# Triangles and Girth in Disk Graphs and Transmission Graphs

## TL;DR

This paper presents efficient algorithms for detecting triangles and computing the girth in disk and transmission graphs, which are important in sensor network modeling, using new batched range searching techniques.

## Contribution

The authors develop algorithms that find shortest triangles and girth in disk and transmission graphs in near-linear expected time, improving on previous methods.

## Key findings

- Shortest triangle detection in O(n log n) expected time
- Girth computation in O(n log n) expected time
- New batched range searching tools introduced

## Abstract

Let $S \subset \mathbb{R}^2$ be a set of $n$ sites, where each $s \in S$ has an associated radius $r_s > 0$. The disk graph $D(S)$ is the undirected graph with vertex set $S$ and an undirected edge between two sites $s, t \in S$ if and only if $|st| \leq r_s + r_t$, i.e., if the disks with centers $s$ and $t$ and respective radii $r_s$ and $r_t$ intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph $T(S)$ is the directed graph with vertex set $S$ and a directed edge from a site $s$ to a site $t$ if and only if $|st| \leq r_s$, i.e., if $t$ lies in the disk with center $s$ and radius $r_s$. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in $D(S)$ and in $T(S)$. These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in $D(S)$ and in $T(S)$ can be found in $O(n \log n)$ expected time, and that the (weighted) girth of $D(S)$ can be found in $O(n \log n)$ expected time. For this, we develop new tools for batched range searching that may be of independent interest.