Join-Reachability Problems in Directed Graphs

TL;DR

This paper introduces the join-reachability graph concept for collections of directed graphs, aiming to efficiently represent common reachability and develop structures for various graph classes.

Contribution

It presents optimal and near-optimal structures for join-reachability in paths, trees, planar graphs, and general directed graphs, addressing explicit and implicit problems.

Findings

Optimal structures for paths and trees

Efficient structures for planar graphs

Scalable solutions for general directed graphs

Abstract

For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs.