst case would be when a list is 'unsorted' in a way that presents the maximum number of challenges and actions to the sorting algorithm (possibly, but not necessarily, when presented with the initial list exactly in the wrong order/reversed). These two limits can each be given by an O-notation, but a single O-notation generally indicates the mean complexity of operation encountered for all inputs.
The joke here is that not only does this run quicker than the mean O(n log n), being terminated early because it is deemed to be 'unnecessary', but its runtime appears to be an hour shorter still because of an act of Congress changing daylight saving time, giving it an end time (in local time) that is an hour less than it would have been under other circumstances. Potentially this would result in an end time that is recorded as earlier than its start time (depending on how the times are handled), and therefore an apparently negative 'runtime'. Daylight saving time is a recurrent theme on xkcd, and it is clear that Randall is not a fan, so Congress making surprise DST changes 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. (This movie has been referenced before in 1076: Groundhog Day.) If the hardware running the algorithm is present in this kind of loop then it may also reset to a previous time before it gets finished, meaning the algorithm would never terminate. This gives rise to a philosophical question about the movie as to whether the whole world is reset after every day, or just the town where the movie takes place. If it is just the town, and you could still connect to their hardware from outside, then from that perspective the algorithm would appear to be taking an interminably long time to run. If the whole world resets, since people (aside from the movie's main character) do not experience the reset, it would only appear to take as long as it does once the last (non-resetting cycle) leads it into the expected following day.
This may be an indirect reference to the halting problem, a famous problem in computer science. The halting problem is undecidable, meaning that no general algorithm can tell whether a given algorithm will halt, but the widely accepted traditional proof of this relies on external action on details of a system considered closed.
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(nk). However, P vs. NP is a Millennium Prize Problem for a reason; most computer scientists expect that P≠NP, so hoping for a breakthrough in proving P=NP is "perpetually optimistic". This may be a reference to optimism bias and the planning fallacy, whereby people tend to assume that the most favourable outcome will be the most likely.
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.]
- [Table Heading]
- Results of algorithm complexity analysis:
- [Row 1]
- Average case
- O(n log n)
- [Row 2]
- Best case
- Algorithm turns out to be unnecessary and is halted, then Congress enacts surprise daylight saving time and we gain an hour
- [Row 3]
- Worst case
- Town in which hardware is located enters a Groundhog Day scenario, algorithm never terminates
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