Constant-Length Labeling Schemes for Deterministic Radio Broadcast

Broadcast is one of the fundamental network communication primitives. One node of a network, called the $\mathit{source}$, has a message that has to be learned by all other nodes. We consider the feasibility of deterministic broadcast in radio networks. If nodes of the network do not have any labels, deterministic broadcast is impossible even in the four-cycle. On the other hand, if all nodes have distinct labels, then broadcast can be carried out, e.g., in a round-robin fashion, and hence $O(\log n)$-bit labels are sufficient for this task in $n$-node networks. In fact, $O(\log \Delta)$-bit labels, where $\Delta$ is the maximum degree, are enough to broadcast successfully. Hence, it is natural to ask if very short labels are sufficient for broadcast. Our main result is a positive answer to this question. We show that every radio network can be labeled using 2 bits in such a way that broadcast can be accomplished by some universal deterministic algorithm that does not know the network topology nor any bound on its size. Moreover, at the expense of an extra bit in the labels, we get the additional strong property that there exists a common round in which all nodes know that broadcast has been completed. Finally, we show that 3-bit labels are also sufficient to solve both versions of broadcast in the case where the labeling scheme does not know which node is the source.