We consider networks of processes which interact with beeps. Various beeping models are used. The basic one, defined by Cornejo and Kuhn [CK10], assumes that a process can choose either to beep or to listen; if it listens it can distinguish between silence or the presence of at least one beep. The aim of this paper is the study of the resolution of paradigms such as collision detection, computation of the degree of a vertex, colouring, or 2-hop-colouring in the framework of beeping models. For each of these problems we present Las Vegas or Monte Carlo algorithms and we analyse their complexities expressed in terms of the number of slots. We present also efficient randomised emulations of more powerful beeping models on the basic one. We illustrate emulation procedures with an efficient degree computation algorithm in the basic beeping model; this algorithm was given initially in a more powerful model.
Maximal Independent Set selection is a fundamental problem in distributed computing. A novel probabilistic algorithm for this problem has recently been proposed by , inspired by the study of the way that developing cells in the fly become specialised. The algorithm they propose is simple and robust, but not as efficient as previous approaches: the expected time complexity is O(log2 n). Here we first show that the approach of cannot achieve better efficiency than this across all networks, no matter how the global probability values are chosen. However, we then propose a new algorithm that incorporates another important feature of the biological system: the probability value at each node is adapted using local feedback from neighbouring nodes. Our new algorithm retains all the advantages of simplicity and robustness, but also achieves the optimal efficiency of O(log n) expected time. The new algorithm also has only a constant message complexity per node. We present the 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, don't 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. We study the problem of , 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 share no resources. To highlight the importance of the discreteness of the model, we contrast it against a continuous variant described in [17]. We present an O(1 @math (T ) @math T @math O( n) @math (T ) @math O( n) @math ( n)$ 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. 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 ( ^3 n) ) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in ( O ( ^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 ( ^2 n) ) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in ( O ( ^2 n) ) time.