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

)/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: • 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].

• 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].

• 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].