利用伪分离器求解网格有向图中的可达性问题
On Solving Reachability in Grid Digraphs using a Psuedoseparator
AI总结：
本文针对网格有向图可达性问题，提出一种基于新型伪分离器的分治算法，将空间复杂度从既有最优的Õ(n^{1/3})进一步降至O(n^{1/4+ε})，同时保持多项式时间。
摘要：
可达性问题要求判定在有向图中是否存在从一个顶点到另一个顶点的路径。在网格有向图中，顶点是二维正方形网格上的点，边只能出现在一个顶点与其直接的水平或垂直相邻顶点之间。Asano和Doerr（CCCG'11）提出了网格有向图可达性的第一个同时时间-空间界，他们以多项式时间和O(n^{1/2 + ε})空间解决了该问题。2018年，Ashida和Nakagawa（SoCG'18）将空间复杂度改进为\ ilde{O}(n^{1/3})。在本文中，我们证明了存在一个多项式时间算法，该算法使用O(n^{1/4 + ε})空间来求解包含n个顶点的网格有向图中的可达性问题。我们定义并构造了一种新的类似分离器的装置，称为伪分离器（pseudoseparator），以开发分治算法。该算法以空间高效的方式求解可达性问题。
英文摘要：
The reachability problem asks to decide if there exists a path from one vertex to another in a digraph. In a grid digraph, the vertices are the points of a two-dimensional square grid, and an edge can occur between a vertex and its immediate horizontal and vertical neighbors only. Asano and Doerr (CCCG'11) presented the first simultaneous time-space bound for reachability in grid digraphs by solving the problem in polynomial time and $O(n^{1/2 + ε})$ space. In 2018, the space complexity was improved to $\tilde{O}(n^{1/3})$ by Ashida and Nakagawa (SoCG'18). In this paper, we show that there exists a polynomial-time algorithm that uses $O(n^{1/4 + ε})$ space to solve the reachability problem in a grid digraph containing $n$ vertices. We define and construct a new separator-like device called pseudoseparator to develop a divide-and-conquer algorithm. This algorithm works in a space-efficient manner to solve reachability.
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