  We address the problem of minimizing power consumption when performing
reliable broadcast on a radio network under the following popular model. Each
node in the network is located on a point in a two dimensional grid, and
whenever a node sends a message, all awake nodes within distance r receive the
message. In the broadcast problem, some node wants to successfully send a
message to all other nodes in the network even when up to a 1/2 fraction of the
nodes within every neighborhood can be deleted by an adversary. The set of
deleted nodes is carefully chosen by the adversary to foil our algorithm and
moreover, the set of deleted nodes may change periodically. This models
worst-case behavior due to mobile nodes, static nodes losing power or simply
some points in the grid being unoccupied. A trivial solution requires each node
in the network to be awake roughly 1/2 the time, and a trivial lower bound
shows that each node must be awake for at least a 1/n fraction of the time. Our
first result is an algorithm that requires each node to be awake for only a
1/sqrt(n) fraction of the time in expectation. Our algorithm achieves this
while ensuring correctness with probability 1, and keeping optimal values for
other resource costs such as latency and number of messages sent. We give a
lower-bound that shows that this reduction in power consumption is
asymptotically optimal when latency and number of messages sent must be
optimal. If we can increase the latency and messages sent by only a log*n
factor we give a Las Vegas algorithm that requires each node to be awake for
only a (log*n)/n expected fraction of the time; we give a lower-bound showing
that this second algorithm is near optimal. Finally, we show how to ensure
energy-efficient broadcast in the presence of Byzantine faults.
