Spanners for directed transmission graphs

Let P ⊂ R² be a planar n-point set such that each point p ∈ P has an associated radius rp > 0. The transmission graph G for P is the directed graph with vertex set P such that for any p, q ∈ P, there is an edge from p to q if and only if d(p, q) ≤ rp. Let t > 1 be a constant. A t-spanner for G is a subgraph H ⊆ G with vertex set P so that for any two vertices p, q ∈ P, we have d_H(p, q) ≤ td_G(p, q), where d_H and d_G denote the shortest path distance in H and G, respectively (with Euclidean edge lengths). We show how to compute a t-spanner for G with O(n) edges in O(n(log n + log Ψ)) time, where Ψ is the ratio of the largest and smallest radius of a point in P. Using more advanced data structures, we obtain a construction that runs in O(n log⁵ n) time, independent of Ψ. We give two applications for our spanners. First, we show how to use our spanner to find a BFS tree in G from any given start vertex in O(n log n) time (in addition to the time it takes to build the spanner). Second, we show how to use our spanner to extend a reachability oracle to answer geometric reachability queries. In a geometric reachability query we ask whether a vertex p in G can “reach” a target q which is an arbitrary point in the plane (rather than restricted to be another vertex q of G in a standard reachability query). Our spanner allows the reachability oracle to answer geometric reachability queries with an additive overhead of O(log n log Ψ) to the query time and O(n log Ψ) to the space.