Grid Graph Reachability

02/01/2019

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by   Rahul Jain, et al.

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The reachability problem is to determine if there exists a path from one vertex to the other in a graph. Grid graphs are the class of graphs where vertices are present on the lattice points of a two-dimensional grid, and an edge can occur between a vertex and its immediate horizontal or vertical neighbor only. Asano et al. presented the first simultaneous time space bound for reachability in grid graphs by presenting an algorithm that solves the problem in polynomial time and O(n^1/2 + ϵ) space. In 2018, the space bound was improved to Õ(n^1/3) by Ashida and Nakagawa. In this paper, we further improve the space bound and present a polynomial time algorithm that uses O(n^1/4 + ϵ) space to solve reachability in a grid graph with n vertices.