Reachability Oracles for Directed Transmission Graphs

Let P ⊂ R be a set of n points in the d dimensions 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 there is an edge from p to q if and only if d(p, q) ≤ rp, for any p, q ∈ P . A reachability oracle is a data structure that decides for any two vertices p, q ∈ G whether G has a path from p to q. The quality of the oracle is measured by the space requirement S(n), the query time Q(n), and the preprocessing time. For transmission graphs of one-dimensional point sets, we can construct in O(n log n) time an oracle with Q(n) = O(1) and S(n) = O(n). For planar point sets, the ratio Ψ between the largest and the smallest associated radius turns out to be an important parameter. We present three data structures whose quality depends on Ψ: the first works only for Ψ < √ 3 and achieves Q(n) = O(1) with S(n) = O(n) and preprocessing time O(n log n); the second data structure gives Q(n) = O(Ψ √ n) and S(n) = O(Ψn); the third data structure is randomized with Q(n) = O(n log Ψ log n) and S(n) = O(n log Ψ log n) and answers queries correctly with high probability.