Spanners and Reachability Oracles for Directed Transmission Graphs

Let P ⊆ ℝ^d be a set of n points, each with an associated radius r_p>0. The transmission graph G for P has vertex set P and an edge from p to q if and only if q lies in the ball with radius r_p around p. Let t>1. A t-spanner H for G is a sparse subgraph of G such that for any two vertices p, q connected by a path of length in G, there is a p-q-path of length at most t in H. We show how to compute a t-spanner for G if d = 2. The running time is O(n(log n + log ψ)), where Ψ is the ratio of the largest and smallest radius of two points in P. We extend this construction to be independent of Ψ at the expense of a polylogarithmic overhead in the running time. As a first application, we prove a property of the t-spanner that allows us to find a BFS tree in G for any given start vertex s 2 P in the same time. After that, we deal with reachability oracles for G. These are data structures that answer reachability queries: given two vertices, is there a directed path between them? The quality of an oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. For d = 1, we show how to compute an oracle with Q(n) = O(1) and S(n) = O(n) in time O(n log n). For d = 2, the radius ratio Ψ again turns out to be an important measure for the complexity of the problem. We present three different data structures whose quality depends on Ψ : (i) if Ψ < 3, we achieve Q(n) = O(1) with S(n) = O(n) and preproccesing time O(n log n); (ii) if Ψ ≥ 3 , we get Q(n) = O(Ψ³n) and S(n) = O(Ψ^53/2); and (iii) if Ψ is polynomially bounded in n, we use probabilistic methods to obtain an oracle with Q(n) = O(n^2/3 log n) and S(n) = O(n^5/3 log n) that answers queries correctly with high probability. We employ our t-spanner to achieve a fast preproccesing time of O(Ψ⁵n^3/2) and O(n^5/3 log² n) in case (ii) and (iii), respectively.