A Note on Reachability and Distance Oracles for Transmission Graphs

Abstract

Let P be a set of n points in the plane, where each point p ∈ P has a transmission radius r(p) > 0. The transmission graph defined by P and the given radii, denoted by Gtr(P ), is the directed graph whose nodes are the points in P and that contains the arcs (p, q) such that |pq| ⩽ r(p).An and Oh [Algorithmica 2022] presented a reachability oracle for transmission graphs. Their oracle uses O(n 5/3 ) storage and, given two query points s, t ∈ P , can decide in O(n 2/3 ) time if there is a path from s to t in Gtr(P ). We show that the clique-based separators introduced by De Berg et al. [SICOMP 2020] can be used to improve the storage of the oracle to O(n √ n) and the query time to O( √ n). Our oracle can be extended to approximate distance queries: we can construct, for a given parameter ε > 0, an oracle that uses O((n/ε) √ n log n) storage and that can report in O((, where d hop (s, t) is the hop-distance from s to t. We also show how to extend the oracle to so-called continuous queries, where the target point t can be any point in the plane.To obtain an efficient preprocessing algorithm, we show that a clique-based separator of a set F of convex fat objects in R d can be constructed in O(n log n) time.