Deterministic Monotone Min-Plus Product and Convolution

Abstract

The Monotone Min-Plus Product problem is a useful primitive that has seen many algorithmic applications over the past decade. It also generalizes various other structured Min-Plus products studied in the literature, such as Bounded Difference Min-Plus Product and Bounded Integer Min-Plus Product. In this problem, we are given two n× n integer matrices A and B, where each row of B is a monotone non-decreasing sequence of integers from {1,…,n}, and the goal is to compute their Min-Plus product, defined as the n× 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 running time 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 𝒪̃(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.