Scalable Wake-up of Multi-Channel Single-Hop Radio Networks

We consider single-hop radio networks with multiple channels as a model of wireless networks. There are $n$ stations connected to $b$ radio channels that do not provide collision detection. A station uses all the channels concurrently and independently. Some $k$ stations may become active spontaneously at arbitrary times. The goal is to wake up the network, which occurs when all the stations hear a successful transmission on some channel. Duration of a waking-up execution is measured starting from the first spontaneous activation. We present a deterministic algorithm for the general problem that wakes up the network in $O(k\log^{1/b} k\log n)$ time, where $k$ is unknown. We give a deterministic scalable algorithm for the special case when $b>d \log \log n$, for some constant $d>1$, which wakes up the network in $O(\frac{k}{b}\log n\log(b\log n))$ time, with $k$ unknown. This algorithm misses time optimality by at most a factor of $O(\log n(\log b +\log\log n))$, because any deterministic algorithm requires $\Omega(\frac{k}{b}\log \frac{n}{k})$ time. We give a randomized algorithm that wakes up the network within $O(k^{1/b}\ln \frac{1}{\epsilon})$ rounds with a probability that is at least $1-\epsilon$, for any $0<\epsilon<1$, where $k$ is known. We also consider a model of jamming, in which each channel in any round may be jammed to prevent a successful transmission, which happens with some known parameter probability $p$, independently across all channels and rounds. For this model, we give two deterministic algorithms for unknown~$k$: one wakes up the network in time $O(\log^{-1}(\frac{1}{p})\, k\log n\log^{1/b} k)$, and the other in time $O(\log^{-1}(\frac{1}{p}) \, \frac{k}{b} \log n\log(b\log n))$ but assuming the inequality $b>\log(128b\log n)$, both with a probability that is at least $1-1/\mbox{poly}(n)$.