# The Homogeneous Broadcast Problem in Narrow and Wide Strips

## TL;DR

This paper investigates the complexity of the homogeneous broadcast problem in wireless networks constrained within narrow and wide strips, providing a near-complete classification based on strip width.

## Contribution

It offers a comprehensive complexity characterization of the broadcast problem in strips, including both regular and hop-bounded versions, as a function of strip width.

## Key findings

- Complexity varies with strip width.
- Almost complete classification of problem complexity.
- Results applicable to both regular and hop-bounded broadcast.

## Abstract

Let $P$ be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $s\in P$ be a given source node. Each node $p$ can transmit information to all other nodes within unit distance, provided $p$ is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source $s$ can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem (in the latter, $s$ must be able to reach every node within a specified number of hops), with the restriction that all points lie inside a strip of width $w$. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width $w$.