Rise and Shine Efficiently! Tight Bounds for Adversarial Wake-up

We study the wake-up problem in distributed networks, where an adversary awakens a subset of nodes at arbitrary times, and the goal is to wake up all other nodes as quickly as possible by sending only few messages. We prove the following lower bounds: * We first consider the setting where each node receives advice from an oracle who can observe the entire network, but does not know which nodes are awake initially. More specifically, we consider the $KT_0$ $LOCAL$ model with advice. We prove that any randomized algorithm must send $\Omega( \frac{n^{2}}{2^{\beta}\log n} )$ messages if nodes receive only $O(\beta)$ bits of advice on average. * For the $KT_1$ assumption, we show that any $(k+1)$-time algorithm requires $\Omega( n^{1+1/k} )$ messages. Our result is the first super-linear (in $n$) lower bound, for a problem that does not require individual nodes to learn a large amount of information about the network topology. To complement our lower bound results, we present several new algorithms: * We give an asynchronous $KT_1$ $LOCAL$ algorithm that solves the wake-up problem with a time and message complexity of $O( n\log n )$ with high probability. * We introduce the notion of \emph{awake distance} $\rho_{\text{awk}}$, which is upper-bounded by the network diameter, and present a synchronous $KT_1$ $LOCAL$ algorithm that takes $O( \rho_{\text{awk}} )$ rounds and sends $O( n^{3/2}\sqrt{\log n} )$ messages with high probability. We also extend these ideas to obtain a near-optimal time- and message complexity of O\( \rho_{awk} \log^3n ) rounds $O( n \log^3n )$ messages. * We give deterministic advising schemes in the asynchronous $KT_0$ $CONGEST$ model (with advice). In particular, we obtain an $O( \rho_{\text{awk}}\log^2n )$-time advising scheme that sends $O( n\log^2n )$ messages, while requiring $O( \log^2n )$ bits of advice per node.