## Random walks on combinatorial objects (1999)

Venue: | Surveys in Combinatorics 1999 |

Citations: | 23 - 8 self |

### BibTeX

@INPROCEEDINGS{Dyer99randomwalks,

author = {Martin Dyer and Catherine Greenhill},

title = {Random walks on combinatorial objects},

booktitle = {Surveys in Combinatorics 1999},

year = {1999},

pages = {101--136},

publisher = {University Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Summary 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 problem of discrete integration over such sets. Recent work is reviewed, and we re-examine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined. In perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area. These are results which imply that there are no polynomial time algorithms of a particular type for a particular problem. 1

### Citations

2384 | Computational Complexity
- Papadimitriou
- 1995
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Citation Context ... some state ω0. For further information on Markov chains, see [2]. Throughout the paper, we refer to the well-known complexity classes P, NP and #P. For their definitions and further information, see =-=[64]-=-. 3 A computational framework Some foundations for approximate counting and uniform sampling were set out in the influential paper of Jerrum, Valiant and Vazirani [50]. However,sRandom Walks on Combin... |

659 | Some optimal inapproximability results
- Håstad
(Show Context)
Citation Context ...any good sampler when ∆ ≥ 25. We omit the proof, but simply remark that the result rests on recent deep work in the complexity of approximate optimization. In particular, it relies on work of H˚astad =-=[43]-=-. However, there is a clear gap between what is known to be possible (∆ ≤ 4) and this impossibility result (∆ ≥ 25). In this region, the following weaker result is proved in [25]. Theorem 8.2 Let ∆ ≥ ... |

522 |
The complexity of computing the permanent
- Valiant
- 1979
(Show Context)
Citation Context ...ty πσ(ω) = W (σ, ω)/Z(σ), with Z(σ) = � ω ′ ∈Ωσ W (σ, ω ′ ) being the corresponding normalising function. It is easy to see that the class of normalising functions so defined is essentially Valiant’s =-=[76]-=- class #P. The definition implies that, for some fixed c ∈ N, |Ωσ| ≤ Z(σ) ≤ 2 |σ|c. If Z(σ) = 0, then Ωσ = ∅ and πσ is the unique (improper) measure on Ωσ. In our definition, two distinct weight funct... |

419 | Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms
- Propp, Wilson
- 1996
(Show Context)
Citation Context ...ctly from a given distribution. We now briefly review results in this area. Further aspects of perfect sampling are considered in Section 7 below. The first paper in this area was by Propp and Wilson =-=[67]-=-. This paper introduced the approach known as coupling from the past (CFTP). This method is particularly efficient if the chain is monotone. In such a chain the state spacesRandom Walks on Combinatori... |

387 | Reversible Markov Chains and Random Walks on Graphs
- Aldous, Fill
(Show Context)
Citation Context ...chain is then τ(ε) = max p0 min t {dTV(pt, π) ≤ ε}, and it is easy to show that the maximum occurs when X0 = ω0, with probability one, for some state ω0. For further information on Markov chains, see =-=[2]-=-. Throughout the paper, we refer to the well-known complexity classes P, NP and #P. For their definitions and further information, see [64]. 3 A computational framework Some foundations for approximat... |

324 |
Lectures on the coupling method
- Lindvall
- 1992
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Citation Context ...ion at time t. As a measure of convergence, the natural choice in this context is variation distance, dTV(pt, π) = 1 2 � ω∈Ω |pt(ω) − π(ω)| = max A⊆Ω (pt(A) − π(A)) . (2) [Some authors, e.g. Lindvall =-=[59]-=-, define this without the 1.] The mixing time 2 of the chain is then τ(ε) = max p0 min t {dTV(pt, π) ≤ ε}, and it is easy to show that the maximum occurs when X0 = ω0, with probability one, for some s... |

286 |
Random generation of combinatorial structures from a uniform distribution
- Jerrum, Valiant, et al.
- 1986
(Show Context)
Citation Context ... from applied probability, but its application in this area has involved some new insights. Formal foundations for work in this area were provided in the seminal paper of Jerrum, Valiant and Vazirani =-=[50]-=-. However, the subject seems subsequently to have outgrown the framework it provided. The present paper makes a modest attempt to update the situation. We begin, in Section 2, by fixing notation which... |

242 | The Markov chain Monte Carlo method: an approach to approximate counting and integration
- Jerrum, Sinclair
- 1996
(Show Context)
Citation Context ...nd approximate integration during the last couple of years. There are several excellent survey articles which introduce the methods and results in this area, including Welsh [78], Jerrum and Sinclair =-=[49]-=- and Jerrum [48]. Therefore we do not give an exhaustive survey here, but concentrate on more recent results from the theoretical computer science community. There is a very extensive body of work on ... |

147 | Path coupling: A technique for proving rapid mixing
- Bubley, Dyer
- 1997
(Show Context)
Citation Context ...is the popular area of perfect sampling. We review perfect sampling separately, in Section 4.1 below. A variant of traditional coupling which is described in Section 6 is path coupling, introduced in =-=[9]-=-. (Bubley’s thesis [14] contains much work on coupling and path coupling, most of which we mention here.) Using path coupling, the authors [29] proved rapid mixing of a new chain for independent sets,... |

117 | A random polynomial-time algorithm for approximating the volume of convex bodies
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- 1991
(Show Context)
Citation Context ...p. 301–302].) More seriously, some problems to which the techniques of [50] have been applied, do not seem to be self-reducible under any re-encoding. Examples are the volume approximation problem of =-=[26]-=-, and approximately counting contingency tables [31, 25]. We therefore propose a modified framework. We emphasise that the techniques are those of [50], and we wish only to emphasise the extent of the... |

116 | Logarithmic Sobolev inequalities for finite Markov chains
- Diaconis, Saloff-Coste
- 1996
(Show Context)
Citation Context ...proved rapid mixing of a Markov chain for sampling generating sets of abelian groups, by comparing the log-Sobolev constants of two Markov chains. They used a comparison theorem from an earlier paper =-=[22]-=-. Other methods for relating the mixing times of Markov chains by comparing their spectral gap or log-Sobolev constant have been proposed by Chung and Graham [16], Randall and Tetali [70] and the auth... |

112 |
Generating random spanning trees
- Broder
- 1989
(Show Context)
Citation Context ...ized stationary stopping time which is within a constant factor of optimal. These algorithms are related to the generation of random spanning trees in a weighted directed graph. Aldous [1] and Broder =-=[6]-=- independently discovered an algorithm for generating random spanning trees of undirected graphs uniformly at random. Propp and Wilson [68] gave faster and more general algorithms for perfect generati... |

111 |
A very simple algorithm for estimating the number of k-colorings of a low-degree graph, Random Structures and Algorithms
- Jerrum
- 1995
(Show Context)
Citation Context ...t was obtained using classical coupling. In the limit, the antiferromagentic Potts model describes proper colourings of graphs. The simple Markov chain for graph colourings described independently by =-=[47]-=- and [71] is rapidly mixing if there are at least 2∆ colours, where ∆ is the maximum degree of the graph. This result was improved in a paper by the authors [30], which describes a Markov chain which ... |

102 |
Perfect simulation for the area-interaction point process
- Kendall
- 1998
(Show Context)
Citation Context ...us consider antimonotone systems. Here there is a partial order which is reversed by transitions of the Markov chain. This concept buildss14 Martin Dyer and Catherine Greenhill on the work of Kendall =-=[58]-=- on repulsive point-processes. Häggström and Nelander [42] showed how to apply CFTP to several systems, including independent sets in graphs, the antiferromagnetic Ising model, and the random cluster ... |

95 | How to get a perfectly random sample from a generic Markov Chain and Generate a random spanning tree of a directed graph - Propp, Wilson - 1998 |

94 | On some tighter inapproximability results
- Berman, Karpinski
- 1998
(Show Context)
Citation Context ...oda [62], who showed that there exists a constant value for which approximate counting is impossible, unless P = NP. This could be made quantitative, using more recent results of Berman and Karpinski =-=[5]-=-, but the best ∆ obtainable in this way seems to be around 8000. However, Dyer, Frieze and Jerrum [25] adopted a different (though related) approach, and proved the following stronger result. Theorem ... |

91 |
L.: Description of a random field by means of conditional probabilities and regularity conditions. Probability theory and applications
- Dobrushin
- 1968
(Show Context)
Citation Context ...[29]. Further results on bounding the spectral gap can be found in Guattery, Leighton and Miller [40] and in Chung and Yau [18]. Another approach used to bound the mixing time is Dobrushin uniqueness =-=[23]-=-. This approach was used by Salas and Sokal [71] in work on graph colourings, and by Peinado and Lengauer [65] to sample structures which arise in computational chemistry. The relationship between Dob... |

88 | Markov chain algorithms for planar lattice structures
- Luby, Randall, et al.
(Show Context)
Citation Context ...his case, we can show that dTV(pt, π) ≤ ε for t ≥ ⌈eD 2 /α⌉⌈ln(ε −1 )⌉. (19)sRandom Walks on Combinatorial Objects 19 This is most easily shown using a martingale argument. We omit the proof, but see =-=[61]-=- for details. Here we need D to be polynomial in the problem size. Example 6.1 We continue Example 5.2, by analysing a modification of the Insert/delete chain. We show that this is rapidly mixing if λ... |

82 | The random walk construction of uniform spanning trees
- Aldous
- 1990
(Show Context)
Citation Context ...niversal randomized stationary stopping time which is within a constant factor of optimal. These algorithms are related to the generation of random spanning trees in a weighted directed graph. Aldous =-=[1]-=- and Broder [6] independently discovered an algorithm for generating random spanning trees of undirected graphs uniformly at random. Propp and Wilson [68] gave faster and more general algorithms for p... |

82 | An interruptible algorithm for perfect sampling via Markov chains
- Fill
- 1998
(Show Context)
Citation Context ...astically by the transitions of the chain. For our purposes we may take this as meaning that there exists a coupling of the chain under which all transitions preserve the ordering. (See, for example, =-=[33]-=- for a more formal definition.) The idea of monotone CFTP is to run two copies of the chain from some time in the past, one starting from the top element of the state space and one from the bottom ele... |

74 | The complexity of counting in sparse, regular, and planar graphs
- Vadhan
- 1997
(Show Context)
Citation Context ...πσ).s4 Martin Dyer and Catherine Greenhill Example 3.1 Let G = (V, E) be a (simple) graph, with maximum degree ∆ and |V | = n, and let I(G) denote the collection of independent sets in G. It is known =-=[75, 38]-=- to be #P-complete to compute |I(G)| exactly for ∆ ≥ 3, and is easily shown to be in P for ∆ < 3. For given λ > 0, the hard-core gas model (see for example [4]) samples from a density where independen... |

69 | Mixing times of lozenge tiling and card shuffling Markov chains
- Wilson
(Show Context)
Citation Context ... show how to sample perfectly in expected polynomial time for these values of λ, ∆. Bubley, Dyer and Jerrum [13] used coupling to give a new approach to sampling points in a convex body. Also, Wilson =-=[79]-=- used coupling to prove upper and lower bounds on the mixing time for lozenge tiling and card shuffling Markov chains. Hernek [44] described a Markov chain for two-rowed contingency tables and, using ... |

67 | On Markov chains for independent sets
- Dyer, Greenhill
- 1997
(Show Context)
Citation Context ...her methods for relating the mixing times of Markov chains by comparing their spectral gap or log-Sobolev constant have been proposed by Chung and Graham [16], Randall and Tetali [70] and the authors =-=[29]-=-. Further results on bounding the spectral gap can be found in Guattery, Leighton and Miller [40] and in Chung and Yau [18]. Another approach used to bound the mixing time is Dobrushin uniqueness [23]... |

60 | On counting independent sets in sparse graphs
- Dyer, Frieze, et al.
(Show Context)
Citation Context ... the techniques of [50] have been applied, do not seem to be self-reducible under any re-encoding. Examples are the volume approximation problem of [26], and approximately counting contingency tables =-=[31, 25]-=-. We therefore propose a modified framework. We emphasise that the techniques are those of [50], and we wish only to emphasise the extent of their applicability. Let size : Σ ∗ → N be such that size(σ... |

59 | Faster random generation of linear extensions
- Bubley, Dyer
- 1999
(Show Context)
Citation Context ... and Vigoda [77].) Another result obtained using path coupling is an O(n 3 log n) bound on the mixing time for a Markov chain for sampling linear extensions of a partial order, due to Bubley and Dyer =-=[11]-=-. This chain is similar to the combinatorial chain of Karzanov and Khachiyan [57], but uses a different distribution to choose the position to update in the partial order. Recently, Wilson [79] built ... |

59 | Analyzing Glauber dynamics by comparison of Markov chains
- Randall, Tetali
- 1998
(Show Context)
Citation Context ...arlier paper [22]. Other methods for relating the mixing times of Markov chains by comparing their spectral gap or log-Sobolev constant have been proposed by Chung and Graham [16], Randall and Tetali =-=[70]-=- and the authors [29]. Further results on bounding the spectral gap can be found in Guattery, Leighton and Miller [40] and in Chung and Yau [18]. Another approach used to bound the mixing time is Dobr... |

57 | Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem
- Salas, Sokal
- 1997
(Show Context)
Citation Context ...ap can be found in Guattery, Leighton and Miller [40] and in Chung and Yau [18]. Another approach used to bound the mixing time is Dobrushin uniqueness [23]. This approach was used by Salas and Sokal =-=[71]-=- in work on graph colourings, and by Peinado and Lengauer [65] to sample structures which arise in computational chemistry. The relationship between Dobrushin uniqueness and path coupling (see below) ... |

48 | Markov chains and polynomial time algorithms
- Kannan
- 1994
(Show Context)
Citation Context ..., and applied their results to a Markov chain for colourings on a grid. One area where the Markov chain Monte Carlo approach has been very successful is in estimating the volume of convex bodies (see =-=[52]-=- for an overview). Typically, the mixing rates of random walks in convex bodies are analysed using isoperimetric inequalities (see, for example [26]). The state of the art is the O ∗ (n 5 ) volume app... |

47 | Simple Markov-chain algorithms for generating bipartite graphs and tournaments
- Kannan, Tetali, et al.
- 1999
(Show Context)
Citation Context ...e of the quantities used to bound the mixing time of a Markov chain are the conductance, spectral gap and log-Sobolev constant of the chain. Using the conductance approach, Kannan, Tetali and Vempala =-=[55]-=- proved rapid convergence of Markov chains for generating bipartite graphs and tournaments. By bounding the spectral gap, Chung, Graham and Yau [17] proved pseudopolynomial convergence of a Markov cha... |

46 |
Exposé de la théorie des chaînes simples constantes de markoff à un nombre fini d’états
- Doeblin
- 1938
(Show Context)
Citation Context ...h that each of Xt, Yt is marginally a copy of M, Pr(Xt = σ1 | Xt−1 = ω1) = P (ω1, σ1), Pr(Yt = σ2 | Yt−1 = ω2) = P (ω2, σ2), (∀t > 0). (5) The following simple but powerful inequality, due to Doeblin =-=[24]-=-, then follows easily from these definitions. Lemma 5.1 (Coupling Lemma) Let Xt, Yt be a coupling for M such that Y0 has the stationary distribution π. Then, if Xt has distribution pt, dTV(pt, π) ≤ Pr... |

41 | A more rapidly mixing Markov chain for graph colourings, Random Structures and Algorithms
- Dyer, Greenhill
- 1998
(Show Context)
Citation Context ... colourings described independently by [47] and [71] is rapidly mixing if there are at least 2∆ colours, where ∆ is the maximum degree of the graph. This result was improved in a paper by the authors =-=[30]-=-, which describes a Markov chain which has better bounds on the mixing time for regular graphs, and when fewer than 3∆ colours are used. The latter chain is also rapidly mixing whenever at least 2∆ co... |

41 |
Sampling contingency tables, Random Structures Algorithms 10
- Dyer, Kannan, et al.
- 1997
(Show Context)
Citation Context ... the techniques of [50] have been applied, do not seem to be self-reducible under any re-encoding. Examples are the volume approximation problem of [26], and approximately counting contingency tables =-=[31, 25]-=-. We therefore propose a modified framework. We emphasise that the techniques are those of [50], and we wish only to emphasise the extent of their applicability. Let size : Σ ∗ → N be such that size(σ... |

41 | The Swendsen-Wang process does not always mix rapidly
- Gore, Jerrum
- 1997
(Show Context)
Citation Context ...as used by Czumaj et. al. [20] to prove rapid mixing for a Markov chain for generating random permutations in parallel. We now describe some negative results which have been obtained. Gore and Jerrum =-=[36]-=- investigated the behaviour of the Swendsen–Wang process on the complete graph. They showed that for certain values of the “coupling constant” K, the process has exponential mixing time. Cooper and Fr... |

39 | Exact sampling from anti-monotone systems
- HÄGGSTRÖM, K
- 1998
(Show Context)
Citation Context ...er has height r. Then E(T ) ≤ 2τ(e −1 )(1 + log(r)), (4) where τ(ε) is the mixing time of the Markov chain [67]. Weaker variations of CFTP been proposed for other situations, for example antimonotone =-=[42]-=- and bounding chains [45, 41]. Fill [33] proposed an alternative algorithm for perfect sampling, known as Fill’s algorithm. His approach is based on rejection sampling, where the coin toss used to dec... |

35 |
On the conductance of order Markov chains
- Karzanov, Khachiyan
- 1991
(Show Context)
Citation Context ... bound on the mixing time for a Markov chain for sampling linear extensions of a partial order, due to Bubley and Dyer [11]. This chain is similar to the combinatorial chain of Karzanov and Khachiyan =-=[57]-=-, but uses a different distribution to choose the position to update in the partial order. Recently, Wilson [79] built on this result to prove an O(n 3 log n) upper bound for the mixing time of the Ka... |

34 |
Algorithms for random generation and counting
- Sinclair
- 1993
(Show Context)
Citation Context ...a modest attempt to update the situation. We begin, in Section 2, by fixing notation which we use throughout. In Section 3 we offer some formal definitions of the central concepts, following [50] and =-=[73]-=-. A review of recent developments in the areas of approximate sampling, approximate counting and perfect sampling is given in Section 4. The coupling method is described in Section 5, and is illustrat... |

33 | Efficient Markovian couplings: examples and counterexamples
- Burdzy, Kendall
- 2000
(Show Context)
Citation Context ...to prove mixing properties of non-Markovian couplings in our setting. Therefore we restrict attention to Markovian couplings, at the (probable) cost of sacrificing equality in (6). Burdzy and Kendall =-=[15]-=- investigated efficient Markovian couplings. Let C(M) be a Markovian coupling, with Q its transition matrix, i.e. the probability of a joint transition from (ω1, ω2) to (σ1, σ2) is Qω1ω2 σ1σ2 . The pr... |

33 |
Optimal algorithms for self-reducible problems
- Schnorr
- 1976
(Show Context)
Citation Context ...3 the ideas of that paper have rather more generality than the results contained within it. Also the paper concentrates its attention on the class of self-reducible problems, as introduced by Schnorr =-=[72]-=-. The restrictive definition of selfreducibility makes this concept difficult to apply, as we discuss below. For these reasons, we re-examine these foundations, to reflect the substantial body of subs... |

32 | Exact sampling and approximate counting techniques
- Huber
- 1999
(Show Context)
Citation Context ... ) ≤ 2τ(e −1 )(1 + log(r)), (4) where τ(ε) is the mixing time of the Markov chain [67]. Weaker variations of CFTP been proposed for other situations, for example antimonotone [42] and bounding chains =-=[45, 41]-=-. Fill [33] proposed an alternative algorithm for perfect sampling, known as Fill’s algorithm. His approach is based on rejection sampling, where the coin toss used to decide whether to reject the out... |

31 |
Percolation and the hard-core lattice gas model
- Berg, Steif
- 1994
(Show Context)
Citation Context ... independent sets in G. It is known [75, 38] to be #P-complete to compute |I(G)| exactly for ∆ ≥ 3, and is easily shown to be in P for ∆ < 3. For given λ > 0, the hard-core gas model (see for example =-=[4]-=-) samples from a density where independent sets of size s have probabilities proportional to λ s . We use this problem as a running example. In our setting, let λ = r/q for integers r, q. Then W (σ, ω... |

31 |
Approximately counting up to four
- Luby, Vigoda
- 1997
(Show Context)
Citation Context ... problem of sampling independent sets is a running example in this paper. Let λ denote the parameter used in the weight function, and let ∆ denote the maximum degree of a given graph. Luby and Vigoda =-=[62]-=-sRandom Walks on Combinatorial Objects 11 described a Markov chain for independent sets which is rapidly mixing for the following values of λ, ∆: when ∆ = 3 the chain is rapidly mixing for λ ≤ 1, and ... |

30 | Mathematical foundations of the markov chain monte carlo method
- Jerrum
- 1998
(Show Context)
Citation Context ...ntegration during the last couple of years. There are several excellent survey articles which introduce the methods and results in this area, including Welsh [78], Jerrum and Sinclair [49] and Jerrum =-=[48]-=-. Therefore we do not give an exhaustive survey here, but concentrate on more recent results from the theoretical computer science community. There is a very extensive body of work on the use of Marko... |

29 | Uniform generation of NP-witnesses using an NP-oracle
- Bellare, Goldreich, et al.
(Show Context)
Citation Context ...hievable. This shows that good sampling is harder than weak perfect sampling. Ideally, we would like to prove a similar result in a more realistic model of computation. Bellare, Goldreich and Petrank =-=[3]-=- showed how to perform strong perfect sampling of NP-witnesses, using an NP-oracle. It follows that, if P = NP, we can perform strong perfect sampling for any relation in NP. Conversely, if strong per... |

29 | Coupling from the past: a users guide
- Propp, Wilson
- 1998
(Show Context)
Citation Context ...pendent sets which is rapidly mixing for the following values of λ, ∆: when ∆ = 3 the chain is rapidly mixing for λ ≤ 1, and when ∆ ≥ 4 the chain is rapidly mixing for λ ≤ 1/(∆ − 3). Propp and Wilson =-=[69]-=- adapted the work of [62] to show how to sample perfectly in expected polynomial time for these values of λ, ∆. Bubley, Dyer and Jerrum [13] used coupling to give a new approach to sampling points in ... |

28 |
Graph orientations with no sink and an approximation for a hard case
- Bubley, Dyer
- 1997
(Show Context)
Citation Context ... a uniform sampling algorithm for graph colourings, which has expected polynomial time if at least (∆ + 1) 2 colours are used, where ∆ is the maximum degree of the graph. By modifying the analysis of =-=[10]-=-, Huber obtained an expected polynomial time uniform sampler for sink free orientations of a graph. Very recently, Huber [46] analysed the Dyer-Greenhill chain for independent sets [29] using bounding... |

28 | Mixing properties of the Swendsen-Wang process on the complete graph and narrow grids
- Cooper, Dyer, et al.
- 2000
(Show Context)
Citation Context ...s model. It was mildly conjectured that this process might be rapidly mixing on any graph. This conjecture has been disproved by two results which we state below. On the other hand, Cooper and Frieze =-=[19]-=- proved two positive results. Using path coupling, they showed that Swendsen–Wang mixes rapidly for graphs with small maximum degree, for small enough valuess12 Martin Dyer and Catherine Greenhill of ... |

25 | Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sampling Paths - Johnson - 1994 |

24 |
The complexity of counting colourings and independent sets in sparse graphs and hypergraphs
- Greenhill
- 2000
(Show Context)
Citation Context ...πσ).s4 Martin Dyer and Catherine Greenhill Example 3.1 Let G = (V, E) be a (simple) graph, with maximum degree ∆ and |V | = n, and let I(G) denote the collection of independent sets in G. It is known =-=[75, 38]-=- to be #P-complete to compute |I(G)| exactly for ∆ ≥ 3, and is easily shown to be in P for ∆ < 3. For given λ > 0, the hard-core gas model (see for example [4]) samples from a density where independen... |

22 | Exact mixing in an unknown Markov chain
- Lovász, Winkler
- 1995
(Show Context)
Citation Context ...ever λ ≤ 2/(∆ − 2), ∆ ≥ 3. There are two results on performing perfect sampling in a Markov chain where the transition probabilities are unknown. The first such result was given by Lovász and Winkler =-=[60]-=-. This was improved upon by Propp and Wilson [68], who presented an algorithm with universal randomized stationary stopping time which is within a constant factor of optimal. These algorithms are rela... |

20 | K.: On exact simulation of Markov random fields using coupling from the past
- H˝aggstrőm, Nelander
- 1999
(Show Context)
Citation Context ... ) ≤ 2τ(e −1 )(1 + log(r)), (4) where τ(ε) is the mixing time of the Markov chain [67]. Weaker variations of CFTP been proposed for other situations, for example antimonotone [42] and bounding chains =-=[45, 41]-=-. Fill [33] proposed an alternative algorithm for perfect sampling, known as Fill’s algorithm. His approach is based on rejection sampling, where the coin toss used to decide whether to reject the out... |