2,128 research outputs found

    Optimal Gossip Algorithms for Exact and Approximate Quantile Computations

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    This paper gives drastically faster gossip algorithms to compute exact and approximate quantiles. Gossip algorithms, which allow each node to contact a uniformly random other node in each round, have been intensely studied and been adopted in many applications due to their fast convergence and their robustness to failures. Kempe et al. [FOCS'03] gave gossip algorithms to compute important aggregate statistics if every node is given a value. In particular, they gave a beautiful O(logn+log1ϵ)O(\log n + \log \frac{1}{\epsilon}) round algorithm to ϵ\epsilon-approximate the sum of all values and an O(log2n)O(\log^2 n) round algorithm to compute the exact ϕ\phi-quantile, i.e., the the ϕn\lceil \phi n \rceil smallest value. We give an quadratically faster and in fact optimal gossip algorithm for the exact ϕ\phi-quantile problem which runs in O(logn)O(\log n) rounds. We furthermore show that one can achieve an exponential speedup if one allows for an ϵ\epsilon-approximation. We give an O(loglogn+log1ϵ)O(\log \log n + \log \frac{1}{\epsilon}) round gossip algorithm which computes a value of rank between ϕn\phi n and (ϕ+ϵ)n(\phi+\epsilon)n at every node.% for any 0ϕ10 \leq \phi \leq 1 and 0<ϵ<10 < \epsilon < 1. Our algorithms are extremely simple and very robust - they can be operated with the same running times even if every transmission fails with a, potentially different, constant probability. We also give a matching Ω(loglogn+log1ϵ)\Omega(\log \log n + \log \frac{1}{\epsilon}) lower bound which shows that our algorithm is optimal for all values of ϵ\epsilon

    Distributed Data Summarization in Well-Connected Networks

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    We study distributed algorithms for some fundamental problems in data summarization. Given a communication graph G of n nodes each of which may hold a value initially, we focus on computing sum_{i=1}^N g(f_i), where f_i is the number of occurrences of value i and g is some fixed function. This includes important statistics such as the number of distinct elements, frequency moments, and the empirical entropy of the data. In the CONGEST~ model, a simple adaptation from streaming lower bounds shows that it requires Omega~(D+ n) rounds, where D is the diameter of the graph, to compute some of these statistics exactly. However, these lower bounds do not hold for graphs that are well-connected. We give an algorithm that computes sum_{i=1}^{N} g(f_i) exactly in {tau_{G}} * 2^{O(sqrt{log n})} rounds where {tau_{G}} is the mixing time of G. This also has applications in computing the top k most frequent elements. We demonstrate that there is a high similarity between the GOSSIP~ model and the CONGEST~ model in well-connected graphs. In particular, we show that each round of the GOSSIP~ model can be simulated almost perfectly in O~({tau_{G}}) rounds of the CONGEST~ model. To this end, we develop a new algorithm for the GOSSIP~ model that 1 +/- epsilon approximates the p-th frequency moment F_p = sum_{i=1}^N f_i^p in O~(epsilon^{-2} n^{1-k/p}) roundsfor p >= 2, when the number of distinct elements F_0 is at most O(n^{1/(k-1)}). This result can be translated back to the CONGEST~ model with a factor O~({tau_{G}}) blow-up in the number of rounds

    Engineering Resilient Collective Adaptive Systems by Self-Stabilisation

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    Collective adaptive systems are an emerging class of networked computational systems, particularly suited in application domains such as smart cities, complex sensor networks, and the Internet of Things. These systems tend to feature large scale, heterogeneity of communication model (including opportunistic peer-to-peer wireless interaction), and require inherent self-adaptiveness properties to address unforeseen changes in operating conditions. In this context, it is extremely difficult (if not seemingly intractable) to engineer reusable pieces of distributed behaviour so as to make them provably correct and smoothly composable. Building on the field calculus, a computational model (and associated toolchain) capturing the notion of aggregate network-level computation, we address this problem with an engineering methodology coupling formal theory and computer simulation. On the one hand, functional properties are addressed by identifying the largest-to-date field calculus fragment generating self-stabilising behaviour, guaranteed to eventually attain a correct and stable final state despite any transient perturbation in state or topology, and including highly reusable building blocks for information spreading, aggregation, and time evolution. On the other hand, dynamical properties are addressed by simulation, empirically evaluating the different performances that can be obtained by switching between implementations of building blocks with provably equivalent functional properties. Overall, our methodology sheds light on how to identify core building blocks of collective behaviour, and how to select implementations that improve system performance while leaving overall system function and resiliency properties unchanged.Comment: To appear on ACM Transactions on Modeling and Computer Simulatio
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