Ce Jin

Bartłomiej Dudek, Nick Fischer, Geri Gokaj et al.

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.

Ce Jin, Jaewoo Park, Barna Saha et al.

The Monotone Min-Plus Product problem is a useful primitive that has seen many algorithmic applications over the past decade. In this problem, we are given two $n\times n$ integer matrices $A$ and $B$, where each row of $B$ is a monotone non-decreasing sequence of integers from $\{1,\dots,n\}$, and the goal is to compute their Min-Plus product, defined as the $n\times n$ matrix $C$ with $C_{i,j} = \min_{k}\{A_{i,k} + B_{k,j}\}$. The fastest known algorithm for this task [Chi, Duan, Xie, and Zhang, STOC'22] runs in $n^{(ω+3)/2+o(1)} = O(n^{2.686})$ time, significantly improving over the brute-force cubic algorithm. However, its main disadvantage is that it requires randomization, which is then inherited by all downstream applications. Our main result is a deterministic algorithm for Monotone Min-Plus product with the same time complexity $n^{(ω+3)/2+o(1)} = O(n^{2.686})$ as its randomized counterpart, improving upon the previous deterministic bound $O(n^{2.875})$ [Gu, Polak, Vassilevska Williams, and Xu, ICALP'21]. Our derandomization also applies to previously studied extensions and variants (e.g., [Dürr, IPL'23]), including rectangular matrices, bounded range $[n^μ]$, and column-monotone matrices. As an immediate consequence, we derandomize state-of-the-art algorithms for multiple problems, including Language Edit Distance, RNA Folding, Optimum Stack Generation, unweighted Tree Edit Distance, Batched Range Mode, and Approximate All-Pairs Shortest Paths. Our techniques also yield a deterministic algorithm for the Monotone Min-Plus Convolution problem that runs in $n^{1.5+o(1)}$ time, nearly matching the best known randomized time complexity $\widetilde{O}(n^{1.5})$ [Chi, Duan, Xie, and Zhang, STOC'22]. This algorithm can be used to derandomize state-of-the-art algorithms for Jumbled Indexing for binary strings and several variants of Knapsack.