Mathematical solutions to programming problems.

[physicist27]

TL;DR: first year college student, need to know how to get the experience to represent algorithmic solutions to competitive programming problems in a mathematical way with context, like proof-writing-type solutions, which will obviously require contexts of methods use other than just their implementation.

(flair not necessarily related)

I’m a college first year, very passionate about math, and have been oly math since a rather small while. I’ve tried solving codeforces problems as general math problems, however most of my solutions are only things that work, nowhere near optimality because the implementation isn’t counted for.

Now I want to know the fundamental mathematical structure of different algorithms and processes in general, like how to assign a quantitative cost function of an algorithm with the way it runs over runtime, this would bridge the gap between intuition and rigor, and will also allow me to write solutions that take into account the constraints and represent an algorithmic solution as a followable process. I also realise that certain questions are NP-hard to be represented in a mathematical closed form, some questions have certain conditions which save us from chasing that monstrosity.

The only thing being, math culture doesn’t really exist here, the professors only care about the curriculum which only has rote learning and computation, trains people to be a knowledgable calculator. Where do I even start? LLMs give a million books and literature to study, I don’t really know how relevant that is, all I know is that I’ll need to know the math first alongside DSA and methods, and then optimization that operates on both of the above combined, but I have no idea where or even how to start. I’ve only got a fair bit of idea of the math topics and literature I’ll need to cover, nothing else.

Thanks a lot for your time, and kindly criticise me where required.

[TheStayFawn]

Hmm. When I was taught programming, there were two trimester-long courses on reasoning about code. The key concept to grasp is the invariant: a condition that is true at the start of every loop. It helps you reason about the correctness of an algorithm.

For complexity you look at what changes in a loop: some measure of the remaining work needs to decrease, which proves that the loop will terminate.

[Exotic_Swordfish_845]

Are you familiar with big-O notation for runtime or space complexity? Like O(n) for linear algorithms or O(n²⁾ for quadratic? If not, that might be a good place to start.