3,803 karma · joined March 21, 2015
If think the owner should give up its monetary value to satisfy your beliefs, you should be willing to do the same.
I think what's actually happening here is that they're doing the intersection on the DFAs and then producing a regex from the resulting DFA. The construction of a regex from a DFA is where things get ugly and weird.
To have sub-constant time it would have to go to 0 as n increases, as otherwise we could bound the time below by a constant. This isn't possible, as a computer will take at least 1 time unit on every input. So the expected time will always be at least 1.
it's not possible for it to behave differently as n increases because it doesn't have enough time to distinguish between different values of n once it's large enough.
You can approximate the runtime with a function that decreases forever, but the actual runtime will be constant for large enough n.
That's the sort of thing I would usually say to explain why we do things like use asymptomatic analysis and ignore polynomial factors. If you have a different solution, that's great, but you're no longer taking about the kind of runtime the article is about.
Of course, showing a constant upper-bound doesn't tell us that it isn't even faster than constant as in the proposition I was responding to. That's why I focused on the constant lower-bound.
All algorithms have a lower-bound on runtime of 1. If a sequence is decreasing and bounded below, it converges to a value [1]. If the sequence is discrete, that means it reaches exactly that limit and never deviates from it.
[1] https://en.wikipedia.org/wiki/Monotone_convergence_theorem
It's actually impossible to construct an algorithm that has its runtime decrease as the input size grows. You can do this for finitely many examples, but we're talking about asymptotics here. With discrete run-times and a lower-bound of 1 operation, the run-time will have to stop decreasing at some point and just stay at the same constant value for all but finitely-many exceptions. This makes it a constant-time algorithm.
A constant run-time is a counter-example to that caption though, as the problem doesn't take longer as the input grows. An example would be checking if a number is divisible by 2. You only need to check 1 bit, so it doesn't take any longer as you add more bits that the algorithm doesn't touch.
I ran the same simulation as you. After 100 iterations, the richest one had $71. After another 1,000 iterations, the richest one had $78 (of a total $133). After another 10,000 iterations, the richest one had 10^-165 dollars. They were still an extreme outlier with 95% of the total wealth, but he had almost nothing.
Joe Rogan being on Spotify is entirely about limiting the availability of his podcast. They're paying him to not make his episodes available outside of Spotify.
If you report a different number, that means some things in your mind changed. Those things may or may not include the level of pain you're experiencing.
Let's take an example with a bit more objectivity. If you ask me to rate how long a tv show is on a scale of 1-10 and I know the show is 30 minutes, how do I turn that into a 1-10 scale? Suppose I mostly watch shows with 12-minute episodes. This is relatively long, so I give it an 8.
Later I start watching hour-long shows. You ask me again how long this show is on a 1-10 scale. Now I put it at a 5. I don't think the show got shorter, and I don't experience it as being shorter. I just changed how I contextualize the length of shows for a 1-10 scale.
It would be silly to say that watching hour-long shows is a good way to shorten the mental experience of watching a 30-minute show.
Before a placebo treatment, you pick something at the higher end. After the treatment, you think maybe it helped, so you pick something on the lower end despite feeling the same.
Maybe you even feel worse than you did at the start of the study but you've gotten used to a higher baseline amount of pain and you're having a relatively good morning compared to that so you report a low number.
This is about how your self-reports change, not necessarily about how your state of mind changes.