Space-efficient algorithms for reachability in directed geometric graphs
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....
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| Published in: | Theoretical computer science Vol. 961; p. 113938 |
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15.06.2023
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| 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(m1/2logn) 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(m1/2logn) 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. |
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| AbstractList | 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(m1/2logn) 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(m1/2logn) 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. |
| ArticleNumber | 113938 |
| Author | Jain, Rahul Bhore, Sujoy |
| Author_xml | – sequence: 1 givenname: Sujoy orcidid: 0000-0003-0104-1659 surname: Bhore fullname: Bhore, Sujoy email: sujoy.bhore@gmail.com organization: Department of Computer Science & Engineering, Indian Institute of Technology Bombay, India – sequence: 2 givenname: Rahul surname: Jain fullname: Jain, Rahul email: rahul.jain@fernuni-hagen.de organization: Fernuniversität in Hagen, Germany |
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| Cites_doi | 10.1016/S0022-0000(70)80006-X 10.1137/S0097539793283151 10.1016/0196-6774(83)90012-3 10.1007/s00454-012-9417-5 10.1007/s00224-009-9172-z 10.1016/0166-218X(89)90031-0 10.1137/0136016 10.2140/pjm.1965.15.835 10.1007/s00224-017-9841-2 10.1016/j.comgeo.2019.101575 10.1016/j.tcs.2022.06.030 10.1016/j.aim.2008.06.002 10.1016/0022-247X(70)90282-9 10.1016/0095-8956(74)90094-X 10.1145/1391289.1391291 10.1007/s00037-012-0047-3 10.1137/0605032 10.1016/0304-3975(82)90058-5 10.1051/ita/2011106 10.1137/0205021 |
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| Snippet | 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... |
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| SubjectTerms | Geometric intersection graphs Reachability Space-efficient algorithms |
| Title | Space-efficient algorithms for reachability in directed geometric graphs |
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