Reachability Oracles for Directed Transmission Graphs

TL;DR

This paper develops efficient data structures called reachability oracles for directed transmission graphs, enabling quick path queries with varying space and time complexities depending on the graph's properties.

Contribution

It introduces new reachability oracle constructions for transmission graphs, especially for planar point sets, with performance depending on the ratio of radii and the dimension.

Findings

One-dimensional oracles achieve constant query time with linear space.

Planar point set oracles depend on the ratio of radii, with some structures requiring polynomial query time.

A randomized oracle offers high-probability correctness with sublinear query time.

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 $|pq| \leq r_p$, 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.