On Solving Reachability in Grid Digraphs using a Psuedoseparator

TL;DR

We study reachability in grid digraphs and present a polynomial-time algorithm that uses $O(n^{1/4+\epsilon})$ space for an $n$-vertex grid digraph. The approach builds a compact auxiliary digraph $\textsf{Aux}_{\alpha}(G)$ that preserves reachability but is non-planar, and then uses a crossing-aware device called a pseudoseparator to enable a divide-and-conquer traversal with provably small space. A key technical contribution is constructing a $h^{1-\beta}$-pseudoseparator of size $O(h^{1/2+\beta/2})$ in $\tilde{O}(h^{1/2+\beta/2})$ space and solving reachability in the auxiliary graph through recursive calls and logspace subroutines, yielding the overall space bound. This improves prior bounds for grid-digraph reachability and highlights a novel use of separator-like ideas in non-planar, grid-derived graphs. The work opens avenues for tighter bounds and potential extensions to related planar and semi-planar graph classes.

Abstract

The reachability problem asks to decide if there exists a path from one vertex to another in a digraph. In a grid digraph, the vertices are the points of a two-dimensional square grid, and an edge can occur between a vertex and its immediate horizontal and vertical neighbors only. Asano and Doerr (CCCG'11) presented the first simultaneous time-space bound for reachability in grid digraphs by solving the problem in polynomial time and $O(n^{1/2 + ε})$ space. In 2018, the space complexity was improved to $\tilde{O}(n^{1/3})$ by Ashida and Nakagawa (SoCG'18). In this paper, we show that there exists a polynomial-time algorithm that uses $O(n^{1/4 + ε})$ space to solve the reachability problem in a grid digraph containing $n$ vertices. We define and construct a new separator-like device called pseudoseparator to develop a divide-and-conquer algorithm. This algorithm works in a space-efficient manner to solve reachability.

Figures (4)

• Figure 1: A grid digraph $G$ divided into subgrids and its corresponding auxiliary digraph $\textsf{Aux}_{\alpha}(G)$
• Figure 2: Edge crossings in an auxiliary grid. On the left, there is one block of the auxiliary digraph that contains edges that cross. The dotted edges are the ones whose existence is made necessary by Lemma \ref{['lem:psg']}. On the right, a subgrid which results in the auxiliary block on the left.
• Figure 3: Illustration of Lemma \ref{['lem:closer']}
• Figure 4:

Theorems & Definitions (28)

• theorem 1: Main Theorem
• definition 1
• definition 2
• lemma 1
• proof
• definition 3
• lemma 2
• proof
• lemma 3
• proof