  The directed graph reachability problem takes as input an $n$-vertex directed
graph $G=(V,E)$, and two distinguished vertices $s$ and $t$. The problem is to
determine whether there exists a path from $s$ to $t$ in $G$. This is a
canonical complete problem for class NL. Asano et al. proposed an
$\tilde{O}(\sqrt{n})$ space and polynomial time algorithm for the directed grid
and planar graph reachability problem. The main result of this paper is to show
that the directed graph reachability problem restricted to grid graphs can be
solved in polynomial time using only $\tilde{O}(n^{1/3})$ space.
