Comment by giomasce 16 hours ago
What Every Programmer Should Know About Enumerative Combinatorics -> Nothing.
It can be interesting to know something (or even a lot) about enumerative combinatorics, and certainly there are some specific programming contexts in which that's a hard prerequisite, but it's not a topic that necessarily concerns every programmer.
OTOH I think it would greatly help programmers, especially beginners, to have fewer click baity titles around.
This is easily verified by the notion that the overwhelmingly vast majority of programmers (myself included) probably know very little of the topic. Seemingly without that causing a lot of issues.
IMHO math in general is overrated for general purpose programming. I had plenty of math in college in the early nineties. I rarely need or use any of it. And when I do, I need to look up a lot of stuff for the simple reason that it's been decades since I last needed that knowledge. Very basic stuff even. Like highschool trigonometry (did some stuff with that a while back). Most programmers are just glorified plumbers that stick things together that others have built. They aren't designing new databases (for example) but simply using them. Which tends to be a lot easier. Though it helps to understand their general design and limitations. And if you are going to build a database, you might want to read up on a thing or two.
There is a wide range of esoteric topics you can dive into and learn a lot about. Diving into some of those in university is useful because it prepares you for a lifetime of needing to learn to wrap your head around random weird shit constantly that you need to understand to do the job. The point is not learning all that stuff upfront but simply learning enough that you can learn more when you need to. So, studying math and some other topics is a good preparation for that. You'll forget most of it if you don't use it. But when you need to, refreshing what you knew isn't that big of a deal.
The skill isn't in knowing that stuff but in being able to master that stuff.
Certain fields need it more than others. Graphics and vide game development needs more math than web app development (well, usually. Sometimes you need to implement a formula), including trigonometry.
I used a bunch of trigonometry when I was making 2D action games, getting characters to move about the screen and move smoothly at all sorts of angles, for one example. I also used Sine functions a lot for UI animations, making things looks like they're hovering or oscillating up and down.
I think one of the benefits of these classes, though, and university classes in general, is that even if you don't use or really remember the specifics decades later, you're at least aware of how these problems can be solved, and can look up and verify potential solutions much quicker than if you hadn't ever been exposed to it at all.
There's value in "being able to master that stuff," and there's value in "having mastered that stuff." The latter lets you trim a lot of possible designs from your search space nearly instantly, letting you focus on routes which are actually viable. The former is only of similar power when you know the design in advance or there aren't many possible solutions.
For a simple example, suppose you need to operate on `n` permutations of an enormous collection of data (far more than fits in RAM or disk), and you need those permutations to be re-usable.
One simple solution is to shuffle the indices `n` times and store the results in your cluster, but even the shuffle process is slow with normal techniques because of inter-machine random-access bandwidth issues. When using those shuffled indices for anything, you're again bandwidth-limited if the task doesn't require access of every index.
With just a tiny bit of a math background, you'll recognize that an O(1)-state shuffle is possible, where you can create some `Permutation` object with a `permute()` method, taking in an index and outputting the corresponding index in your hypothetical shuffle. That permutation will be CPU-bound rather than bandwidth-bound.
The problem with "being able to master stuff" is that your search process in the design space is slow. If I went and told you that an O(1)-state shuffle existed and would be good for the problem, sure, you'd be able to go code that up without issues. What's the chance that you'd even know to try though?
> wide range of esoteric topics ... prepares you for a lifetime of learning
That's part of it, but each of those esoteric topics also give new ideas something to latch onto. Our brains are associative, and being able to look at a new thing and tie it to a few esoteric concepts is a bit of a superpower, even if the association is weak. The difference between knowing nothing other than how to learn and knowing what's vaguely potentially possible or not is weeks or months of research. It's the difference between having to do the dumb, slow thing and being the person promoted for saving $1m/yr fixing whatever you wrote. You can get by for a long time, maybe your entire career, just making shit work, but if you're looking for more money or prestige then there are better routes.
> What's the chance that you'd even know to try though?
> [...] esoteric topics also give new ideas something to latch onto. Our brains are associative, and being able to look at a new thing and tie it to a few esoteric concepts is a bit of a superpower, even if the association is weak. The difference between knowing nothing other than how to learn and knowing what's vaguely potentially possible or not is weeks or months of research.
The only point I'd add to your paragraph is that this applies to every domain when you're on the job, not just math. I live in constant terror of discovering that other disciplines solved my problem like a hundred years ago.
What "O(1)-state shuffle" could you possibly be talking about? It takes `O(nlogn)` space to store a permutation of list of length n. Any smaller and some permutations will be unrepresentable. I am very aware of this because shuffling a deck of cards correctly on a computer requires at least 200 random bits.
If the requirements are softer than "n random permutations", there might be a lot of potential solutions. It is very easy to come up with "n permutations" if you have no requirements on the randomness of them. Pick the lowest `k` such that `n < k!`, permute the first k elements leaving the rest in place, and now you have n distinct permutations storeable in `O(log(n)` (still not O(1) but close).
I know this is not really your point, but misusing `O(1)` is a huge pet peeve of mine.
It's O(1) if you don't need access to every permutation (common in various monte carlo applications). 64-128 bits of entropy is good enough for a lot of applications, and that's all you get from any stdlib prng, so that's what I was comparing it to.
Those sorts of applications would tend to not work well with a solution leaving most elements in the same place or with the same relative ordering.
> IMHO math in general is overrated for general purpose programming. I had plenty of math in college in the early nineties. I rarely need or use any of it. And when I do, I need to look up a lot of stuff
The value isn't in knowing how to do math, it's in knowing when.
The value of a math class is far less in learning and remembering exactly how and, far more in learning what you can do with it so you can spot possible solutions when they arise. Expanding your mental toolkit.
Yep, also with little in the way of motivating examples and a pile of mathematics to sift through, its hard to devote the time to this.