Towards Power-Sensitive Communication on a Multiple-Access Channel

We are given $n$ stations of which $k$ are active, while the remaining $n-k$ are asleep. The active stations communicate via a multiple-access channel. If a subset $Q$ of active stations transmits in the same round, all active stations can recognize from the signal strength how many stations have transmitted (i.e., they learn the size of set $Q$), even though they may not be able to decode the contents of transmitted messages. The goal is to let each active station to learn about the set of all active stations. It is well known that $\Theta(k\log_{k+1} n)$ rounds are enough, even for non-adaptive deterministic algorithms. A natural interesting generalization arises when we are required to identify a subset of $m\leq k$ active stations. We show that while for randomized or for adaptive deterministic algorithms $O(m \log_{m+1} n)$ rounds are sufficient, the non-adaptive deterministic counterpart still requires $\Theta(k\log_{k+1} n)$ rounds, therefore, finding any subset of active stations is not easier than finding all of them by a non-adaptive deterministic algorithm. We prove our results in the more general framework of combinatorial search theory, where the problem of identifying active stations on a multiple-access channel can be viewed as a variant of the well-known counterfeit coin problem.