Global Information Sharing under Network Dynamics

TL;DR

This paper investigates the complexity of spreading multiple tokens in dynamic networks, establishing lower bounds for token dissemination under adaptive adversaries and providing various upper bounds in different models.

Contribution

It introduces tight lower bounds for token dissemination in strongly adaptive dynamic networks, resolving an open problem and analyzing various dissemination models.

Findings

Omega(nk/log n + n) rounds are necessary for token spreading.

Lower bounds apply to well-mixed and arbitrary initial distributions.

The results clarify the limits of token-forwarding algorithms in dynamic networks.

Abstract

We study how to spread $k$ tokens of information to every node on an $n$-node dynamic network, the edges of which are changing at each round. This basic {\em gossip problem} can be completed in $O(n + k)$ rounds in any static network, and determining its complexity in dynamic networks is central to understanding the algorithmic limits and capabilities of various dynamic network models. Our focus is on token-forwarding algorithms, which do not manipulate tokens in any way other than storing, copying and forwarding them. We first consider the {\em strongly adaptive} adversary model where in each round, each node first chooses a token to broadcast to all its neighbors (without knowing who they are), and then an adversary chooses an arbitrary connected communication network for that round with the knowledge of the tokens chosen by each node. We show that $\Omega(nk/\log n + n)$ rounds are needed for any randomized (centralized or distributed) token-forwarding algorithm to disseminate the $k$ tokens, thus resolving an open problem raised in~\cite{kuhn+lo:dynamic}. The bound applies to a wide class of initial token distributions, including those in which each token is held by exactly one node and {\em well-mixed} ones in which each node has each token independently with a constant probability. We also show several upper bounds in varying models.