TY - GEN

T1 - Space-Efficient Algorithms for Reachability in Directed Geometric Graphs

AU - Bhore, Sujoy

AU - Jain, Rahul

N1 - Publisher Copyright:
© Sujoy Bhore and Rahul Jain.

PY - 2021/12/1

Y1 - 2021/12/1

N2 - 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(m1/2 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(m1/2 log n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ϵ > 0, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n1/4+ϵ) 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.

AB - 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(m1/2 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(m1/2 log n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ϵ > 0, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n1/4+ϵ) 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.

KW - Geometric intersection graphs

KW - Reachablity

KW - Space-efficient algorithms

UR - http://www.scopus.com/inward/record.url?scp=85122475564&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2021.63

DO - 10.4230/LIPIcs.ISAAC.2021.63

M3 - Conference contribution

AN - SCOPUS:85122475564

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 32nd International Symposium on Algorithms and Computation, ISAAC 2021

A2 - Ahn, Hee-Kap

A2 - Sadakane, Kunihiko

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 32nd International Symposium on Algorithms and Computation, ISAAC 2021

Y2 - 6 December 2021 through 8 December 2021

ER -