7,902 research outputs found
Convergence of some leader election algorithms
We start with a set of n players. With some probability P(n,k), we kill n-k
players; the other ones stay alive, and we repeat with them. What is the
distribution of the number X_n of phases (or rounds) before getting only one
player? We present a probabilistic analysis of this algorithm under some
conditions on the probability distributions P(n,k), including stochastic
monotonicity and the assumption that roughly a fixed proportion alpha of the
players survive in each round.
We prove a kind of convergence in distribution for X_n-log_a n, where the
basis a=1/alpha; as in many other similar problems there are oscillations and
no true limit distribution, but suitable subsequences converge, and there is an
absolutely continuous random variable Z such that the distribution of X_n can
be approximated by Z+log_a n rounded to the nearest larger integer.
Applications of the general result include the leader election algorithm
where players are eliminated by independent coin tosses and a variation of the
leader election algorithm proposed by W.R. Franklin. We study the latter
algorithm further, including numerical results.Comment: 27 pages, 13 figures, 5 table
Convergence of some leader election algorithms
We start with a set of players. With some probability , we kill players; the other ones stay alive, and we repeat with them. What is the distribution of the number of \emph{phases} (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions , including stochastic monotonicity and the assumption that roughly a fixed proportion \al of the players survive in each round. We prove a kind of convergence in distribution for ; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable such that d\l(X_n, \lceil Z + \log_{1/\!\alpha} (n)\rceil\r) \to 0, where is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithm further, including numerical results
Randomization Adaptive Self-Stabilization
We present a scheme to convert self-stabilizing algorithms that use
randomization during and following convergence to self-stabilizing algorithms
that use randomization only during convergence. We thus reduce the number of
random bits from an infinite number to a bounded number. The scheme is
applicable to the cases in which there exits a local predicate for each node,
such that global consistency is implied by the union of the local predicates.
We demonstrate our scheme over the token circulation algorithm of Herman and
the recent constant time Byzantine self-stabilizing clock synchronization
algorithm by Ben-Or, Dolev and Hoch. The application of our scheme results in
the first constant time Byzantine self-stabilizing clock synchronization
algorithm that uses a bounded number of random bits
Automated Synthesis of Distributed Self-Stabilizing Protocols
In this paper, we introduce an SMT-based method that automatically
synthesizes a distributed self-stabilizing protocol from a given high-level
specification and network topology. Unlike existing approaches, where synthesis
algorithms require the explicit description of the set of legitimate states,
our technique only needs the temporal behavior of the protocol. We extend our
approach to synthesize ideal-stabilizing protocols, where every state is
legitimate. We also extend our technique to synthesize monotonic-stabilizing
protocols, where during recovery, each process can execute an most once one
action. Our proposed methods are fully implemented and we report successful
synthesis of well-known protocols such as Dijkstra's token ring, a
self-stabilizing version of Raymond's mutual exclusion algorithm,
ideal-stabilizing leader election and local mutual exclusion, as well as
monotonic-stabilizing maximal independent set and distributed Grundy coloring
Efficient size estimation and impossibility of termination in uniform dense population protocols
We study uniform population protocols: networks of anonymous agents whose
pairwise interactions are chosen at random, where each agent uses an identical
transition algorithm that does not depend on the population size . Many
existing polylog time protocols for leader election and majority
computation are nonuniform: to operate correctly, they require all agents to be
initialized with an approximate estimate of (specifically, the exact value
). Our first main result is a uniform protocol for
calculating with high probability in time and
states ( bits of memory). The protocol is
converging but not terminating: it does not signal when the estimate is close
to the true value of . If it could be made terminating, this would
allow composition with protocols, such as those for leader election or
majority, that require a size estimate initially, to make them uniform (though
with a small probability of failure). We do show how our main protocol can be
indirectly composed with others in a simple and elegant way, based on the
leaderless phase clock, demonstrating that those protocols can in fact be made
uniform. However, our second main result implies that the protocol cannot be
made terminating, a consequence of a much stronger result: a uniform protocol
for any task requiring more than constant time cannot be terminating even with
probability bounded above 0, if infinitely many initial configurations are
dense: any state present initially occupies agents. (In particular,
no leader is allowed.) Crucially, the result holds no matter the memory or time
permitted. Finally, we show that with an initial leader, our size-estimation
protocol can be made terminating with high probability, with the same
asymptotic time and space bounds.Comment: Using leaderless phase cloc
Leader election: A Markov chain approach
A well-studied randomized election algorithm proceeds as follows: In each
round the remaining candidates each toss a coin and leave the competition if
they obtain heads. Of interest is the number of rounds required and the number
of winners, both related to maxima of geometric random samples, as well as the
number of remaining participants as a function of the number of rounds. We
introduce two related Markov chains and use ideas and methods from discrete
potential theory to analyse the respective asymptotic behaviour as the initial
number of participants grows. One of the tools used is the approach via the
R\'enyi-Sukhatme representation of exponential order statistics, which was
first used in the leader election context by Bruss and Gr\"ubel in
\cite{BrGr03}
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