# Faster Min-Plus Product for Monotone Instances

In this paper, we show that the time complexity of monotone min-plus product of two n× n matrices is Õ(n^(3+ω)/2)=Õ(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 Õ(n^(3+ω)/2) time, which greatly improves the previous time bound of Õ(n^(12+ω)/5)=Õ(n^2.875) [Gu, Polak, Vassilevska Williams and Xu 2021]. Then by simple reductions, this means the following problems also have Õ(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 Õ(n^2.824) [Bringmann, Grandoni, Saha and Vassilevska Williams 2016] and Õ(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 Õ(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 Õ(n^1.5). Previously, this task requires running time Õ(n^(9+√(177))/12) = O(n^1.859) [Chan and Lewenstein 2015].

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