Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection
Hey there, little explorer! 🚀
Imagine you have some special toys, like building blocks, and some rules for how they fit together. Like, if you put a red block and a blue block together, what happens?
Scientists (grown-ups who love puzzles!) want to find the fastest way to check if these rules always work for ALL the blocks. It's like checking if "red + blue = blue + red" is always true, no matter which blocks you pick!
This paper is super exciting because they found a NEW, super-duper fast way to check one of these rules, called "distributivity." It's like finding a shortcut to make sure all your blocks play nicely together, much quicker than before! 🏎️💨
So, they made computers smarter and faster at checking rules! Yay! 🎉
arXiv:2603.28843v1 Announce Type: new Abstract: We revisit the complexity of verifying basic identities, such as associativity and distributivity, on a given finite algebraic structure. In particular, while Rajagopalan and Schulman (FOCS'96, SICOMP'00) gave a surprising randomized algorithm to verify associativity of an operation $\odot: S\times S\to S$ in optimal time $O(|S|^2)$, they left the open problem of finding any subcubic algorithm for verifying distributivity of given operations $\odot,\oplus: S\times S\to S$. Our results are as follows: * We resolve the open problem by Rajagopalan and Schulman by devising an algorithm verifying distributivity in strongly subcubic time $O(|S|^\omega)$, together with a matching conditional lower bound based on the Triangle Detection Hypothesis. *
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Abstract:We revisit the complexity of verifying basic identities, such as associativity and distributivity, on a given finite algebraic structure. In particular, while Rajagopalan and Schulman (FOCS'96, SICOMP'00) gave a surprising randomized algorithm to verify associativity of an operation $\odot: S\times S\to S$ in optimal time $O(|S|^2)$, they left the open problem of finding any subcubic algorithm for verifying distributivity of given operations $\odot,\oplus: S\times S\to S$. Our results are as follows:
- We resolve the open problem by Rajagopalan and Schulman by devising an algorithm verifying distributivity in strongly subcubic time $O(|S|^\omega)$, together with a matching conditional lower bound based on the Triangle Detection Hypothesis.
- We propose arithmetic progression detection in small universes as a consequential algorithmic challenge: We show that unless we can detect $4$-term arithmetic progressions in a set $X\subseteq{1,\dots, N}$ in time $O(N^{2-\epsilon})$, then (a) the 3-uniform 4-hyperclique hypothesis is true, and (b) verifying certain identities requires running time~$|S|^{3-o(1)}$.
- A careful combination of our algorithmic and hardness ideas allows us to \emph{fully classify} a natural subclass of identities: Specifically, any 3-variable identity over binary operations in which no side is a subexpression of the other is either: (1) verifiable in randomized time $O(|S|^2)$, (2) verifiable in randomized time $O(|S|^\omega)$ with a matching lower bound from triangle detection, or (3) trivially verifiable in time $O(|S|^3)$ with a matching lower bound from hardness of 4-term arithmetic progression detection.
- We obtain near-optimal algorithms for verifying whether a given algebraic structure forms a field or ring, and show that \emph{counting} the number of distributive triples is conditionally harder than verifying distributivity.
Comments: To appear at STOC 2026
Subjects:
Data Structures and Algorithms (cs.DS)
Cite as: arXiv:2603.28843 [cs.DS]
(or arXiv:2603.28843v1 [cs.DS] for this version)
https://doi.org/10.48550/arXiv.2603.28843
arXiv-issued DOI via DataCite
Submission history
From: Bartlomiej Dudek [view email] [v1] Mon, 30 Mar 2026 17:42:29 UTC (84 KB)
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