In a recent breakthrough paper, Chi et al. (STOC'22) introduce an $\tilde{O}(n^{\frac{3 + \omega}{2}})$ time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions $n \times n$ and entries bounded by $O(n)$. This greatly improves upon the previous $\tilde O(n^{\frac{12 + \omega}{5}})$ time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: $M$-bounded Single Source Replacement Path, Batch Range Mode, $k$-Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al.

翻译：在近期一项突破性工作中，Chi等人（STOC'22）提出了一种$\tilde{O}(n^{\frac{3 + \omega}{2}})$时间复杂度的算法，用于计算两个维度为$n \times n$且元素取值不超过$O(n)$的方阵之间的单调最小加乘积。该算法极大地改进了此前$\tilde O(n^{\frac{12 + \omega}{5}})$时间复杂度的算法，进而改进了其应用场景的界。其他若干应用涉及矩形矩阵间的单调最小加乘积，尽管Chi等人的算法似乎可适用于矩形情形，但其推广并不直接。本文提出了Chi等人算法的推广版本，用于求解值域受多项式约束的矩形矩阵的单调最小加乘积。随后，我们利用这一更快的算法改进了以下矩形单调最小加乘积应用问题的运行时间：$M$有界单源替换路径、批处理范围众数、$k$-Dyck编辑距离以及全对最短路径的2近似算法。此外，我们还利用Chi等人的算法改进了无权树编辑距离的运行时间。