Reachability Problems for Transmission Graphs

TL;DR

This paper introduces a novel data structure for efficiently answering reachability queries in transmission graphs, with performance depending solely on the number of points, not on the total edges, improving computational efficiency.

Contribution

It presents the first data structure that enables reachability queries in transmission graphs with size and query time depending only on the number of points.

Findings

Data structure size: O(n^{5/3})

Query time: O(n^{2/3})

Performance independent of total edges

Abstract

Let $P$ be a set of $n$ points in the plane where each point $p$ of $P$ is associated with a radius $r_p>0$.The transmission graph $G=(P,E)$ of $P$ is defined as the directed graph such that $E$ contains an edge from $p$ to $q$ if and only if $|pq|\leq r_p$ for any two points $p$ and $q$ in $P$, where $|pq|$ denotes the Euclidean distance between $p$ and $q$. In this paper, we present a data structure of size $O(n^{5/3})$ such that for any two points in $P$, we can check in $O(n^{2/3})$ time if there is a path in $G$ between the two points. This is the first data structure for answering reachability queries whose performance depends only on $n$ but not on the number of edges.