Math

This is a collection of my thoughts on AI and problem-solving, which have remained relatively constant over the past few years, and some comments on recent efforts in this direction. Recently, Deepmind claimed that they had made significant progress on the IMO Grand Challenge, and their automated systems called AlphaProof and the buffed version of their original AlphaGeometry, called AlphaGeometry 2 now perform at the IMO silver level. Having monitored this field in the past on-and-off, this set off some flags in my head, so I read through the whole post here, as well as the Zulip discussion on the same here....

The other day I had a discussion with someone about how to implement FFT-based convolution/polynomial multiplication - they were having a hard time squeezing their library implementation into the time limit on this problem, and it soon turned into a discussion on how to optimize it as much as possible. It turned out that the bit-reversing part of their iterative implementation was taking a pretty large amount of time, so I suggested not using bit-reversal at all, as is done in a few libraries....

This post was originally written on Codeforces; relevant discussion can be found here. Introduction A lot of people shy away from solving (mathematical) recurrences just because the theory is not very clear/approachable due to not having an advanced background in math. As a consequence, the usual ways of solving recurrences tend to be: Find the first few terms on OEIS. Guess terms from the rate of growth of the recurrence (exponential rate of growth means you can sometimes estimate the exponential terms going from largest to smallest — though this fails in cases where there is a double-root of the characteristic equation) Use some theorem whose validity you can’t prove (the so-called characteristic equation method) Overkill using generating functions But this doesn’t have to be the case, because there is a nice method you can apply to solve equations reliably....

This post was originally written on Codeforces; relevant discussion can be found here. Motivation On a computer science discord server, someone recently asked the following question: Is the master theorem applicable for the following recurrence? \(T(n) = 7T(\lfloor n / 20 \rfloor) + 2T(\lfloor n / 8 \rfloor) + n\) There was some discussion related to how \(T\) is monotonically increasing (which is hard to prove), and then someone claimed that there is a solution using induction for a better bound....

This post was originally written on Codeforces; relevant discussion can be found here. Disclaimer: This post (and all of my other posts, unless specified otherwise) is 100% ChatGPT-free — there has been no use of any AI/ML-based application while coming up with the content of this post. The reason and an appeal There is a lot of AI-generated content out there these days that sounds plausible and useful but is absolute garbage and contributes nothing beyond a list of superficial sentences — even if it has content that is true (which is a big IF, by the way), it generates content that you could have just looked up on your favorite search engine....

This post was originally written on Codeforces; relevant discussion can be found here. TL;DR The Floyd-Warshall algorithm is a special case of the more general case of aggregation over paths in a graph — in this post, we look at a few examples and point to some references for further study. The algebraic path problem constitutes a family of problems that can be solved using similar techniques. This and this paper develop the theory of semirings in this context, and this treats some computational aspects of it....

This post was originally written on Codeforces; relevant discussion can be found here. There were a few instances on CF according to which it seems that quite a few people aren’t very comfortable with floor and ceiling functions (and inequalities in general) — for instance, here and here. This inspired me to write a hopefully short post on how to systematically work with these topics. Note that I will not define what a real number is, or do anything that seems too non-elementary....

This post was originally written on Codeforces, relevant discussion can be found here. This post is intended to be an introduction to permutations for beginners, as there seem to be no resources other than cryptic (for beginners) editorials that talk about some cycles/composition, and people unfamiliar with permutations are left wondering what those things are. We’ll break the post into three major parts based on how we most commonly look at permutations....

This post was originally written on Codeforces; relevant discussion can be found here. Disclaimer: This is not an introduction to greedy algorithms. Rather, it is only a way to formalize and internalize certain patterns that crop up while solving problems that can be solved using a greedy algorithm. Note for beginners: If you’re uncomfortable with proving the correctness of greedy algorithms, I would refer you to this tutorial that describes two common ways of proving correctness — “greedy stays ahead” and “exchange arguments”....

This post was originally written on Codeforces; relevant discussion can be found here. Recently someone asked me to explain how to solve a couple of problems which went like this: “Find the expected time before XYZ happens”. Note that here the time to completion is a random variable, and in probability theory, such random variables are called “stopping times” for obvious reasons. It turned out that these problems were solvable using something called martingales which are random processes with nice invariants and a ton of properties....