Asymptotically Tight Bounds on the Time Complexity of Broadcast and its Variants in Dynamic Networks

TL;DR

This paper establishes tight linear bounds on the time complexity of broadcast problems in dynamic networks with adversarial control, improving understanding of fundamental data dissemination limits under various models.

Contribution

It provides the first linear upper bounds for broadcast variants in dynamic networks with adversarial conditions, extending previous bounds and solving open problems.

Findings

Linear upper bounds for broadcast in dynamic networks.

Matching lower bounds demonstrating tightness.

Extension to multiple variants with new bounds.

Abstract

Data dissemination is a fundamental task in distributed computing. This paper studies broadcast problems in various innovative models where the communication network connecting $n$ processes is dynamic (e.g., due to mobility or failures) and controlled by an adversary. In the first model, the processes transitively communicate their ids in synchronous rounds along a rooted tree given in each round by the adversary whose goal is to maximize the number of rounds until at least one id is known by all 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 $\lceil{(1 + \sqrt 2) n-1}\rceil \approx 2.4n$. We extend these results to the setting where the adversary gives in each round $k$-disjoint forests and their goal is to maximize the number of rounds until there is a set of $k$ ids such that each process knows of at least one of them. We give a $\left\lceil{\frac{3(n-k)}{2}}\right\rceil-1$ lower bound and a $\frac{\pi^2+6}{6}n+1 \approx 2.6n$ upper bound for this problem. Finally, we study the setting where the adversary gives in each round a directed graph with $k$ roots and their goal is to maximize the number of rounds until there exist $k$ ids that are known by all processes. We give a $\left\lceil{\frac{3(n-3k)}{2}}\right\rceil+2$ lower bound and a $\lceil { (1+\sqrt{2})n}\rceil+k-1 \approx 2.4n+k$ upper bound for this problem. For the two latter problems no upper or lower bounds were previously known.