Consider n nodes connected to a single coordinator. Each node receives an individual online data stream of numbers and, at any point in time, the coordinator has to know the k nodes currently observing the largest values, for a given k between 1 and n. We design and analyze an algorithm that solves this problem while bounding the amount of messages exchanged between the nodes and the coordinator. Our algorithm employs the idea of using filters which, intuitively speaking, leads to few messages to be sent, if the new input is "similar" to the previous ones. The algorithm uses a number of messages that is on expectation by a factor of O((log + k) log n) larger than that of an offline algorithm that sets filters in an optimal way, where is upper bounded by the largest value observed by any node.
introduce the Continuous Monitoring Model @cite_11 focusing on systems comprised of a coordinator and @math nodes generating or observing data streams. Consequently, the goal shifts to continuously (that is, in every time step) compute a function depending on the information available across all @math data streams up to the current time at a dedicated coordinator. The major objective becomes the minimization of the overall number of messages exchanged between the nodes and the coordinator. We refer to this model and enhance it by a broadcast channel as proposed by in @cite_4 .
@cite_4 @cite_11
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We study what we call functional monitoring problems. We have k players each tracking their inputs, say player i tracking a multiset Ai(t) up until time t, and communicating with a central coordinator. The coordinator's task is to monitor a given function f computed over the union of the inputs ∪iAi(t), continuously at all times t. The goal is to minimize the number of bits communicated between the players and the coordinator. A simple example is when f is the sum, and the coordinator is required to alert when the sum of a distributed set of values exceeds a given threshold τ. Of interest is the approximate version where the coordinator outputs 1 if f ≥ τ and 0 if f ≤ (1 - e)τ. This defines the (k, f, τ, e) distributed, functional monitoring problem. Functional monitoring problems are fundamental in distributed systems, in particular sensor networks, where we must minimize communication; they also connect to problems in communication complexity, communication theory, and signal processing. Yet few formal bounds are known for functional monitoring. We give upper and lower bounds for the (k, f, τ, e) problem for some of the basic f's. In particular, we study frequency moments (F0, F1, F2). For F0 and F1, we obtain continuously monitoring algorithms with costs almost the same as their one-shot computation algorithms. However, for F2 the monitoring problem seems much harder. We give a carefully constructed multi-round algorithm that uses "sketch summaries" at multiple levels of detail and solves the (k, F2, τ, e) problem with communication O(k2 e+ (√k e)3). Since frequency moment estimation is central to other problems, our results have immediate applications to histograms, wavelet computations, and others. Our algorithmic techniques are likely to be useful for other functional monitoring problems as well. Wireless systems comprised of rechargeable nodes have a significantly prolonged lifetime and are sustainable. A distinct characteristic of these systems is the fact that the nodes can harvest energy throughout the duration in which communication takes place. As such, transmission policies of the nodes need to adapt to these harvested energy arrivals. In this paper, we consider optimization of point-to-point data transmission with an energy harvesting transmitter which has a limited battery capacity, communicating in a wireless fading channel. We consider two objectives: maximizing the throughput by a deadline, and minimizing the transmission completion time of the communication session. We optimize these objectives by controlling the time sequence of transmit powers subject to energy storage capacity and causality constraints. We, first, study optimal offline policies. We introduce a directional water-filling algorithm which provides a simple and concise interpretation of the necessary optimality conditions. We show the optimality of an adaptive directional water-filling algorithm for the throughput maximization problem. We solve the transmission completion time minimization problem by utilizing its equivalence to its throughput maximization counterpart. Next, we consider online policies. We use stochastic dynamic programming to solve for the optimal online policy that maximizes the average number of bits delivered by a deadline under stochastic fading and energy arrival processes with causal channel state feedback. We also propose near-optimal policies with reduced complexity, and numerically study their performances along with the performances of the offline and online optimal policies under various different configurations. 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. One of the main operations in wireless sensor networks is the surveillance of a set of events (targets) that occur in the field. In practice, a node monitors an event accurately when it is located closer to it, while the opposite happens when the node is moving away from the target. This detection accuracy can be represented by a probabilistic distribution. Since the network nodes are usually randomly deployed, some of the events are monitored by a few nodes and others by many nodes. In applications where there is a need of a full coverage and of a minimum allowed detection accuracy, a single node may not be able to sufficiently cover an event by itself. In this case, two or more nodes are needed to collaborate and to cover a single target. Moreover, all the nodes must be connected with a base station that collects the monitoring data. In this paper we describe the problem of the minimum sampling quality, where an event must be sufficiently detected by the maximum possible amount of time. Since the probability of detecting a single target using randomly deployed static nodes is quite low, we present a localized algorithm based on mobile nodes. Our algorithm sacrifices a part of the energy of the nodes by moving them to a new location in order to satisfy the desired detection accuracy. It divides the monitoring process in rounds to extend the network lifetime, while it ensures connectivity with the base station. Furthermore, since the network lifetime is strongly related to the number of rounds, we propose two redeployment schemes that enhance the performance of our approach by balancing the number of sensors between densely covered areas and areas that are poorly covered. Finally, our evaluation results show an over 10 times improvement on the network lifetime compared to the case where the sensors are static. Our approaches, also, outperform a virtual forces algorithm when connectivity with the base station is required. The redeployment schemes present a good balance between network lifetime and convergence time.