In a recent breakthrough paper, Chi et al. (STOC'22) introduce an $\tilde{O}(n^{\frac{3 + ω}{2}})$ 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 $\tilde O(n^{\frac{12 + ω}{5}})$ 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：在最近一篇突破性论文中，Chi等人（STOC'22）提出了一种$\tilde{O}(n^{\frac{3 + ω}{2}})$时间算法，用于计算两个维度为$n \times n$、元素值以$O(n)$为界的方阵之间的单调最小加性乘积。这显著改进了先前$\tilde O(n^{\frac{12 + ω}{5}})$时间算法，并因此改善了其应用场景的边界。若干其他应用涉及矩形矩阵之间的单调最小加性乘积，尽管Chi等人的算法似乎适用于矩形情形，但其推广并非直接可得。本文提出将Chi等人的算法推广至解决具有多项式有界值的矩形矩阵单调最小加性乘积问题。随后，我们利用这一加速算法改进了以下矩形单调最小加性乘积应用的运行时间：$M$有界单源替换路径、批量区间众数、$k$-Dyck编辑距离以及全对最短路径的2-近似解。此外，我们借助Chi等人的算法提升了无权树编辑距离的计算效率。