Faster Min-Plus Product for Monotone Instances

In this paper, we show that the time complexity of monotone minplus product of two n x n matrices is (O) over tilde (n((3)(+omega)()/2)) = (O) over tilde (n(2.687)), where omega < 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 columnmonotone with integer elements bounded by O(n), computing the min-plus product C where C-i,C-j = min(k){A(i,k) + B-k,B-j} takes <(O)over tilde>(n((3)(+omega)()/2)) time, which greatly improves the previous time bound of (O) over tilde (n((12+omega)/5)) = (O) over tilde (n(2.875)) [Gu, Polak, Vassilevska Williams and Xu 2021]. Then by simple reductions, this means the case that A is arbitrary and the columns or rows of B are bounded-difference can also be solved in (O) over tilde (n((3)(+omega)()/2)) time, whose previous result gives time complexity of (O) over tilde (n(2.922)) [Bringmann, Grandoni, Saha and Vassilevska Williams 2016]. So the case that both of A and B are bounded-difference also has (O) over tilde (n((3)(+omega)()/2)) time algorithm, whose previous results give time complexities of (O) over tilde (n(2.824)) [Bringmann, Grandoni, Saha and Vassilevska Williams 2016] and (O) over tilde (n(2.779)) [Chi, Duan and Xie 2022]. Many problems are reducible to these problems, such as language edit distance, RNA-folding, scored parsing problem on BD grammars [Bringmann, Grandoni, Saha and Vassilevska Williams 2016]. Thus, their complexities are all improved. 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) over tilde (n(1.5)). Previously, this task requires running time (O) over tilde (n((9+root 177)/12)) = O(n(1.859)) [Chan and Lewenstein 2015].