Broadcast is one of the fundamental network communication primitives. One node of a network, called the
s
ource, 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
c
ollision 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 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 Δ)-bit labels, where Δ 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. Finally, we show that 3-bit labels are also sufficient to solve both versions of broadcast in the case where it is not known
a priori
which node is the source.
Unconditional C-strong property of Faber–Schauder system
