Sleeping on the job
Valerie Jean King, Cynthia A. Phillips, Jared Saia, Maxwell Young · 2008
We address the problem of minimizing power consumption when broadcasting a message from one node to all the other nodes in a radio network. To enable power savings for such a problem, we introduce a compelling new data streaming problem that we call the Bad Santa problem. Our results on this problem apply for any situation where: 1) a node can listen to a set of n nodes, out of which at least half are non-faulty and know the correct message; and 2) each of these n nodes sends according to some predetermined schedule which assigns each of them its own unique time slot. In this situation, we show that in order to receive the correct message with probability 1, it is necessary and sufficient for the listening node to listen to a Θ(√n) expected number of time slots. Moreover, if we allow for repetitions of transmissions so that each sending node sends the message O(log* n) times (i.e. in O(log* n) rounds each consisting of the n time slots), then listening to O(log* n) expected number of time slots suffices. We show that this is near optimal.