Beeping a Maximal Independent Set

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 an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, 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 of its neighbors 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 find 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 O(log^3 n) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in O(log^3 n) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in O(log^2 n) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in O(log^2 n) time.Comment: arXiv admin note: substantial text overlap with arXiv:1108.192

On the Computational Power of Radio Channels

Radio networks can be a challenging platform for which to develop distributed algorithms, because the network nodes must contend for a shared channel. In some cases, though, the shared medium is an advantage rather than a disadvantage: for example, many radio network algorithms cleverly use the shared channel to approximate the degree of a node, or estimate the contention. In this paper we ask how far the inherent power of a shared radio channel goes, and whether it can efficiently compute "classicaly hard" functions such as Majority, Approximate Sum, and Parity. Using techniques from circuit complexity, we show that in many cases, the answer is "no". We show that simple radio channels, such as the beeping model or the channel with collision-detection, can be approximated by a low-degree polynomial, which makes them subject to known lower bounds on functions such as Parity and Majority; we obtain round lower bounds of the form Omega(n^{delta}) on these functions, for delta in (0,1). Next, we use the technique of random restrictions, used to prove AC^0 lower bounds, to prove a tight lower bound of Omega(1/epsilon^2) on computing a (1 +/- epsilon)-approximation to the sum of the nodes\u27 inputs. Our techniques are general, and apply to many types of radio channels studied in the literature