Transmitting once to Elect a Leader on Wireless Networks

Abstract

Distributed wireless network’s devices are battery-powered most of the time. Transmitting a message uses more energy than receiving one which spends more energy than internal computations. Therefore in this paper, we will focus on the energy complexity of leader election, a fundamental distributed computing problem. As the message’s size impacts on the energy consumption, we highlight that our algorithms have almost optimal time complexities: each device is allowed to send only once $1-bit$ message and to listen to the network during at most 2 time slots. We will firstly work on Radio Networks on which the devices can detect when a node transmits alone: RNstrongCD where both senders and receivers have collision detection capability, RNsenderCD, RNreceiverCD and RNnoCD. If the nodes know their number n, our algorithm elects a leader in optimal $O(\log n)$ time slots with a probability of $1-1/poly(n)$. Then, if all nodes do not know n but know its upper bound u such that $\log u = \Theta (\log n)$, it has $O(\log ^{2}{n})$ time complexity on RNnoCD and RNsenderCD. On RNreceiverCD and RNstrongCD, it has $O(\log ^{(1+\alpha )}{n})$ time complexity where $\alpha \in ]0, 1[$ is constant. For the Beeping Networks model on which the devices cannot detect single transmissions, it has $O(n^{\alpha })$ time complexity with probability $1-1/poly(n)$.