On the Wake-Up Problem in Radio Networks

Abstract

Radio networks model wireless communication when processing units communicate using one wave frequency. This is captured by the property that multiple messages arriving simultaneously to a node interfere with one another and none of them can be read reliably. We present improved solutions to the problem of waking up such a network. This requires activating all nodes in a scenario when some nodes start to be active spontaneously, while every sleeping node needs to be awaken by receiving successfully a message from a neighbor. Our contributions concern the existence and efficient construction of universal radio synchronizers, which are combinatorial structures introduced in [6] as building blocks of efficient wake-up algorithms. First we show by counting that there are (n,g)-universal synchronizers for $g(k)={\mathcal O}(k \ {\rm log}\ k \ {\rm log}\ n)$. Next we show an explicit construction of (n,g)-universal-synchronizers for $g(k) = {\mathcal O}(k^{2}{\rm polylog}\ n)$. By way of applications, we obtain an existential wake-up algorithm which works in time  ${\mathcal O}(n {\rm log}^{2}n)$ and an explicitly instantiated algorithm that works in time ${\mathcal O}(n{\it \Delta} {\rm polylog}\ n)$, where n is the number of nodes and ${\it \Delta}$ is the maximum in-degree in the network. Algorithms for leader-election and synchronization can be developed on top of wake-up ones, as shown in [7], such that they work in time slower by a factor of ${\mathcal O}({\rm log} \ n)$ than the underlying wake-up ones.