2022/04/09 by Shucheng Chi, Chi, Shucheng, Ran Duan +5 · 3 citations
Computer Science · Engineering · #Coding theory and cryptography #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #F.2.2 #FOS: Computer and information sciences #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2204.04500
openalex publication_date 2022/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we show that the time complexity of monotone min-plus product of two n× n matrices is O(n(3+ω)/2)=O(n2.687), where ω< 2.373 is the fast matrix multiplication exponent [Alman and Vassilevska Williams 2021]. That is, when A is an arbitrary integer matrix and B is either row-monotone or column-monotone with integer elements bounded by O(n), computing the min-plus product C where Ci,j=mink\Ai,k+Bk,j\ takes O(n(3+ω)/2) time, which greatly improves the previous time bound of O(n(12+ω)/5)=O(n2.875) [Gu, Polak, Vassilevska Williams and Xu 2021]. Then by simple reductions, this means the following problems also have O(n(3+ω)/2) time algorithms: (1) A and B are both bounded-difference, that is, the difference between any two adjacent entries is a constant. The previous results give time complexities of O(n2.824) [Bringmann, Grandoni, Saha and Vassilevska Williams 2016] and O(n2.779) [Chi, Duan and Xie 2022]. (2) A is arbitrary and the columns or rows of B are bounded-difference. Previous result gives time complexity of O(n2.922) [Bringmann, Grandoni, Saha and Vassilevska Williams 2016]. (3) The problems reducible to these problems, such as language edit distance, RNA-folding, scored parsing problem on BD grammars. [Bringmann, Grandoni, Saha and Vassilevska Williams 2016]. Finally, we also consider the problem of min-plus convolution between two integral sequences which are monotone and bounded by O(n), and achieve a running time upper bound of O(n1.5). Previously, this task requires running time O(n(9+√(177))/12) = O(n1.859) [Chan and Lewenstein 2015].