## Abstract

Let \(P \subset \mathbb {R}^d\) be a set of *n* points in *d* dimensions such that each point \(p \in P\) has an *associated radius*\(r_p > 0\). 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 \(|pq| \le r_p\), for any \(p, q \in P\). A *reachability oracle* is a data structure that decides for any two vertices \(p, q \in G\) 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 \(\Psi \) between the largest and the smallest associated radius turns out to be an important parameter. We present three data structures whose quality depends on \(\Psi \): the first works only for \(\Psi < \sqrt{3}\) and achieves \(Q(n) = O(1)\) with \(S(n) = O(n)\) and preprocessing time \(O(n\log n)\); the second data structure gives \(Q(n) = O(\Psi ^3 \sqrt{n})\) and \(S(n) = O(\Psi ^3 n^{3/2})\); the third data structure is randomized with \(Q(n) = O(n^{2/3}\log ^{1/3} \Psi \log ^{2/3} n)\) and \(S(n) = O(n^{5/3}\log ^{1/3} \Psi \log ^{2/3} n)\) and answers queries correctly with high probability.

## Notes

### Acknowledgements

We like to thank Günter Rote and the anonymous reviewers for valuable comments, in particular for pointing out a drastic simplification for the one-dimensional reachability oracle.

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