$\tilde{O}(n^{1/3})$-Space Algorithm for the Grid Graph Reachability Problem

TL;DR

This paper presents a novel algorithm that solves the directed grid graph reachability problem using significantly less space, specifically $ ilde{O}(n^{1/3})$, while maintaining polynomial time complexity.

Contribution

It introduces the first $ ilde{O}(n^{1/3})$-space polynomial time algorithm for directed grid graph reachability, improving upon previous $ ilde{O}( oot{2}{}n)$ space bounds.

Findings

Achieves $ ilde{O}(n^{1/3})$ space complexity for grid graph reachability

Maintains polynomial time complexity

Improves space bounds over previous algorithms

Abstract

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.