# On Solving Reachability in Grid Digraphs using a Psuedoseparator

## TL;DR

This paper introduces a polynomial-time, space-efficient algorithm for the reachability problem in grid digraphs, utilizing a novel pseudoseparator to improve space bounds to approximately O(n^{1/4}).

## Contribution

It presents a new divide-and-conquer approach using pseudoseparators, achieving better space complexity for grid digraph reachability.

## Key findings

- Achieves O(n^{1/4 + ε}) space complexity for reachability.
- Introduces the concept of pseudoseparator for divide-and-conquer algorithms.
- Provides a polynomial-time algorithm with improved space bounds.

## Abstract

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 $O(n^{1/2 + \epsilon})$ 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 $O(n^{1/4 + \epsilon})$ 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.