Radio Network Lower Bounds Made Easy

Theoreticians have studied distributed algorithms in the synchronous radio network model for close to three decades. A significant fraction of this work focuses on lower bounds for basic communication problems such as

wake-up

(symmetry breaking among an unknown set of nodes) and

broadcast

(message dissemination through an unknown network topology). In this paper, we introduce a new technique for proving this type of bound, based on reduction from a probabilistic hitting game, that simplifies and strengthens much of this existing work. In more detail, in this single paper we prove new expected time and high probability lower bounds for wake-up and global broadcast in single and multi-channel versions of the radio network model both with and without collision detection. In doing so, we are able to reproduce results that previously spanned a half-dozen papers published over a period of twenty-five years. In addition to simplifying these existing results, our technique, in many places, also improves the state of the art: of the eight bounds we prove, four strictly strengthen the best known previous result (in terms of time complexity and/or generality of the algorithm class for which it holds), and three provide the first known non-trivial bound for the case in question. The fact that the same technique can easily generate this diverse collection of lower bounds indicates a surprising unity underlying communication tasks in the radio network model—revealing that deep down, below the specifics of the problem definition and model assumptions, communication in this setting reduces to finding efficient strategies for a simple game.