Faster random generation of linear extensions

@article{Bubley1998FasterRG,
  title={Faster random generation of linear extensions},
  author={Russ Bubley and Martin E. Dyer},
  journal={Discret. Math.},
  year={1998},
  volume={201},
  pages={81-88}
}
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183 Citations
Highly Influential Citations
13
Background Citations
46
Methods Citations
47

183 Citations

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Citation Type

Fast perfect sampling from linear extensions
The Mixing of Markov Chains on Linear Extensions in Practice
TLDR
The empirical results suggest that the Markov chain approach to sample linear extensions can be made to scale well in practice, provided that the actual mixing times can be realized by instance-sensitive bounds or termination rules.
A Sequential Importance Sampling Algorithm for Counting Linear Extensions
TLDR
This technique is based on improving the so-called Knuth counting algorithm by incorporating an importance function into the node selection technique giving a sequential importance sampling (SIS) method, and defines and reports performance on two importance functions.
Surveys in Combinatorics, 1999: Random Walks on Combinatorial Objects
Approximate sampling from combinatorially-defined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated…
Bounds on mixing time of finite Markov chains.
We provide a general framework for computing mixing times of finite Markov chains when its minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our…
Spectral Gap for Random-to-Random Shuffling on Linear Extensions
TLDR
This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by n2/(n + 2) and a mixing time of O(n2log n) as for the usual random-to-random shuffling.
Mixing times of lozenge tiling and card shuffling Markov chains
We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its…
Path coupling: A technique for proving rapid mixing in Markov chains
  • Russ Bubley, M. Dyer
  • Mathematics
    Proceedings 38th Annual Symposium on Foundations of Computer Science
  • 1997
TLDR
A new approach to the coupling technique, which is called path coupling, for bounding mixing rates, is illustrated, which may allow coupling proofs which were previously unknown, or provide significantly better bounds than those obtained using the standard method.
An approximation algorithm for random generation of capacities
TLDR
This paper proposes the 2-layer approximation method, which generates a subset of linear extensions, eliminating those with very low probability, and shows that this method has similar performance compared to the Markov chain but is much less time consuming.
L ∞ -Discrepancy Analysis of Polynomial-Time Deterministic Samplers Emulating Rapidly Mixing Chains
TLDR
A deterministic sampling algorithm based on deterministic random walk, such as the rotor-router model (a.k.a. Propp machine) is proposed, which provides samples with a “distribution” with a point-wise distance at most e from the target distribution, in time polynomial in the input size and e − 1.
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References

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The general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomised approximate counting algorithms for these graphs with very good asymptotic behaviour.
Markov chains for linear extensions, the two-dimensional case
TLDR
It is shown that monotone coupling from the past can be applied in the case of linear extensions of two-dimensional orders using Markou chains and for width two orders a mixing rate of O(n3 logn) is proved.
Path coupling: A technique for proving rapid mixing in Markov chains
  • Russ Bubley, M. Dyer
  • Mathematics
    Proceedings 38th Annual Symposium on Foundations of Computer Science
  • 1997
TLDR
A new approach to the coupling technique, which is called path coupling, for bounding mixing rates, is illustrated, which may allow coupling proofs which were previously unknown, or provide significantly better bounds than those obtained using the standard method.
Counting linear extensions
We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #P-complete, settling a long-standing open question. This result is…
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A new rapidly mixing Markov chain for independent sets is defined and a polynomial upper bound for the mixing time of the new chain is obtained for a certain range of values of the parameter ?, which is wider than the range for which the mixingTime of the Luby?Vigoda chain is known to be polynomially bounded.
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Generating Linear Extensions Fast
TLDR
The algorithm presented here is the first constant amortized time algorithm for generating a "naturally defined" class of combinatorial objects for which the corresponding counting problem is #P-complete.
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