Exponential Separations in the Energy Complexity of Leader Election

Abstract

Energy is often the most constrained resource for battery-powered wireless devices, and most of the energy is often spent on transceiver usage (i.e., transmitting and receiving packets) rather than computation. In this article, we study the energy complexity of fundamental problems 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 (in which only transmitters detect collisions), Receiver-CD (in which 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 algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD: Randomized: No-CD = Sender-CD > Receiver-CD = Strong-CD and for deterministic algorithms, there is another exponential gap in energy complexity, but in the reverse direction : Deterministic: No-CD = Receiver-CD > Sender-CD = Strong-CD Precisely, 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 number of devices, which is unknown to the devices at the beginning; the deterministic complexity of Leader Election is Θ(log N ) in Receiver-CD but Θ(log log N ) in Sender-CD, where N is the size of the ID space. There is a tradeoff between time and energy. We provide a new upper bound on the time-energy tradeoff curve for randomized algorithms. A critical component of this algorithm is a new deterministic Leader Election algorithm for dense instances, when n = Θ( N ), with inverse Ackermann energy complexity.