Beeping a Maximal Independent Set

Abstract

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that nodes wake up asynchronously. At each time slot a node can either beep (i.e., emit a signal) or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one neighbor beeping.

We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and compute an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in $\mathcal{O}(\log^3 n)$ time. Second, if sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in $\mathcal{O}(\log^3 n)$ time. Third, if in addition to this wakeup assumption we allow beeping nodes to receive feedback to identify if at least one neighboring node is beeping concurrently (i.e., sender-side collision detection) we can find an MIS in $\mathcal{O}(\log^2 n)$ time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to compute an MIS in $\mathcal{O}(\log^2 n)$ time.