Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection
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arXiv
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| Autori principali: | , , , , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866908923440660480 |
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| author | Dudek, Bartłomiej Fischer, Nick Gokaj, Geri Jin, Ce Künnemann, Marvin Mao, Xiao Redžić, Mirza |
| author_facet | Dudek, Bartłomiej Fischer, Nick Gokaj, Geri Jin, Ce Künnemann, Marvin Mao, Xiao Redžić, Mirza |
| contents | 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 $O(|S|^ω)$, 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 $O(N^{2-ε})$, 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 $O(|S|^ω)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28843 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection Dudek, Bartłomiej Fischer, Nick Gokaj, Geri Jin, Ce Künnemann, Marvin Mao, Xiao Redžić, Mirza Data Structures and Algorithms 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 $O(|S|^ω)$, 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 $O(N^{2-ε})$, 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 $O(|S|^ω)$ 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. |
| title | Classifying Identities: Subcubic Distributivity Checking and Hardness from Arithmetic Progression Detection |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2603.28843 |