Enabling Minimal Dominating Set in Highly Dynamic Distributed Systems

TL;DR

This paper introduces a new approach for computing minimal dominating sets in highly dynamic, weakly connected distributed systems, providing a new definition, necessary conditions, and a fair time complexity measure.

Contribution

It proposes a new definition of minimal dominating set for time-varying graphs, establishes topological conditions for deterministic algorithms, and introduces a fair time complexity measure.

Findings

New definition of minimal dominating set for dynamic graphs

Topological conditions for algorithm existence

A time complexity measure accounting for network dynamics

Abstract

We address the problem of computing a Minimal Dominating Set in highly dynamic distributed systems. We assume weak connectivity, i.e., the network may be disconnected at each time instant and topological changes are unpredictable. We make only weak assumptions on the communication: every process is infinitely often able to communicate with other processes (not necessarily directly). Our contribution is threefold. First, we propose a new definition of minimal dominating set suitable for the context of time-varying graphs that seems more relevant than existing ones. Next, we provide a necessary and sufficient topological condition for the existence of a deterministic algorithm for minimal dominating set construction in our settings. Finally, we propose a new measure of time complexity in time-varying graph in order to to allow fair comparison between algorithms. Indeed, this measure takes account of communication delays attributable to dynamicity of the graph and not to the algorithms.