Reachability Oracles for Directed Transmission Graphs

Let P⊂ R^d be a set of n points in d dimensions such that each point p∈ P has an associated radiusr_p> 0. The transmission graphG for P is the directed graph with vertex set P such that there is an edge from p to q if and only if | pq| ≤ r_p, 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(nlog 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(nlog n) ; the second data structure gives Q(n)=O(Ψ3n) and S(n) = O(Ψ ³n^{3 / 2}) ; the third data structure is randomized with Q(n) = O(n^{2 / 3}log ^{1 / 3}Ψ log ^{2 / 3}n) and S(n) = O(n^{5 / 3}log ^{1 / 3}Ψ log ^{2 / 3}n) and answers queries correctly with high probability.