Constant-Length Labeling Schemes for Deterministic Radio Broadcast

Broadcast is one of the fundamental network communication primitives. One node of a network, called the source, has a message that has to be learned by all other nodes. We consider broadcast in radio networks, modeled as simple undirected connected graphs with a distinguished source. Nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbours, or stay silent and listen. At the receiving end, a node v hears a message from a neighbour w in a given round if v listens in this round and if w is its only neighbour that transmits in this round. If more than one neighbour of a node v transmits in a given round, we say that a collision occurs at v. We do not assume collision detection: in case of a collision, node v does not hear anything (except the background noise that it also hears when no neighbour transmits). We are interested in 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 fourcycle. 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 ∆)-bit labels, where ∆ 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 can get the following additional strong property of our algorithm: there exists a common round in which all nodes know that broadcast has been completed. keywords: broadcast, radio network, labeling scheme, feasibility §Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, Ontario, Canada, M5S 3G4, Canada. [email protected] ∗Indian Institute of Information Technology Vadodara, Sector-28, Gandhinagar, Gujarat 382028, India. [email protected] †Department of Computer Science, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada. [email protected] ‡Département d’informatique, Université du Québec en Outaouais, Gatineau, Québec J8X 3X7, Canada. [email protected]. Partially supported by NSERC discovery grant 8136–2013 and by the Research Chair in Distributed Computing at the Université du Québec en Outaouais. ar X iv :1 71 0. 03 17 8v 1 [ cs .D C ] 9 O ct 2 01 7