4,733 research outputs found
A probabilistic analysis of a leader election algorithm
A {\em leader election} algorithm is an elimination process that divides
recursively into tow subgroups an initial group of n items, eliminates one
subgroup and continues the procedure until a subgroup is of size 1. In this
paper the biased case is analyzed. We are interested in the {\em cost} of the
algorithm, i.e. the number of operations needed until the algorithm stops.
Using a probabilistic approach, the asymptotic behavior of the algorithm is
shown to be related to the behavior of a hitting time of two random sequences
on [0,1]
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
Leader Election in Anonymous Rings: Franklin Goes Probabilistic
We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size
The Computational Power of Beeps
In this paper, we study the quantity of computational resources (state
machine states and/or probabilistic transition precision) needed to solve
specific problems in a single hop network where nodes communicate using only
beeps. We begin by focusing on randomized leader election. We prove a lower
bound on the states required to solve this problem with a given error bound,
probability precision, and (when relevant) network size lower bound. We then
show the bound tight with a matching upper bound. Noting that our optimal upper
bound is slow, we describe two faster algorithms that trade some state
optimality to gain efficiency. We then turn our attention to more general
classes of problems by proving that once you have enough states to solve leader
election with a given error bound, you have (within constant factors) enough
states to simulate correctly, with this same error bound, a logspace TM with a
constant number of unary input tapes: allowing you to solve a large and
expressive set of problems. These results identify a key simplicity threshold
beyond which useful distributed computation is possible in the beeping model.Comment: Extended abstract to appear in the Proceedings of the International
Symposium on Distributed Computing (DISC 2015
Counterexample Generation in Probabilistic Model Checking
Providing evidence for the refutation of a property is an essential, if not the most important, feature of model checking. This paper considers algorithms for counterexample generation for probabilistic CTL formulae in discrete-time Markov chains. Finding the strongest evidence (i.e., the most probable path) violating a (bounded) until-formula is shown to be reducible to a single-source (hop-constrained) shortest path problem. Counterexamples of smallest size that deviate most from the required probability bound can be obtained by applying (small amendments to) k-shortest (hop-constrained) paths algorithms. These results can be extended to Markov chains with rewards, to LTL model checking, and are useful for Markov decision processes. Experimental results show that typically the size of a counterexample is excessive. To obtain much more compact representations, we present a simple algorithm to generate (minimal) regular expressions that can act as counterexamples. The feasibility of our approach is illustrated by means of two communication protocols: leader election in an anonymous ring network and the Crowds protocol
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