Exponential separations in the energy complexity of leader election

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 Leader Election and Approximate Counting 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 Leader Election algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD: No-CD = Sender-CD >> Receiver-CD = Strong-CD and for deterministic Leader Election algorithms, there is another exponential gap in energy complexity, but in the reverse direction: No-CD = Receiver-CD >> Sender-CD = Strong-CD In particular, the randomized energy complexity of Leader Election is Θ(log^∗ n) in Sender-CD but Θ(log(log^∗ n)) in Receiver-CD, where n is the (unknown) number of devices. Its deterministic complexity is Θ(log N) in Receiver-CD but Θ(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 Leader Election and Approximate Counting. A critical component of this algorithm is a new deterministic Leader Election algorithm for dense instances, when n = Θ(N), with inverse-Ackermann-type (O(α(N))) energy complexity.