Finding the Size and the Diameter of a Radio Network Using Short Labels

The number of nodes of a network, called its size, and the largest distance between nodes of a network, called its diameter, are among the most important network parameters. Knowing the size and/or diameter is a prerequisite of many distributed network algorithms. A radio network is a collection of nodes, with wireless transmission and receiving capabilities. It is modeled as a simple 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 round $i$, if $v$ listens in round $i$, and if $w$ is its only neighbor that transmits in round $i$. If $v$ listens in a round, and multiple neighbors of $v$ transmit in this round, a collision occurs at $v$. If $v$ transmits in a round, it does not hear anything. If listening nodes can distinguish collision from silence, we say that the network has collision detection capability, otherwise there is no collision detection. We consider the tasks of size discovery and diameter discovery: finding the size (resp. the diameter) of an unknown radio network with collision detection. All nodes have to output the size (resp. the diameter) of the network, using a deterministic algorithm. Nodes have labels which are binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on the following problems: 1. What is the shortest labeling scheme that permits size discovery in all radio networks of maximum degree $\Delta$? 2. What is the shortest labeling scheme that permits diameter discovery in all radio networks? We show that the minimum length of a labeling scheme that permits size discovery is $\Theta(\log\log \Delta)$. By contrast, we show that diameter discovery can be done using a labeling scheme of constant length.