Polynomial Counting in Anonymous Dynamic Networks with Applications to Anonymous Dynamic Algebraic Computations

Starting with Michail, Chatzigiannakis, and Spirakis work [Michail et al., 2013], the problem of Counting the number of nodes in {Anonymous Dynamic Networks} has attracted a lot of attention. The problem is challenging because nodes are indistinguishable (they lack identifiers and execute the same program) and the topology may change arbitrarily from round to round of communication, as long as the network is connected in each round. The problem is central in distributed computing as the number of participants is frequently needed to make important decisions, such as termination, agreement, synchronization, and many others. A variety of algorithms built on top of mass-distribution techniques have been presented, analyzed, and also experimentally evaluated; some of them assumed additional knowledge of network characteristics, such as bounded degree or given upper bound on the network size. However, the question of whether Counting can be solved deterministically in sub-exponential time remained open. In this work, we answer this question positively by presenting Methodical Counting, which runs in polynomial time and requires no knowledge of network characteristics. Moreover, we also show how to extend Methodical Counting to compute the sum of input values and more complex functions without extra cost. Our analysis leverages previous work on random walks in evolving graphs, combined with carefully chosen alarms in the algorithm that control the process and its parameters. To the best of our knowledge, our Counting algorithm and its extensions to other algebraic and Boolean functions are the first that can be implemented in practice with worst-case guarantees

An Early-stopping Protocol for Computing Aggregate Functions in Sensor Networks

International audienceIn this paper, we study algebraic aggregate com- putations 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 collision- free 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 algo- rithms 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

Initializing Sensor Networks of Non-uniform Density in the Weak Sensor Model

Assumptions about node density in the sensor networks literature are frequently too strong. Neither adversarially chosen nor uniform random deployment seem realistic in many intended applications of sensor nodes. We define smooth distributions of sensor nodes to be those where the minimum density is guaranteed to achieve connectivity in random deployments, but higher densities may appear in certain areas. We study basic problems for smooth distribution of nodes. Most notably, we present a Weak Sensor Model-compliant distributed protocol for hop-optimal network initialization (NI), a fundamental problem in sensor networks. In order to prove lower bounds, we observe that all nodes must communicate with some other node in order to join the network, and we call the problem of achieving such a communication the group therapy (GT) problem. We show a tight lower bound for the GT problem in radio networks for any class of protocols, and a stronger lower bound for the important class of randomized uniform-oblivious protocols. Given that any NI protocol also solves GT, these lower bounds apply to NI. We also show that the same lower bound holds for a related problem that we call independent set , when nodes are distributed uniformly, even in expectation