用于单调递归的更快的Min-Plus乘积
Faster Min-Plus Product for Monotone Instances
摘要：
本文证明了两个$n\times n$矩阵的单调最小乘积的时间复杂度为$\tilde{O}(n^{(3+ω)/2})=\tilde{O}(n^{2.687})$，其中$ω< 2.373$是快速矩阵乘法指数[Alman and Vassilevska Williams 2021]。也就是说，当$A$是任意整数矩阵并且$B$是行单调或列单调的且整数元素以$O(n)$为界时，计算最小加积$C$，其中$C_{i,j}=\min_k\{A_{i,k}+B_{k,j}\}$需要$\tilde{O}(n^{(3+ω)/2})$时间，这大大改进了$\tilde{O}(n^{(12+ω)/5})=\tilde{O}(n^{2.875})$的先前时间界[Gu，Polak，Vassilevska Williams and Xu 2021]。然后通过简单的约简，说明以下问题也有$\tilde{O}(n^{(3+ω)/2})$时间算法：（1）$A$和$B$都是有界差分，即任意两个相邻项之差为常数。之前的结果给出了$\tilde{O}(n^{2.824})$ [Bringmann，Grandoni，Saha and Vassilevska Williams 2016]和$\tilde{O}(n^{2.779})$ [Chi，Duan and Xie 2022]的时间复杂度。(2)$A$是任意的，$B$的列或行是有界差的。之前的结果给出了$\tilde{O}(n^{2.922})$的时间复杂度[Bringmann，Grandoni，Saha and Vassilevska Williams 2016]。(3)这些问题可以归结为BD文法的语言编辑距离、RNA折叠、评分分析等问题。[Bringmann，Grandoni，Saha and Vassilevska Williams 2016].最后，我们还考虑了$O(n)$有界的两个单调整数序列之间的最小加卷积问题，得到了$\tilde{O}(n^{1.5})$的一个运行时间上界。以前，此任务需要运行时间$\tilde{O}(n^{(9+\sqrt{177})/12}) = O(n^{1.859})$ [Chan and Lewenstein 2015]。
英文摘要：
In this paper, we show that the time complexity of monotone min-plus product of two $n\times n$ matrices is $\tilde{O}(n^{(3+ω)/2})=\tilde{O}(n^{2.687})$, 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 $C_{i,j}=\min_k\{A_{i,k}+B_{k,j}\}$ takes $\tilde{O}(n^{(3+ω)/2})$ time, which greatly improves the previous time bound of $\tilde{O}(n^{(12+ω)/5})=\tilde{O}(n^{2.875})$ [Gu, Polak, Vassilevska Williams and Xu 2021]. Then by simple reductions, this means the following problems also have $\tilde{O}(n^{(3+ω)/2})$ time algorithms: (1) $A$ 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 $\tilde{O}(n^{2.824})$ [Bringmann, Grandoni, Saha and Vassilevska Williams 2016] and $\tilde{O}(n^{2.779})$ [Chi, Duan and Xie 2022]. (2) $A$ is arbitrary and the columns or rows of $B$ are bounded-difference. Previous result gives time complexity of $\tilde{O}(n^{2.922})$ [Bringmann, Grandoni, Saha and Vassilevska Williams 2016]. (3) 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 $\tilde{O}(n^{1.5})$. Previously, this task requires running time $\tilde{O}(n^{(9+\sqrt{177})/12}) = O(n^{1.859})$ [Chan and Lewenstein 2015].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
