Algorithmica

pp 1–18

# Reachability Oracles for Directed Transmission Graphs

Article

## 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 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 $$|pq| \le r_p$$, 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.

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## Authors and Affiliations

• Haim Kaplan
• 1
• Wolfgang Mulzer
• 2
• Liam Roditty
• 3
• Paul Seiferth
• 2
1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
3. 3.Department of Computer ScienceBar Ilan UniversityRamat GanIsrael