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 that wakes up a 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 a network in O ( 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 ? ( k b log ? n k ) time. We give a randomized algorithm that wakes up a network within O ( k 1 / b ln ? 1 ? ) rounds with a probability that is at least 1 - ? , for any 0 < ? < 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 a network in time O ( log - 1 ? ( 1 p ) k log ? n log 1 / b ? k ) , and the other in time O ( log - 1 ? ( 1 p ) k b log ? n log ? ( b log ? n ) ) when the inequality b log ? ( 128 b log ? n ) holds, both with probabilities that are at least 1 - 1 / poly ( n ) .

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