1,236 research outputs found

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

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    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

    Polynomial Anonymous Dynamic Distributed Computing Without a Unique Leader

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    Counting the number of nodes in {Anonymous Dynamic Networks} is enticing from an algorithmic perspective: an important computation in a restricted platform with promising applications. Starting with Michail, Chatzigiannakis, and Spirakis [Michail et al., 2013], a flurry of papers sped up the running time guarantees from doubly-exponential to polynomial [Dariusz R. Kowalski and Miguel A. Mosteiro, 2018]. There is a common theme across all those works: a distinguished node is assumed to be present, because Counting cannot be solved deterministically without at least one. In the present work we study challenging questions that naturally follow: how to efficiently count with more than one distinguished node, or how to count without any distinguished node. More importantly, what is the minimal information needed about these distinguished nodes and what is the best we can aim for (count precision, stochastic guarantees, etc.) without any. We present negative and positive results to answer these questions. To the best of our knowledge, this is the first work that addresses them

    Optimal Computation in Leaderless and Multi-Leader Disconnected Anonymous Dynamic Networks

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    We give a simple characterization of which functions can be computed deterministically by anonymous processes in disconnected dynamic networks, depending on the number of leaders in the network. In addition, we provide efficient distributed algorithms for computing all such functions assuming minimal or no knowledge about the network. Each of our algorithms comes in two versions: one that terminates with the correct output and a faster one that stabilizes on the correct output without explicit termination. Notably, these are the first deterministic algorithms whose running times scale linearly with both the number of processes and a parameter of the network which we call "dynamic disconnectivity". We also provide matching lower bounds, showing that all our algorithms are asymptotically optimal for any fixed number of leaders. While most of the existing literature on anonymous dynamic networks relies on classical mass-distribution techniques, our work makes use of a recently introduced combinatorial structure called "history tree", also developing its theory in new directions. Among other contributions, our results make definitive progress on two popular fundamental problems for anonymous dynamic networks: leaderless Average Consensus (i.e., computing the mean value of input numbers distributed among the processes) and multi-leader Counting (i.e., determining the exact number of processes in the network). In fact, our approach unifies and improves upon several independent lines of research on anonymous networks, including Nedic et al., IEEE Trans. Automat. Contr. 2009; Olshevsky, SIAM J. Control Optim. 2017; Kowalski-Mosteiro, ICALP 2019, SPAA 2021; Di Luna-Viglietta, FOCS 2022.Comment: 35 pages, 1 figure. arXiv admin note: text overlap with arXiv:2204.0212

    Computing Outside the Box: Average Consensus over Dynamic Networks

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    International audienceNetworked systems of autonomous agents, and applications thereof, often rely on the control primitive of average consensus, where the agents are to compute the average of private initial values. To provide reliable services that are easy to deploy, average consensus should continue to operate when the network is subject to frequent and unpredictable change, and should mobilize few computational resources, so that deterministic, low powered, and anonymous agents can partake in the network.In this stringent adversarial context, we investigate the implementation of average consensus by distributed algorithms over networks with bidirectional, but potentially short-lived, communication links. Inspired by convex recurrence rules for multi-agent systems, and the Metropolis average consensus rule in particular, we design a deterministic distributed algorithm that achieves asymptotic average consensus, which we show to operate in polynomial time in a synchronous temporal model.The algorithm is easy to implement, has low space and computational complexity, and is fully distributed, requiring neither symmetry-breaking devices like unique identifiers, nor global control or knowledge of the network. In the fully decentralized model that we adopt, to our knowledge, no other distributed average consensus algorithm has a better temporal complexity.Our approach distinguishes itself from classical convex recurrence rules in that the agent’s values may sometimes leave their previous convex hull. As a consequence, our convergence bound requires a subtle analysis, despite the syntactic simplicity of our algorithm

    Moment Closure - A Brief Review

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    Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in mathematics, physics, chemistry and quantitative biolog

    A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

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    For even kk, the matchings connectivity matrix Mk\mathbf{M}_k encodes which pairs of perfect matchings on kk vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of Mk\mathbf{M}_k over Z2\mathbb{Z}_2 is Θ(2k)\Theta(\sqrt 2^k) and used this to give an O((2+2)pw)O^*((2+\sqrt{2})^{\mathsf{pw}}) time algorithm for counting Hamiltonian cycles modulo 22 on graphs of pathwidth pw\mathsf{pw}. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within Mk\mathbf{M}_k, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of Mk\mathbf{M}_k is given; no stronger structural insights such as the existence of large permutation submatrices in Mk\mathbf{M}_k are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes pp) parameterized by pathwidth. To apply this technique, we prove that the rank of Mk\mathbf{M}_k over the rationals is 4k/poly(k)4^k / \mathrm{poly}(k). We also show that the rank of Mk\mathbf{M}_k over Zp\mathbb{Z}_p is Ω(1.97k)\Omega(1.97^k) for any prime p2p\neq 2 and even Ω(2.15k)\Omega(2.15^k) for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time O((6ϵ)pw)O^*((6-\epsilon)^{\mathsf{pw}}) for any ϵ>0\epsilon>0 unless SETH fails. This bound is tight due to a O(6pw)O^*(6^{\mathsf{pw}}) time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes p2p\neq 2 in time O(3.97pw)O^*(3.97^\mathsf{pw}), indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201

    Master index volumes 51–60

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    Faster all-pairs shortest paths via circuit complexity

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    We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two n×nn \times n matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense nn-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time n32Ω(logn)1/2\frac{n^3}{2^{\Omega(\log n)^{1/2}}} and is correct with high probability. On the word RAM, the algorithm runs in n3/2Ω(logn)1/2+n2+o(1)logMn^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log M time for edge weights in ([0,M]Z){}([0,M] \cap {\mathbb Z})\cup\{\infty\}. Prior algorithms used either n3/(logcn)n^3/(\log^c n) time for various c2c \leq 2, or O(Mαnβ)O(M^{\alpha}n^{\beta}) time for various α>0\alpha > 0 and β>2\beta > 2. The new algorithm applies a tool from circuit complexity, namely the Razborov-Smolensky polynomials for approximately representing AC0[p]{\sf AC}^0[p] circuits, to efficiently reduce a matrix product over the (min,+)(\min,+) algebra to a relatively small number of rectangular matrix products over F2{\mathbb F}_2, each of which are computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in n3/2logδnn^3/2^{\log^{\delta} n} time for some δ>0\delta > 0, which utilizes the Yao-Beigel-Tarui translation of AC0[m]{\sf AC}^0[m] circuits into "nice" depth-two circuits.Comment: 24 pages. Updated version now has slightly faster running time. To appear in ACM Symposium on Theory of Computing (STOC), 201
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