## Boltzmann Samplers For The Random Generation Of Combinatorial Structures (2004)

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Venue: | Combinatorics, Probability and Computing |

Citations: | 67 - 2 self |

### BibTeX

@ARTICLE{Duchon04boltzmannsamplers,

author = {Philippe Duchon and Philippe Flajolet and Guy Louchard and Gilles Schaeffer},

title = {Boltzmann Samplers For The Random Generation Of Combinatorial Structures},

journal = {Combinatorics, Probability and Computing},

year = {2004},

volume = {13},

pages = {2004}

}

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### Abstract

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class -- an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

### Citations

2195 | The Art of Computer Programming - Knuth - 2000 |

2045 | An introduction to probability theory and its applications, Volume I - Feller - 1957 |

1454 | An Introduction to Probability Theory - Feller - 1971 |

623 | Non-Uniform Random Variate Generation - Devroye - 1986 |

531 |
Efficient String Matching: An Aid to Bibliographic Search
- Aho, Corasick
- 1975
(Show Context)
Citation Context ...istribution of parameter 2x; if the value N = n is obtained, draw uniformly at random any of the possible words of size n. For the labelled case, consider the class K of all cyclic permutations, K = f=-=[1]; [-=-1 2]; [1 2 3]; [1; 3; 2]; : : :g. There are Kn = (n 1)! cyclic permutations of size n over the canonical set of \labels" f1; : : : ; ng. The EGF is (3) b K(z) = X n1 (n 1)! z n n! = X n1 z n n = ... |

446 | Asymptotics and special functions - Olver - 1974 |

348 | The on-line encyclopedia of integer sequences. Published electronically at http://www.research.att.com/~njas/sequences/. tel-00482196, version 1 - 9 - Sloane - 2010 |

345 |
Combinatorial Enumeration
- Goulden, Jackson
- 1983
(Show Context)
Citation Context ...pecied in terms of a basic collection of general-purpose combinatorial constructions. These constructions are precisely the ones that surface recurrently in modern theories of combinatorial analysis [=-=4, 28, 30, 60, 61]-=- and in systematic approaches to random generation of combinatorial structures [29, 51]. As a consequence, one obtains with surprising ease Boltzmann samplers covering an extremely wide range of combi... |

317 |
Singularity analysis of generating functions
- Flajolet, Odlyzko
- 1990
(Show Context)
Citation Context ...cient. The discussion is fundamentally based on the types of singularities that the generating functions exhibit. This is an otherwise well-researched topic as it is central to asymptotic enumeration =-=[26, 28-=-, 52]. Denition 2. A function f(z) analytic at 0 and a withsnite radius of analyticitys > 0 is said to be {singular if it satises the two conditions: (i) The function admits as its only singularity o... |

292 |
Graphical enumeration
- Harary, Palmer
- 1973
(Show Context)
Citation Context ...and n if the object considered 2 This terminology is standard in combinatorial enumeration and graph theory; see, e.g., the books of Bergeron et al., Goulden{Jackson, Harary{Palmer, Stanley, and Wilf =-=[4, 30, 34, 60, 61, 69]-=- or the preprints by Flajolet & Sedgewick [28]. BOLTZMANN SAMPLERS FOR RANDOM GENERATION 5 has size n. Permutations written as sequences of distinct integers are typical labelled objects while words o... |

248 | Seminumerical Algorithms - Knuth - 1998 |

223 |
Univariate Discrete Distributions
- Johnson, Kemp, et al.
- 2005
(Show Context)
Citation Context ...) 1 k k : 18 P. DUCHON, P. FLAJOLET, G. LOUCHARD, G. SCHAEFFER (This is the same as in Equation (4); the distribution occurs in statistical ecology and economy and forms the subject of Chapter 7 of [=-=38-=-].) Then cycles under the exponential Boltzmann model can be drawn like in the case of sets upon replacing the Poisson law by the log-law: C(x) = Loga( b A(x)) =) A(x) : These constructions are summa... |

185 | Advanced Combinatorics - Comtet - 1974 |

178 | An Introduction to Probability Theory and Its Applications. 3rd ed - Feller - 1968 |

176 |
Combinatorial Species and Tree-like Structures
- Bergeron, Labelle, et al.
- 1998
(Show Context)
Citation Context ...pecied in terms of a basic collection of general-purpose combinatorial constructions. These constructions are precisely the ones that surface recurrently in modern theories of combinatorial analysis [=-=4, 28, 30, 60, 61]-=- and in systematic approaches to random generation of combinatorial structures [29, 51]. As a consequence, one obtains with surprising ease Boltzmann samplers covering an extremely wide range of combi... |

144 | Combinatorial aspects of continued fractions
- Flajolet
- 1980
(Show Context)
Citation Context ...1,2,3,3,4) is then expressed by an innite specication (not a priori covered by the standard paradigm): S(ZS(Z 2 S(Z 3 S( )))): The OGF is consequently given by the continued fraction (see also [23]), O(z) = 1 1 z 1 z 2 1 z 3 : At top level, the singular Boltzmann sampler of Theorem 7 applies (write O = S(Q) and O(z) = (1 Q(z)) 1 ), this even though O is notsnitely speciable. The root o... |

129 |
On the altitude of nodes in random trees
- Meir, Moon
- 1978
(Show Context)
Citation Context ...a polar singularity, corresponding to the singular exponent 1. Trees, secondary structures, and noncrossing graphs (Example 2, 3, and 4), which are recursively dened have singular exponent 1 2 ; see [=-=24, 4-=-9] and Section 8 below. Many properties go along with the conditions of Denition 2. Most notably, the counting sequence associated with a generating function f(z) that is -singular systematically obey... |

109 | Asymptotic enumeration methods - Odlyzko - 1995 |

105 |
Cutsem. A calculus for the random generation of labelled combinatorial structures
- Flajolet, Zimmermann, et al.
- 1994
(Show Context)
Citation Context ...(see, e.g., [4, 28, 30]) we thus estimate to well over a hundred the number of classical combinatorial structures that are amenable to ecient Boltzmann sampling. In contrast with the recursive method =-=[13, 29, 51]-=-, memory requirements are kept to a minimum since only a table of constants of size O(1) is required. For the reader's convenience, we gather in Figure 10 the best strategies that have been developed ... |

78 | Random mapping statistics
- Flajolet, Odlyzko
- 1989
(Show Context)
Citation Context ...e set and restrict attention to degree-constrained mappings f such that for each x in the domain, the cardinality of f ( 1) (x) lies ins(In the combinatorics literature, such mappings are surveyed in =-=[2, 25]-=-.) For instance, in asniteseld, a non-zero element has either 0 or 2 predecessors under the mapping f ; x 7! x 2 , so that (neglecting one exceptional value) a quadratic function may be regarded as an... |

77 |
The On-Line Encyclopedia of Integer Sequences, 2000, published electronically at http://www.research.att.com/ ~ njas/sequences
- Sloane
(Show Context)
Citation Context ...eometric), with appropriate parameters: F (x) = Pois x 1 x =) Geom 1 (x) =) Z : The corresponding counting sequence, 1; 1; 3; 13; 73; 501; : : :, appears as A000262 in Sloane's encyclopedia [58]. This example is closely related to linear forests and posets as described in Burris' book (see [6], pp. 23-24 and Ch. 4). At this stage, it may be of interest to note that many classical probabili... |

68 |
The complexity of nonuniform random number generation. Algorithms and complexity
- Knuth, Yao
- 1976
(Show Context)
Citation Context ...immermann [12, 13]. In a companion paper, we shall explore another route and describe purely discrete Boltzmann samplers which are solely based on binary coinsips in the style of Knuth and Yao's work =-=[40] and -=-have the additional feature of \automatically" detecting when accuracy is insucient. 6. Exact-size and approximate-size sampling Our primary objective in this article is the fast random generatio... |

60 | Mathematics for the analysis of algorithms. Birkhauser - Greene, Knuth - 1982 |

60 |
Statistical mechanics of combinatorial partitions and their limit shapes, Funct Anal Appl 30
- Vershik
- 1996
(Show Context)
Citation Context ... types of objects that are systematically amenable to Boltzmann sampling. Ideas from a statistical physics point of view on combinatorics, of which great use was made by Vershik and his collaborators =-=[10, 67]-=-, then provided crucial insight regarding the new class of algorithms for random generation that is presented here. Another important ingredient is the collection of rejection algorithms developed by ... |

58 | Conceptual proofs of l log l criteria for mean behavior of branching processes
- Lyons, Pemantle, et al.
- 1995
(Show Context)
Citation Context ...rity (with exponent = 1). For instance, the class K of connected mappings is dened by fK = C(T ); T = Z ? P(T )g : 5 This construction is akin to the \size-biased" Galton{Watson process exposed =-=in [4-=-7]. It is interesting to note that we are here led naturally to it by a systematic use of formal transformations. BOLTZMANN SAMPLERS FOR RANDOM GENERATION 33 The derived specication for K is then fK ... |

56 | Average-Case Analysis of Algorithms and
- FLAJOLET, VITTER
- 1987
(Show Context)
Citation Context ...on provides a geometrically converging approximation scheme that makes it possible to determine generating function values for all coherent values of x (by analyticity and dominated convergence). See =-=[27, 2-=-9] for a detailed discussion of this topic and the corresponding decision procedures. BOLTZMANN SAMPLERS FOR RANDOM GENERATION 11 Theorem 1. Dene as speciable an unlabelled class that can besnitely sp... |

55 |
Ordered cycle lengths in a random permutation
- Shepp, Lloyd
- 1966
(Show Context)
Citation Context ...in permutations. This corresponds to P = P(C1 (Z)) and is obtained by a (PoissonLog) process: P (x) = Pois(log(1 x) 1 ) =) (Loga(x) =) Z) : This example is loosely related to the Shepp{Lloyd model [5=-=7-=-] that generates permutations by ordered cycle lengths, as a potentially innite sequence of Poisson variables of parameters x=1, x 2 =2, and so on. The interest of this construction is to give rise to... |

55 | Analytic combinatorics of non-crossing configurations - Flajolet, Noy - 1999 |

51 | A generalization of Stirlingâ€™s formula - Hayman - 1956 |

46 |
F.: Large Deviations
- Hollander
- 2000
(Show Context)
Citation Context ... polyominos, and planar maps [17, 45, 56]. There are also similarities with the technique of \shifting the mean" (see Greene and Knuth's book [33, p. 78-80]) as well as the theory of large deviat=-=ions [11] and-=- \exponential families" of probability theory|we have beneted from discussions with Alain Denise on these aspects. Finally, the principles of analytic combinatorics (see [28]) provide essential c... |

40 |
Finite range random walk on free groups and homogeneous trees.Ann. Probab
- Lalley
- 1993
(Show Context)
Citation Context ...t dened near the origin. Let (z) be the spectral radius of J(z). For small enough positive x, the matrix J(x) is Perron{Frobenius by irreducibility. A local analysis of the Drmota{Lalley{Woods type [1=-=6, 41, 70] bas-=-ed on \failure" of the implicit function theorem in its analytic version establishes the following: each F j has a singularity at which is determined as the smallest positive root 38 P. DUCHON, ... |

39 | Factorization of the eighth Fermat number - Brent, Pollard - 1981 |

38 |
Statistical Mechanics, 2nd ed
- Huang
- 1987
(Show Context)
Citation Context ...x x, (5) P x (N = n) = Cnx n C(x) ; or P x (N = n) = Cnx n n! b C(x) ; for the ordinary and exponential model, respectively. The law is well quantied by the following lemma. (See, e.g., Huang's book [=-=37]-=- for similar calculations from the statistical mechanics angle.) Proposition 1. The random size of the object produced under the ordinary Boltzmann model of parameter x hassrst and second moments sati... |

35 |
On some new sequences generalizing the Catalan and Motzkin numbers
- Stein, Waterman
- 1979
(Show Context)
Citation Context ...ced by a critical Boltzmann sampler. Example 3. Secondary structures. This example is inspired by works of Watermanset al., themselves motivated by the problem of enumerating secondary RNA structures =-=[36, 62]. -=-Tosx ideas, consider rooted binary trees where edges contain 2 or 3 atoms and leaves (\loops") contain 4 or 5 atoms. A specication is W = (Z 4 +Z 5 ) + (Z 2 +Z 3 ) 2 (W W). A Bernoulli switch wi... |

33 | Coloring rules for finite trees, probabilities of monadic second order sentences. Random Structures Algorithms 10 - Woods - 1997 |

30 |
A logical approach to asymptotic combinatorics I: First-order properties
- Compton
- 1987
(Show Context)
Citation Context ...2] have shown the possibility of drawing at random independent sets of graphs according to a Boltzmann distribution, at least for certain values of the control parameter x = e . Closer to us, Compton =-=[7, 8]-=- has made an implicit use of what we call Boltzmann models for the analysis of 0-1 laws and limit laws in logic; see also the account by Burris [6]. Vershik has initiated in a series of papers (see [6... |

30 | Uniform Random Generation of Decomposable Structures Using Floating-Point Arithmetic
- Denise, Zimmermann
- 1997
(Show Context)
Citation Context ...e be decreased from O(n 2 ) to O(n 1+" ) thanks to the recent work of van der Hoeven [65], but this does not aect our basic conclusions. ) An alternative, initiated by Denise, Dutour, and Zimmerm=-=ann [12, 13], con-=-sists in treating integers as real numbers and approximating them using real arithmetics (\ oating-point" implementations), possibly supplementing the technique by adaptive precision routines. In... |

29 |
Computation of generating functions for biological molecules
- Howell, S, et al.
(Show Context)
Citation Context ...ced by a critical Boltzmann sampler. Example 3. Secondary structures. This example is inspired by works of Watermanset al., themselves motivated by the problem of enumerating secondary RNA structures =-=[36, 62]. -=-Tosx ideas, consider rooted binary trees where edges contain 2 or 3 atoms and leaves (\loops") contain 4 or 5 atoms. A specication is W = (Z 4 +Z 5 ) + (Z 2 +Z 3 ) 2 (W W). A Bernoulli switch wi... |

29 | Sampling adsorbing staircase walks using a new markov chain decomposition method
- Martin, Randall
- 2000
(Show Context)
Citation Context ... have previously employed an adaptation of the recursive method, but it is limited to sizes perhaps in the order of a few hundreds. Example 11. Weighted Dyck paths and adsorbing staircase walks. In [=-=48-=-], Martin and Randall examine (under the name of adsorbing walks) the generation of Dyck paths of length 2n, where a path receives a weight proportional to k if it hits the horizontal axis k times. T... |

28 |
Random sampling of large planar maps and convex polyhedra
- Schaeffer
- 1999
(Show Context)
Citation Context ...tion that is presented here. Another important ingredient is the collection of rejection algorithms developed by Duchon, Louchard, and Schaeer for certain types of trees, polyominos, and planar maps [=-=17, 45, 56]. There ar-=-e also similarities with the technique of \shifting the mean" (see Greene and Knuth's book [33, p. 78-80]) as well as the theory of large deviations [11] and \exponential families" of probab... |

24 | Combinatorics on Words, Vol 17 of Encyclopedia of mathematics and its applications - Lothaire - 1983 |

21 | Systems of functional equations
- Drmota
- 1997
(Show Context)
Citation Context ...t dened near the origin. Let (z) be the spectral radius of J(z). For small enough positive x, the matrix J(x) is Perron{Frobenius by irreducibility. A local analysis of the Drmota{Lalley{Woods type [1=-=6, 41, 70] bas-=-ed on \failure" of the implicit function theorem in its analytic version establishes the following: each F j has a singularity at which is determined as the smallest positive root 38 P. DUCHON, ... |

18 |
theoretic density and logical limit laws
- Burris, Number
- 2001
(Show Context)
Citation Context ...e control parameter x = e . Closer to us, Compton [7, 8] has made an implicit use of what we call Boltzmann models for the analysis of 0-1 laws and limit laws in logic; see also the account by Burris =-=[6]-=-. Vershik has initiated in a series of papers (see [67] and references therein) a programme that can be described in our terms assrst developing the probabilistic study of combinatorial objects under ... |

17 |
Random mappings with constraints on coalescence and number of origins
- Arney, Bender
- 1982
(Show Context)
Citation Context ...ter 2x; if the value N = n is obtained, draw uniformly at random any of the possible words of size n. For the labelled case, consider the class K of all cyclic permutations, K = f[1]; [1 2]; [1 2 3]; =-=[1; 3; 2]; :-=- : :g. There are Kn = (n 1)! cyclic permutations of size n over the canonical set of \labels" f1; : : : ; ng. The EGF is (3) b K(z) = X n1 (n 1)! z n n! = X n1 z n n = log 1 1 z : The probability... |

17 |
A Generalisation of Stirling f s Formula." Journal fur die reine und angewandte Mathematik 196
- Hayman
(Show Context)
Citation Context ...om the theory of admissibility, an area originally developed for the purpose of estimating asymptotically Taylor coecients of entire functions. This theory was started in an important paper of Hayman =-=[35-=-] and is lucidly exposed in Odlyzko's survey [52, Sec. 12]. A function is said to be H-admissible if, in addition to the mean value condition (17) and the variance condition (19), it satises the follo... |

13 |
Labelled Formal Languages and Their Uses
- Greene
- 1983
(Show Context)
Citation Context ...ed by a sampler of C n to obtain an object of Cn . (Only the distributions of sizes under C and C are dierent.) The pointing operator is related to an operator studied systematically by Greene [32]=-= (his \box-=-" operation) and it plays a central r^ole in the recursive method BOLTZMANN SAMPLERS FOR RANDOM GENERATION 31 (where it has been used under the name of \Theta operator"). For Boltzmann sampl... |

12 | Random Sampling from Boltzmann Principles
- Duchon, Flajolet, et al.
- 2002
(Show Context)
Citation Context ... to the control parameter x. Section 8 oers a few concluding remarks. An extended abstract summarizing several of the results described here has been presented at the ICALP'2002 Conference in Malaga [=-=1-=-8]. 2. Boltzmann models and samplers We consider a class C of combinatorial objects of sorts, with j j the size function mapping C to Z0 . By Cn is meant the subclass of C comprising all the objects ... |

12 | Statistical Mechanics, 2nd edn - Huang - 1987 |

11 | The randomness recycler: a new technique for perfect sampling
- Fill, Huber
- 2000
(Show Context)
Citation Context ...being generated.) Exponential weights of the Boltzmann type are naturally essential to the simulated annealing approach to combinatorial optimization. In the latter area, for instance, Fill and Huber =-=[22]-=- have shown the possibility of drawing at random independent sets of graphs according to a Boltzmann distribution, at least for certain values of the control parameter x = e . Closer to us, Compton [7... |

11 |
How easy is collision search? Application to DES
- Quisquater, Delescaille
- 1990
(Show Context)
Citation Context ...xceptional value) a quadratic function may be regarded as an element of the set of mappings constrained by = f0; 2g. Mappings are of interest in computational number theory as well as in cryptography =-=[55]-=-, and the eighth Fermat number, F 8 = 2 2 8 + 1 wassrst factored by Brent and Pollard [5] in 1981 by means of an algorithm that precisely exploits statistical properties of degree-constrained mappings... |