$k$-shot Broadcasting in Ad Hoc Radio Networks

TL;DR

This paper investigates the efficiency of distributed broadcasting in unknown radio networks under limited transmission constraints, providing tight bounds for oblivious protocols and lower bounds for adaptive strategies.

Contribution

It introduces tight bounds for oblivious $k$-shot broadcasting protocols and establishes new lower bounds for adaptive protocols using a novel transmission tree method.

Findings

Oblivious $k$-shot protocols have a broadcasting time of $ heta(n^2/k)$ for $k o ext{small}$.

Adaptive protocols have a lower bound of $ ext{Omega}(n^{(1+k)/k})$ on broadcasting time.

The results delineate the limits of broadcasting efficiency under different protocol adaptiveness.

Abstract

We study distributed broadcasting protocols with few transmissions (`shots') in radio networks where the topology is unknown. In particular, we examine the case in which a bound $k$ is given and a node may transmit at most $k$ times during the broadcasting protocol. Initially, we focus on oblivious algorithms for $k$-shot broadcasting, that is, algorithms where each node decides whether to transmit or not with no consideration of the transmission history. Our main contributions are (a) a lower bound of $\Omega(n^2/k)$ on the broadcasting time of any oblivious $k$-shot broadcasting algorithm and (b) an oblivious broadcasting protocol that achieves a matching upper bound, namely $O(n^2/k)$, for every $k \le \sqrt{n}$ and an upper bound of $O(n^{3/2})$ for every $k > \sqrt{n}$. We also study the general case of adaptive broadcasting protocols where nodes decide whether to transmit based on all the available information, namely the transmission history known by each. We prove a lower bound of $\Omega\left(n^{\frac{1+k}{k}}\right)$ on the broadcasting time of any protocol by introducing the \emph{transmission tree} construction which generalizes previous approaches.