  The reachability problem is to determine if there exists a path from one
vertex to another 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 $O(n^{1/2 + \epsilon})$ space. In 2018, the
space bound was improved to $\tilde{O}(n^{1/3})$ by Ashida and Nakagawa. In
this paper, we show that reachability in an $n$ vertex grid graph can be
decided by an algorithm using $O(n^{1/4 + \epsilon})$ space and polynomial time
simultaneously.
