Broadcasting on Paths and Cycles

TL;DR

This paper analyzes the time needed for multiple agents performing random walks on paths and cycles to fully broadcast a message, providing tight asymptotic bounds for different numbers of agents.

Contribution

It offers the first tight bounds on broadcasting times on paths and cycles for all agent counts, advancing understanding of random walk communication processes.

Findings

Derived tight asymptotic bounds for paths and cycles

Results hold for the entire range of agent numbers

Enhances understanding of message spreading in random walk models

Abstract

Consider the following broadcasting process run on a connected graph $G=(V,E)$. Suppose that $k \ge 2$ agents start on vertices selected from $V$ uniformly and independently at random. One of the agents has a message that she wants to communicate to the other agents. All agents perform independent random walks on $G$, with the message being passed when an agent that knows the message meets an agent that does not know the message. The broadcasting time $\xi(G,k)$ is the time it takes to spread the message to all agents. We provide tight bounds for $\xi(P_n,k)$ and $\xi(C_n,k)$ that hold asymptotically almost surely for the whole range of the parameter~$k$.