Space-Efficient Algorithms for Reachability in Geometric Graphs

TL;DR

This paper develops space-efficient algorithms for the Reachability problem in various geometric graph classes, improving space complexity for intersection graphs of Jordan regions, chordal graphs, and penny graphs.

Contribution

It introduces novel space-efficient algorithms for Reachability in geometric graphs, including polynomial-time methods with sublinear space bounds.

Findings

Reachability in Jordan region graphs solved in polynomial time with $O(m^{1/2}\log n)$ space.

Chordal graphs reachability achieved with similar space complexity.

Penny graphs have a polynomial-time algorithm using $O(n^{1/4+\epsilon})$ space.

Abstract

The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem. For intersection graphs of Jordan regions, we show how to obtain a "good" vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and $O(m^{1/2}\log n)$ space, where $n$ is the number of Jordan regions, and $m$ is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial-time and $O(m^{1/2}\log n)$ space algorithm, where $n$ and $m$ are the number of vertices and edges, respectively. However, we use a more involved technique for unit contact disk graphs (penny graphs) and obtain a better algorithm. We show that for every $\epsilon> 0$, there exists a polynomial-time algorithm that can solve Reachability in an $n$ vertex directed penny graph, using $O(n^{1/4+\epsilon})$ space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.