Generating a Random Sink-free Orientation in Quadratic Time

A sink-free orientation of a finite undirected graph is a choice of orientation for each edge such that every vertex has out-degree at least 1. Bubley and Dyer (1997) use Markov Chain Monte Carlo to sample approximately from the uniform distribution on sink-free orientations in time $O(m^3 \log (1 / \varepsilon))$, where $m$ is the number of edges and $\varepsilon$ the degree of approximation. Huber (1998) uses coupling from the past to obtain an exact sample in time $O(m^4)$. We present a simple randomized algorithm inspired by Wilson's cycle popping method which obtains an exact sample in mean time at most $O(nm)$, where $n$ is the number of vertices.

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Widening and clustering techniques allowing the use of monotone CFTP algorithm

Monte Carlo Methods and Applications ◽

10.1515/mcma-2015-0111 ◽

2015 ◽

Vol 21 (4) ◽

Author(s):

Mohamed Yasser Bounnite ◽

Abdelaziz Nasroallah

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Monte Carlo Simulations ◽

State Space ◽

Stationary Distribution ◽

Clustering Techniques ◽

Clustering Technique ◽

The Past ◽

Coupling From The Past

AbstractThe standard Coupling From The Past (CFTP) algorithm is an interesting tool to sample from exact stationary distribution of a Markov chain. But it is very expensive in time consuming for large chains. There is a monotone version of CFTP, called MCFTP, that is less time consuming for monotone chains. In this work, we propose two techniques to get monotone chain allowing use of MCFTP: widening technique based on adding two fictitious states and clustering technique based on partitioning the state space in clusters. Usefulness and efficiency of our approaches are showed through a sample of Markov Chain Monte Carlo simulations.

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THE MEAN WAITING TIME FOR A PATTERN

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964899131012 ◽

1999 ◽

Vol 13 (1) ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Sheldon M. Ross

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Waiting Time ◽

Random Variables ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Mean Waiting Time ◽

The Mean ◽

Mean Time

Consider a sequence of independent and identically distributed random variables along with a specified set of k-vectors. We present an expression for E [T], the mean time until the last k observed random variables fall within this set. Not only can this expression often be used to obtain bounds on E[T], it also gives rise to an efficient way of approximating E[T] by a simulation. Specific lower and upper bounds for E[T] are also derived. These latter bounds are given in terms of a parameter, and a Markov chain Monte Carlo approach to approximate this parameter by a simulation is indicated. The results of this paper are illustrated by considering the problem of determining the mean time until a sequence of k-valued random variables has a run of size k that encompasses each value.

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A kind of dual form for coupling from the past algorithm, to sample from Markov chain steady-state probability

Monte Carlo Methods and Applications ◽

10.1515/mcma-2019-2050 ◽

2019 ◽

Vol 25 (4) ◽

pp. 317-327

Author(s):

Abdelaziz Nasroallah ◽

Mohamed Yasser Bounnite

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Steady State ◽

Performance Comparison ◽

State Probability ◽

Steady State Probability ◽

Dual Form ◽

The Past ◽

Coupling From The Past ◽

History Of

Abstract The standard coupling from the past (CFTP) algorithm is an interesting tool to sample from exact Markov chain steady-state probability. The CFTP detects, with probability one, the end of the transient phase (called burn-in period) of the chain and consequently the beginning of its stationary phase. For large and/or stiff Markov chains, the burn-in period is expensive in time consumption. In this work, we propose a kind of dual form for CFTP called D-CFTP that, in many situations, reduces the Monte Carlo simulation time and does not need to store the history of the used random numbers from one iteration to another. A performance comparison of CFTP and D-CFTP will be discussed, and some numerical Monte Carlo simulations are carried out to show the smooth running of the proposed D-CFTP.

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