Space-efficient algorithms for reachability in directed geometric graphs

Sujoy Bhore, Rahul Jain

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(m1/2log⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">m</mi></mrow><mrow is="true"><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mn is="true">2</mn></mrow></msup><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> 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(m1/2log⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">m</mi></mrow><mrow is="true"><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mn is="true">2</mn></mrow></msup><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> 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<math><mi is="true">ϵ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></math>, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n1/4+ϵ)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mn is="true">4</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">ϵ</mi></mrow></msup><mo stretchy="false" is="true">)</mo></math> 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.