An O ( n ϵ ) Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs

Abstract

Given a graph G and two vertices s and t in it, graph reachability is the problem of checking whether there exists a path from s to t in G . We show that reachability in directed layered planar graphs can be decided in polynomial time and O ( n ϵ ) space, for any ϵ > 0. The previous best-known space bound for this problem with polynomial time was approximately O (√ n ) space (Imai et al. 2013). Deciding graph reachability in SC (Steve's class) is an important open question in complexity theory, and in this article, we make progress toward resolving this question.