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 resolve several fundamental questions in the area of distributed functional monitoring, initiated by Cormode, Muthukrishnan, and Yi (SODA, 2008). In this model there are @math sites each tracking their input and communicating with a central coordinator that continuously maintain an approximate output to a function @math computed over the union of the inputs. The goal is to minimize the communication. We show the randomized communication complexity of estimating the number of distinct elements up to a @math factor is @math , improving the previous @math bound and matching known upper bounds up to a logarithmic factor. For the @math -th frequency moment @math , @math , we improve the previous @math communication bound to @math . We obtain similar improvements for heavy hitters, empirical entropy, and other problems. We also show that we can estimate @math , for any @math , using @math communication. This greatly improves upon the previous @math bound of Cormode, Muthukrishnan, and Yi for general @math , and their @math bound for @math . For @math , our bound resolves their main open question. Our lower bounds are based on new direct sum theorems for approximate majority, and yield significant improvements to problems in the data stream model, improving the bound for estimating @math in @math passes from @math to @math , giving the first bound for estimating @math in @math passes of @math bits of space that does not use the gap-hamming problem. We propose a new randomized coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel, and PROXimal; this is the first time such a method is proposed. In the special case when the number of processors is equal to the number of coordinates, the method converges at the rate @math , where @math is the iteration counter, @math is a data-weighted average degree of separability of the loss function, @math is the average of Lipschitz constants associated with the coordinates and individual functions in the sum, and @math is the distance of the initial point from the minimizer. We show that the method can be implemented without the need to perform full-dimensional vector operations, which is the major bottleneck of accelerated coordinate descent. The fact that the method depends on the average degree of separability, and not on the maximum degree... The notion of distributed functional monitoring was recently introduced by Cormode, Muthukrishnan and Yi to initiate a formal study of the communication cost of certain fundamental problems arising in distributed systems, especially sensor networks. In this model, each of k sites reads a stream of tokens and is in communication with a central coordinator, who wishes to continuously monitor some function f of *** , the union of the k streams. The goal is to minimize the number of bits communicated by a protocol that correctly monitors f (*** ), to within some small error. As in previous work, we focus on a threshold version of the problem, where the coordinator's task is simply to maintain a single output bit, which is 0 whenever f (*** ) ≤ *** (1 *** *** ) and 1 whenever f (*** ) *** *** . Following , we term this the (k ,f ,*** ,*** ) functional monitoring problem. In previous work, some upper and lower bounds were obtained for this problem, with f being a frequency moment function, e.g., F 0 , F 1 , F 2 . Importantly, these functions are monotone . Here, we further advance the study of such problems, proving three new classes of results. First, we provide nontrivial monitoring protocols when f is either H , the empirical Shannon entropy of a stream, or any of a related class of entropy functions (Tsallis entropies). These are the first nontrivial algorithms for distributed monitoring of non-monotone functions. Second, we study the effect of non-monotonicity of f on our ability to give nontrivial monitoring protocols, by considering f = F p with deletions allowed, as well as f = H . Third, we prove new lower bounds on this problem when f = F p , for several values of p . Massive data sets often arise as physically distributed, parallel data streams. We present algorithms for estimating simple functions on the union of such data streams, while using only logarithmic space per stream. Each processor observes only its own stream, and communicates with the other processors only after observing its entire stream. This models the set-up in current network monitoring products. Our algorithms employ a novel coordinated sampling technique to extract a sample of the union; this sample can be used to estimate aggregate functions on the union. The technique can also be used to estimate aggregate functions over the distinct “labels” in one or more data streams, e.g., to determine the zeroth frequency moment (i.e., the number of distinct labels) in one or more data streams. Our space and time bounds are the best known for these problems, and our logarithmic space bounds for coordinated sampling contrast with polynomial lower bounds for independent sampling. We relate our distributed streams model to previously studied non-distributed (i.e., merged) streams models, presenting tight bounds on the gap between the distributed and merged models for deterministic algorithms. We propose a new stochastic coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel and PROXimal; this is the first time such a method is proposed. In the special case when the number of processors is equal to the number of coordinates, the method converges at the rate @math , where @math is the iteration counter, @math is an average degree of separability of the loss function, @math is the average of Lipschitz constants associated with the coordinates and individual functions in the sum, and @math is the distance of the initial point from the minimizer. We show that the method can be implemented without the need to perform full-dimensional vector operations, which is the major bottleneck of existing accelerated coordinate descent methods. The fact that the method depends on the average degree of separability, and not on the maximum degree of separability, can be attributed to the use of new safe large stepsizes, leading to improved expected separable overapproximation (ESO). These are of independent interest and can be utilized in all existing parallel stochastic coordinate descent algorithms based on the concept of ESO. We consider continuous maintenance of a random sample of distinct elements from a massive data stream, whose input elements are observed at multiple distributed sites that communicate via a central coordinator. At any point, when a query is received at the coordinator, it responds with a random sample from the set of all distinct elements observed at the different sites so far. We present the first algorithms for distinct random sampling from a distributed stream. We also present a lower bound on the expected number of messages that must be transmitted by any distributed algorithm, showing that our algorithm is message optimal to within a factor of four. We present extensions to sliding windows, and experimental results showing the performance of our algorithm on real-world data sets. In this paper, we study the interference mitigation scheme in a network with multiple co-located wireless body area networks (WBANs). Each WBAN consists of a coordinator and multiple sensor nodes. Interference happens when multiple nodes transmit to their coordinators at the same time. Our objective is twofold: firstly we want to construct an interference-free time slot schedule for all the nodes in the network; secondly we want to minimize the transmission cycle of all the nodes. Towards such goal, we map different time slots to distinct colors and propose a WBAN distributed coloring (DC) algorithm to find a color assignment for each node in the network. To implement the algorithm, the coordinators need to exchange messages for multiple rounds to achieve a non-conflict coloring scheme distributively. The simulation results show that on average the proposed algorithm has a significant performance gain over existing schemes. 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. We consider n wireless ad hoc network nodes with one antenna each and equidistantly placed on a line. The transmission power of each node is just large enough to reach its next neighbor. For this setting we show that a message can be broadcasted to all nodes in time O(log n) without increasing each node's transmission power. Our algorithm needs O(log n) messages and consumes a total energy which is only a constant factor larger than the standard approach where nodes sequentially transmit the broadcast message to their next neighbors. We obtain this by synchronizing the nodes on the fly and using MIMO (multiple input multiple output) techniques. To achieve this goal we analyze the communication capacity of multiple antennas positioned on a line and use a communication model which is based on electromagnetic fields in free space. We extend existing communication models which either reflect only the sender power or neglect the locations by concentrating only on the channel matrix. Here, we compute the scalar channel matrix from the locations of the antennas and thereby only consider line-of-sight-communication without obstacles, reflections, diffractions or scattering. First, we show that this communication model reduces to the SINR power model if the antennas are uncoordinated. We show that n coordinated antennas can send a signal which is n times more powerful than the sum of their transmission powers. Alternatively, the power can be reduced to an arbitrarily small polynomial with respect to the distance. For coordinated antennas we show how the well-known power gain for MISO (multiple input single output) and SIMO (single input multiple output) can be described in this model. Furthermore, we analyze the channel matrix and prove that in the free space model no diversity gain can be expected for MIMO. Finally, we present the logarithmic time broadcast algorithm which takes advantage of the MISO power gain by self-coordinating wireless nodes. Motivated by applications in blockchains and sensor networks, we consider a model of @math nodes trying to reach consensus on their majority bit. Each node @math is assigned a bit at time zero, and is a finite automaton with @math bits of memory (i.e., @math states) and a Poisson clock. When the clock of @math rings, @math can choose to communicate, and is then matched to a uniformly chosen node @math . The nodes @math and @math may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that when @math , consensus can be reached at linear communication cost, but this is impossible if @math . We also study a synchronous variant of the model, where our upper and lower bounds on @math for achieving linear communication cost are @math and @math , respectively. A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low. 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 propose a novel stochastic gradient method---semi-stochastic coordinate descent (S2CD)---for the problem of minimizing a strongly convex function represented as the average of a large number of smooth convex functions: @math . Our method first performs a deterministic step (computation of the gradient of @math at the starting point), followed by a large number of stochastic steps. The process is repeated a few times, with the last stochastic iterate becoming the new starting point where the deterministic step is taken. The novelty of our method is in how the stochastic steps are performed. In each such step, we pick a random function @math and a random coordinate @math ---both using nonuniform distributions---and update a single coordinate of the decision vector only, based on the computation of the @math partial derivative of @math at two different points. Each random step of the method constitutes an unbiased estimate of the gradient of @math and moreover, the squared norm of the steps goes to zero in expectation, meaning that the stochastic estimate of the gradient progressively improves. The complexity of the method is the sum of two terms: @math evaluations of gradients @math and @math evaluations of partial derivatives @math , where @math is a novel condition number. We consider broadcasting in radio networks, modeled as undirected graphs, whose nodes know only their own label and labels of their neighbors. In every step every node acts either as a transmitter or as a receiver. A node acting as a transmitter sends a message which can potentially reach all of its neighbors. A node acting as a receiver in a given step gets a message if and only if exactly one of its neighbors transmits in this step. Bar-Yehuda, Goldreich, and Itai [J. Comput. System Sci., 45 (1992), pp. 104--126] considered broadcasting in this model. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks of diameter 3. This claim turns out to be incorrect in this model (although it is valid in a more pessimistic model [R. Bar-Yehuda, O. Goldreich, and A. Itai, Errata Regarding "On the time complexity of broadcast in radio networks: An exponential gap between determinism and randomization," http: www.wisdom.weizmann.ac.il mathusers oded p .html, 2002]). We construct an algorithm that broadcasts in logarithmic time on all graphs from the Bar-Yehuda, Goldreich, and Itai paper (BGI). Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter @math . On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time @math on these graphs. In view of the randomized algorithm from BGI, running in expected time @math on all @math -node graphs of diameter D (cf. also a recent @math -time algorithm from [D. Kowalski and A. Pelc, Proceedings of the 22nd Annual ACM Symposium on Principles of Distributed Computing, Boston, 2003, pp. 73--82; A. Czumaj and W. Rytter, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, Cambridge, MA, 2003, pp. 492--501]), our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting, under the considered model of radio communication. In this paper we study the problem of data exchange, where each node in the system has a number of linear combinations of the data packets. Communicating over a public channel, the goal is for all nodes to reconstruct the entire set of the data packets in minimal total number of bits exchanged over the channel. We present a novel divide and conquer based architecture that determines the number of bits each node should transmit. This along with the well known fact, that it is sufficient for the nodes to broadcast linear combinations of their local information, provides a polynomial time deterministic algorithm for reconstructing the entire set of the data packets at all nodes in minimal amount of total communication. We present a simple, message-optimal algorithm for maintaining a random sample from a large data stream whose input elements are distributed across multiple sites that communicate via a central coordinator. At any point in time, the set of elements held by the coordinator represent a uniform random sample from the set of all the elements observed so far. When compared with prior work, our algorithms asymptotically improve the total number of messages sent in the system. We present a matching lower bound, showing that our protocol sends the optimal number of messages up to a constant factor with large probability. We also consider the important case when the distribution of elements across different sites is non-uniform, and show that for such inputs, our algorithm significantly outperforms prior solutions. Recent advances in miniaturization and low-cost, low-power design have led to active research in large-scale networks of small, wireless, low-power sensors and actuators. Time synchronization is critical in sensor networks for diverse purposes including sensor data fusion, coordinated actuation, and power-efficient duty cycling. Though the clock accuracy and precision requirements are often stricter than in traditional distributed systems, strict energy constraints limit the resources available to meet these goals.We present Reference-Broadcast Synchronization, a scheme in which nodes send reference beacons to their neighbors using physical-layer broadcasts. A reference broadcast does not contain an explicit timestamp; instead, receivers use its arrival time as a point of reference for comparing their clocks. In this paper, we use measurements from two wireless implementations to show that removing the sender's nondeterminism from the critical path in this way produces high-precision clock agreement (1.85 ± 1.28μsec, using off-the-shelf 802.11 wireless Ethernet), while using minimal energy. We also describe a novel algorithm that uses this same broadcast property to federate clocks across broadcast domains with a slow decay in precision (3.68 ± 2.57μsec after 4 hops). RBS can be used without external references, forming a precise relative timescale, or can maintain microsecond-level synchronization to an external timescale such as UTC. We show a significant improvement over the Network Time Protocol (NTP) under similar conditions. This paper concerns the worst-case complexity of Gauss-Seidel method for solving a positive semi-definite linear system; or equivalently, that of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function. The known provable complexity of C-CD can be @math times slower than gradient descent (GD) and @math times slower than randomized coordinate descent (R-CD). However, these gaps seem rather puzzling since so far they have not been observed in practice; in fact, C-CD usually converges much faster than GD and sometimes comparable to R-CD. Thus some researchers believe the gaps are due to the weakness of the proof, but not that of the C-CD algorithm itself. In this paper we show that the gaps indeed exist. We prove that there exists an example for which C-CD takes at least @math or @math operations, where @math is the condition number, @math is a well-studied quantity that determines the convergence rate of R-CD, and @math hides the dependency on @math . This implies that C-CD can indeed be @math times slower than GD, and @math times slower than R-CD in the worst case. Our result establishes one of the few examples in continuous optimization that demonstrates a large gap between a deterministic algorithm and its randomized counterpart. Based on the example, we establish several almost tight complexity bounds of C-CD for quadratic problems. One difficulty with the analysis of the constructed example is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitutes a lower bound for the convergence rate. This paper focuses on status age, which is a metric for measuring the freshness of a continually updated piece of information (i.e., status) as observed at a remote monitor. In paper, we study a system in which a sensor sends random status updates over a dynamic network to a monitor. For this system, we consider the impact of having messages take different routes through the network on the status age. First, we consider a network with plentiful resources (i.e., many nodes that can provide numerous alternate paths), so that packets need not wait in queues at each node in a multihop path. This system is modeled as a single queue with an infinite number of servers, specifically as an @math queue. Packets routed over a dynamic network may arrive at the monitor out of order, which we account for in our analysis for the @math model. We then consider a network with somewhat limited resources, so that packets can arrive out of order but also must wait in a queue. This is modeled as a single queue with two servers, specifically an @math queue. We present the exact approach to computing the analytical status age, and we provide an approximation that is shown to be close to the simulated age. We also compare both models with @math , which corresponds to severely limited network resources, and we demonstrate the tradeoff between the status age and the unnecessary network resource consumption. There is a rising interest in accelerating stream processing through modern parallel hardware, yet it remains a challenge as how to exploit the available resources to achieve higher throughput without sacrificing latency due to the increased length of processing pipeline and communication path and the need for central coordination. To achieve these objectives, we introduce a novel top-down data flow model for stream join processing (arguably, one of the most resource-intensive operators in stream processing), called SplitJoin, that operates by splitting the join operation into independent storing and processing steps that gracefully scale with respect to the number of cores. Furthermore, SplitJoin eliminates the need for global coordination while preserving the order of input streams by re-thinking how streams are channeled into distributed join computation cores and maintaining the order of output streams by proposing a novel distributed punctuation technique. Throughout our experimental analysis, SplitJoin offered up to 60 improvement in throughput while reducing latency by up to 3.3X compared to state-of-the-art solutions. In the model of continuous distributed monitoring, a number of observers each see a stream of observations. Their goal is to work together to compute a function of the union of their observations. This can be as simple as counting the total number of observations, or more complex non-linear functions such as tracking the entropy of the induced distribution. Assuming that it is too costly to simply centralize all the observations, it becomes quite challenging to design solutions which provide a good approximation to the current answer, while bounding the communication cost of the observers, and their other resources such as their space usage. This survey introduces this model, and describe a selection results in this setting, from the simple counting problem to a variety of other functions that have been studied.