Deploying Wireless Networks with Beeps

Abstract

We present the discrete beeping communication model, which assumes nodes have minimal knowledge about their environment and severely limited communication capabilities. Specifically, nodes have no information regarding the local or global structure of the network, do not have access to synchronized clocks and are woken up by an adversary. Moreover, instead on communicating through messages they rely solely on carrier sensing to exchange information. This model is interesting from a practical point of view, because it is possible to implement it (or emulate it) even in extremely restricted radio network environments. From a theory point of view, it shows that complex problems (such as vertex coloring) can be solved efficiently even without strong assumptions on properties of the communication model.

We study the problem of interval coloring, a variant of vertex coloring specially suited for the studied beeping model. Given a set of resources, the goal of interval coloring is to assign every node a large contiguous fraction of the resources, such that neighboring nodes have disjoint resources. A k-interval coloring is one where every node gets at least a 1/k fraction of the resources.

To highlight the importance of the discreteness of the model, we contrast it against a continuous variant described in [17]. We present an ${\mathcal O}(1)$ time algorithm that with probability 1 produces a ${\mathcal O}(\Delta)$-interval coloring. This improves an ${\mathcal O}(\log n)$ time algorithm with the same guarantees presented in [17], and accentuates the unrealistic assumptions of the continuous model. Under the more realistic discrete model, we present a Las Vegas algorithm that solves ${\mathcal O}(\Delta)$-interval coloring in ${\mathcal O}(\log n)$ time with high probability and describe how to adapt the algorithm for dynamic networks where nodes may join or leave. For constant degree graphs we prove a lower bound of Ω(logn) on the time required to solve interval coloring for this model against randomized algorithms. This lower bound implies that our algorithm is asymptotically optimal for constant degree graphs.