  Energy is often the most constrained resource for battery-powered wireless
devices and the lion's share of energy is often spent on transceiver usage
(sending/receiving packets), not on computation. In this paper we study the
energy complexity of LeaderElection and ApproximateCounting in several models
of wireless radio networks. It turns out that energy complexity is very
sensitive to whether the devices can generate random bits and their ability to
detect collisions. We consider four collision-detection models: Strong-CD (in
which transmitters and listeners detect collisions), Sender-CD and Receiver-CD
(in which only transmitters or only listeners detect collisions), and No-CD (in
which no one detects collisions.)
  The take-away message of our results is quite surprising. For randomized
LeaderElection algorithms, there is an exponential gap between the energy
complexity of Sender-CD and Receiver-CD, and for deterministic LeaderElection
algorithms there is another exponential gap, but in the reverse direction.
  In particular, the randomized energy complexity of LeaderElection is
$\Theta(\log^* n)$ in Sender-CD but $\Theta(\log(\log^* n))$ in Receiver-CD,
where $n$ is the (unknown) number of devices. Its deterministic complexity is
$\Theta(\log N)$ in Receiver-CD but $\Theta(\log\log N)$ in Sender-CD, where
$N$ is the (known) size of the devices' ID space.
  There is a tradeoff between time and energy. We give a new upper bound on the
time-energy tradeoff curve for randomized LeaderElection and
ApproximateCounting. A critical component of this algorithm is a new
deterministic LeaderElection algorithm for dense instances, when $n=\Theta(N)$,
with inverse-Ackermann-type ($O(\alpha(N))$) energy complexity.
