Conflict Resolution and Membership Problem in Beeping Channels

Abstract

Consider a group of nodes connected through multiple-access channels and the only observable feedback on the channel is a binary value: either one or more nodes have transmitted (busy), or no node has transmitted (idle). The channel model thus described is called Beeping Model and captures computation in hardware using a group of sequential circuit modules connected by a logic-OR gate. It has also been used to study chemical signaling mechanisms between biological cells and carrier-sensing based wireless communication.

In this paper, we study the distributed complexity of two fundamental problems in the Beeping Model. In both problems, there is a set of nodes each with a unique identifier i ∈ {1,2,…,n}. A subset of the nodes A ⊆ {1,2,…,n} is called active nodes. In the Membership Problem, every node needs to find out the identifiers of all active nodes. In the Conflict Resolution Problem, the goal is to let every active node use the channel alone (without collision) at least once.

We derive two results that characterize the distributed complexity of these problems. First, we prove that in the Beeping Model the two above problems are equally hard. This is in stark contrast to traditional channel models with ternary feedback in which the membership problem is strictly harder than conflict resolution. The equivalence result also leads to a randomized lower bound for conflict resolution, which shows a relative powerlessness of randomization in the beeping model. Secondly, we give a new deterministic algorithm for the problems that achieves the best known parallelization among all practical algorithms.