Finding the Size of a Radio Network with Short Labels

The number of nodes of a network, called its size, is one of the most important network parameters. Knowing the size (or a good upper bound on it) is a prerequisite of many distributed network algorithms, ranging from broadcasting and gossiping, through leader election, to rendezvous and exploration. A radio network is a collection of stations, called nodes, with wireless transmission and receiving capabilities. It is modeled as a simple connected undirected graph whose nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbors, or stay silent and listen. At the receiving end, a node v hears a message from a neighbor w in a given round, if v listens in this round, and if w is its only neighbor that transmits in this round. If v listens in a round, and two or more neighbors of v transmit in this round, a collision occurs at v. If v transmits in a round, it does not hear anything in this round. Two scenarios are considered in the literature: if listening nodes can distinguish collision from silence (the latter occurs when no neighbor transmits), we say that the network has the collision detection capability, otherwise there is no collision detection. We consider the task of size discovery: finding the size of an unknown radio network with collision detection. All nodes have to output the size of the network, using a deterministic algorithm. Nodes have labels which are (not necessarily distinct) binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on the following problem: What is the shortest labeling scheme that permits size discovery in all radio networks of maximum degree Δ? Our main result states that the minimum length of such a labeling scheme is Θ(loglogΔ). The upper bound is proven by designing a size discovery algorithm using a labeling scheme of length O (loglogΔ), for all networks of maximum degree Δ. The matching lower bound is proven by constructing a class of graphs (in fact even of trees) of maximum degree Δ, for which any size discovery algorithm must use a labeling scheme of length at least Ω(loglogΔ) on some graph of this class.