A Note on Reachability and Distance Oracles for Transmission Graphs

TL;DR

This paper improves data structures for reachability and approximate distance queries in transmission graphs, reducing storage and query time, and extends methods to continuous queries and convex fat objects.

Contribution

It introduces a more efficient reachability oracle for transmission graphs using clique-based separators, with improved storage and query times, and extends to approximate distances and continuous queries.

Findings

Reduced storage to O(n√n)

Query time improved to O(√n)

Extended to approximate and continuous queries

Abstract

Let $P$ be a set of $n$ points in the plane, where each point $p\in P$ has a transmission radius $r(p)>0$. The transmission graph defined by $P$ and the given radii, denoted by $\mathcal{G}_{\mathrm{tr}}(P)$, is the directed graph whose nodes are the points in $P$ and that contains the arcs $(p,q)$ such that $|pq|\leq r(p)$. 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 $s,t\in P$, can decide in $O(n^{2/3})$ time if there is a path from $s$ to $t$ in $\mathcal{G}_{\mathrm{tr}}(P)$. We show that the clique-based separators introduced by De Berg \emph{et al.} [SICOMP 2020] can be used to improve the storage of the oracle to $O(n\sqrt{n})$ and the query time to $O(\sqrt{n})$. Our oracle can be extended to approximate distance queries: we can construct, for a given parameter $\varepsilon>0$, an oracle that uses $O((n/\varepsilon)\sqrt{n}\log n)$ storage and that can report in $O((\sqrt{n}/\varepsilon)\log n)$ time a value $d_{\mathrm{hop}}^*(s,t)$ satisfying $d_{\mathrm{hop}}(s,t) \leq d_{\mathrm{hop}}^*(s,t) < (1+\varepsilon)\cdot d_{\mathrm{hop}}(s,t) + 1$, where $d_{\mathrm{hop}}(s,t)$ 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 $\Bbb{R}^d$ can be constructed in $O(n\log n)$ time.