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. since the improvement requires additional optimization.

翻译：Chi et al. (STOC' 22) 在最近一份突破性论文中, Chi et al. (STOC' 22) 引入了 $\ tilde{O} (n\ frac{ 3+\ omega}2 ⁇ ) 时间算法, 在两个维度的平方基体( $\ timen n$) 和 $O( n) 和 $( n) 美元( n) 。 这大大改进了以前的 $( \\ frac{ { 12+\ omega} +\ omga} 时间算法, 从而改进了其应用范围。 其他几个应用程序涉及在矩形矩阵矩阵矩阵之间, 即使 Chi et al. 等的算法似乎适用于矩形的两平方基等方基等, 但总则不简单化。 在本文中, 我们用这个较快的算法来改进矩形矩阵矩阵矩阵矩阵矩阵矩阵的运行时间。 我们接下来使用这个较快的运行时间来改进以下的矩形M- Min- Pintravelillateal- dashaltravelopal dust Paltravelopdal dash