Faster Min-Plus Product for Monotone Instances

Abstract

In this paper, we show that the time complexity of monotone min-plus product of two $n\times n$ matrices is $\tilde{O}(n^{(3+ω)/2})=\tilde{O}(n^{2.687})$, 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 $C_{i,j}=\min_k\{A_{i,k}+B_{k,j}\}$ takes $\tilde{O}(n^{(3+ω)/2})$ time, which greatly improves the previous time bound of $\tilde{O}(n^{(12+ω)/5})=\tilde{O}(n^{2.875})$ [Gu, Polak, Vassilevska Williams and Xu 2021]. Then by simple reductions, this means the following problems also have $\tilde{O}(n^{(3+ω)/2})$ 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 $\tilde{O}(n^{2.824})$ [Bringmann, Grandoni, Saha and Vassilevska Williams 2016] and $\tilde{O}(n^{2.779})$ [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 $\tilde{O}(n^{2.922})$ [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 $\tilde{O}(n^{1.5})$. Previously, this task requires running time $\tilde{O}(n^{(9+\sqrt{177})/12}) = O(n^{1.859})$ [Chan and Lewenstein 2015].