Brief Announcement: Broadcasting Time in Dynamic Rooted Trees is Linear

TL;DR

This paper proves a linear upper bound on the broadcast time in dynamic rooted trees, improving previous bounds and providing new insights into the problem's complexity.

Contribution

It introduces the first linear upper bound for broadcast time in dynamic networks with adversarial tree changes, based on adjacency matrix analysis.

Findings

Established a linear upper bound of approximately 2.4n rounds.

Improved upon the previous upper bound of O(n log log n).

Analyzed the evolution of the adjacency matrix over time.

Abstract

We study the broadcast problem on dynamic networks with $n$ processes. The processes communicate in synchronous rounds along an arbitrary rooted tree. The sequence of trees is given by an adversary whose goal is to maximize the number of rounds until at least one process reaches all other processes. Previous research has shown a $\lceil{\frac{3n-1}{2}}\rceil-2$ lower bound and an $O(n\log\log n)$ upper bound. We show the first linear upper bound for this problem, namely $\lceil{(1 + \sqrt 2) n-1}\rceil \approx 2.4n$. Our result follows from a detailed analysis of the evolution of the adjacency matrix of the network over time.