Abstract—In this paper, we study algebraic aggregate computations in Sensor Networks. The main contribution is the presentation of an early-stopping protocol that computes the average function under a harsh model of the conditions under which sensor nodes operate. This protocol is shown to be time-optimal in presence of unfrequent failures. The approach followed saves time and energy by relying the computation on a small network of delegate nodes that can be rebuilt fast in case of node failures and communicate using a collisionfree schedule. Delegate nodes run simultaneously two protocols, namely, a collection/dissemination tree-based algorithm, which is shown to be optimal, and a mass-distribution algorithm. Both algorithms are analyzed under a model where the frequency of failures is a parameter. Other aggregate computation algorithms can be easily derived from this protocol. To the best of our knowledge, this is the first optimal early-stopping algorithm for aggregate computations in Sensor Networks.

Abstract. In Sensor Networks, the lack of topology information and the availability of only one communication channel has led research work to the use of randomization to deal with collision of transmissions. However, the scarcest resource in this setting is the energy supply, and radio communication dominates the sensor node energy consumption. Hence, redundant trials of transmission as used in randomized protocols may be counter-effective. Additionally, most of the research work in Sensor Networks is either heuristic or includes unreallistic assumptions. Hence, provable results for many basic problems still remain to be given. In this paper, we study upper and lower bounds for deterministic communication primitives under the harsh constraints of sensor nodes. 1

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Sensor nodes are very weak computers that get distributed at random on a surface. Once deployed, they must wake up and form a radio network. Sensor network bootstrapping research thus has three parts: One must model the restrictions on sensor nodes; one must prove that the connectivity graph of the sensors has a subgraph that would make a good network; and one must give a distributed protocol for finding such a network subgraph that can be implemented on sensor nodes. Although many particular restrictions on sensor nodes are implicit or explicit in many papers, there remain many inconsistencies and ambiguities from paper to paper. The lack of a clear model means that solutions to the network bootstrapping problem in both the theory and systems literature all violate constraints on sensor nodes. For example, random geometric graph results on sensor networks predict the existence of subgraphs on the connectivity graph with good route-stretch, but these results do not address the degree of such a graph, and sensor networks must have constant degree. Furthermore, proposed protocols for actually finding such graphs require that nodes have too much memory, whereas others assume the existence of a contention-resolution mechanism. We present a formal Weak Sensor model that summarizes the literature on sensor node restrictions, taking the most restrictive choices when possible. We show that sensor connectivity graphs have lowdegree