1,236 research outputs found
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
Polynomial Anonymous Dynamic Distributed Computing Without a Unique Leader
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
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
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
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
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . 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
, 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 is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-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
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of 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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