It’s okay, you’re not the first to point it out to me!
I’d be interested to hear how you would explain it when you’re at a more appropriate device.
I’d be interested to hear how you would explain it when you’re at a more appropriate device.
The O(g(n)) part says f is asymptotically bounded by g(n) up to some constant factor.
The fact that f looks innocuous but is secretly actually this complicated thing is perhaps the stumbling block for people. The actual O(*) part is just "line stays at or below other line once you go far enough out" (with a slight tweak to allow you to pre-stretch the bounding line).
As for what people care about, it depends on what they're doing and what they know about the input distribution. If you're concerned about DOS attacks or response time in a game, then worse case is of primary importance. But if you're paying for wall time then you measure a hash table lookup by its O(1) average time, not its O(n) worst case time.
Maybe this comes across as nitpicky to some, but that doesn't make the text any less incorrect, unfortunately. It's an extremely common misconception, however, and I would say that a huge proportion of students that go through an introductory course on this subject come out thinking the same thing.
> It’s okay, you’re not the first to point it out to me!
It would be nice if you corrected it - there are already too many people making this mistake online.
I really just want to find the way of describing this that won’t net me comments like yours. It is very disheartening to spend so much time on something, and really try to do the topic justice, to be met with a torrent of “this is wrong, that’s wrong, this is also wrong.” Please remember I am a human being trying to do my best.
And what made you think they were right?
> I really just want to find the way of describing this that won’t net me comments like yours.
Do research. There's a large literature on algorithmic complexity ... maybe start with https://en.wikipedia.org/wiki/Big_O_notation
> It is very disheartening to spend so much time on something, and really try to do the topic justice, to be met with a torrent of “this is wrong, that’s wrong, this is also wrong.” Please remember I am a human being trying to do my best.
non sequitur
For example: "the function x^2 is O(x^3)" is a valid sentence in big-O notation, and is true.
Big O is commonly used in other places besides analysis of algorithms, such as when truncating the higher-order terms in a Taylor series approximation.
Another example is in statistics and learning theory, where we see claims like "if we fit the model with N samples from the population, then the expected error is O(1/sqrt(N))." Notice the word expected - this is an average-case, not worst-case, analysis.
Using "always" is definitely wrong, but that isn't really what makes this wrong. BonoboTP caught another thing that I missed which tells me that you misunderstand some of these concepts on a somewhat fundamental level. Quote:
> You may sometimes see the Greek letter "theta", Θ, instead of the letter O in big O notation. This is used when the best and worst case have the same order of growth. We can't use it for bubble sort because the best case is O(n) and the worst case is O(n^2).
Big O, Ω, and Θ are ways to describe asymptotic bounds on runtime. You can apply them in best, average, or worst case analyses. For example, linear search is O(1), Ω(1), and Θ(1) in the best case but it is O(n), Ω(n), and Θ(n) in the worst case.
> I really just want to find the way of describing this that won’t net me comments like yours. It is very disheartening to spend so much time on something, and really try to do the topic justice, to be met with a torrent of “this is wrong, that’s wrong, this is also wrong.” Please remember I am a human being trying to do my best.
I'm sorry.. but what? You posted some learning material on the web which is wrong. I am just correcting you, and in a civil manner, mind you. I'm not exactly sure what you want me to do. Should we all just pretend that there aren't errors in what you posted so we don't hurt your feelings?
It's not clear to me based on what you're written what I should change. With the "always" wording changed, what else do I need to do for this to be correct? Ideally I want to avoid talking about asymptotic bounds.
HN (and forums) can be blunt in this way (cf. Cunningham's Law). When you put something up, most of the reaction will be about how it's wrong. Things being correct is just the default, people won't comment on how correct something is. Try to take it not personally and more like a helpful piece of info. Ideally you'd want all statements to be correct. Textbook authors often incentivize hardcore nitpicking by mentioning the submitters in the acknowledgments or even cash. Try to take it in that manner.