Exponential Separations in the Energy Complexity of Leader Election

TL;DR

This paper investigates the energy complexity of leader election in wireless networks, revealing exponential gaps based on collision detection capabilities and providing new algorithms with optimal tradeoffs and dense-instance solutions.

Contribution

It uncovers exponential separations in energy complexity depending on collision detection models and introduces new algorithms with optimal tradeoffs and dense-instance solutions.

Findings

Exponential energy complexity gaps between collision detection models.

New algorithms achieving optimal time-energy tradeoffs.

Deterministic leader election for dense instances with inverse-Ackermann energy complexity.

Abstract

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.