## Random Sampling from Boltzmann Principles (2002)

Citations: | 12 - 2 self |

### BibTeX

@MISC{Duchon02randomsampling,

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

title = {Random Sampling from Boltzmann Principles},

year = {2002}

}

### OpenURL

### Abstract

This extended abstract proposes a surprisingly simple framework for the random generation of combinatorial configurations based on Boltzmann models. Random generation of possibly complex structured objects is performed by placing an appropriate measure spread over the whole of a combinatorial class. The resulting algorithms can be implemented easily within a computer algebra system, be analysed mathematically with great precision, and, when suitably tuned, tend to be efficient in practice, as they often operate in linear time.

### Citations

632 | Non-uniform Random Variate Generation
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(Show Context)
Citation Context ...{Uses a Boltzmann generator "gC"} choose an x such that 0 repeat gamma := gC(x); until |gamma| = n; end. This is plainly based on rejection, a supple technique of subtle eectiveness in many =-=contexts [4]. Ran-=-dom generation of \approximate size" is also easily obtained by suitably weakening the halting condition of the repeat loop (thereby increasing the probability of success). Precisely, the size of... |

346 |
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(Show Context)
Citation Context ...e of \anonymous" letters, say fa; bg for a binary alphabet. This terminology is standard in combinatorial enumeration and graph theory (see, e.g., the books of Goulden-Jackson, Harary-Palmer, Sta=-=nley [11, 14, 30]-=- or [9]). 4 P. DUCHON, P. FLAJOLET, G. LOUCHARD, G. SCHAEFFER 3. Ordinary Boltzmann Generators A combinatorial construction builds a new class C from structurally simpler classes A; B, in such a way t... |

292 |
Graphical Enumeration
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(Show Context)
Citation Context ...e of \anonymous" letters, say fa; bg for a binary alphabet. This terminology is standard in combinatorial enumeration and graph theory (see, e.g., the books of Goulden-Jackson, Harary-Palmer, Sta=-=nley [11, 14, 30]-=- or [9]). 4 P. DUCHON, P. FLAJOLET, G. LOUCHARD, G. SCHAEFFER 3. Ordinary Boltzmann Generators A combinatorial construction builds a new class C from structurally simpler classes A; B, in such a way t... |

179 | Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications - Bergeron, Labelle, et al. - 1998 |

110 | Asymptotic enumeration methods
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(Show Context)
Citation Context ...e \variance condition". (These two conditions obviously depend on the singular type of the generating function C(x). This is an otherwise well researched topic as it is central to asymptotic coun=-=ting [24].) We have, str-=-aight from Chebyshev's inequalities: Proposition 3. Let " be asxed relative tolerance on size. An object from C of size N 2 [n(1 "); n(1 + ")] is to be generated uniformly at random in ... |

106 |
Cutsem, A calculus for the random generation of labelled combinatorial structures, Theoret
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(Show Context)
Citation Context ...d here draw their origin from many sources. First what has come to be known as the \recursive method" method originating with Nijenhuis and Wilf and formalized by Flajolet, Zimmermann, and Van Cu=-=tsem [10]-=- served as a key conceptual guide. Ideas from a statistical physics point of view on combinatorics, of which great use was made by Vershik and his collaborators [2, 31], then provided important intuit... |

77 |
The On-Line Encyclopedia of Integer Sequences, 2006, Published electronically at www.research.att.com/˜njas/sequences
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(Show Context)
Citation Context ... sequences, F = P(S1 (Z)). The EGF is exp z 1 z . The random generation algorithm is a compound of the form (PoissonGeometric), with appropriate parameters. (See A000262 in Sloane's encyclopedia [28=-=].) -=- As these example show, one can compile automatically Boltzmann generators from formal specications (\grammars") describing combinatorial models. As a matter of fact, we plan to develop a general... |

69 |
The complexity of nonuniform random number generation
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(Show Context)
Citation Context ...ws. RANDOM SAMPLING FROM BOLTZMANN PRINCIPLES 9 Note. We even believe the mean bit-complexity to remain linear for a multitape Turing machine equipped with a source of random bits. (E.g., Knuth-Yao [=-=17-=-] show that one can build in this context a Bernoulli source of parameter 1=, say.) 5.1. Controlling Sizes. Recall that our primary goal was random generation of objects of somesxed size n, or at leas... |

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
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(Show Context)
Citation Context ...ajolet, Zimmermann, and Van Cutsem [10] served as a key conceptual guide. Ideas from a statistical physics point of view on combinatorics, of which great use was made by Vershik and his collaborators =-=[2, 31]-=-, then provided important intuition regarding the new class of algorithms for random generation that is presented here. Another important ingredient is the collection of rejection algorithms developed... |

56 | Automatic average{case analysis of algorithms
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(Show Context)
Citation Context ...erness" in the theory of contextfree grammars. This condition also guarantees that the equations dening generating function equations are well-posed and contracting for all coherent values of x. =-=See [8]-=- on this topic. 3 A brutal rejection method based on generating random words andsltering out those that satisfy the condition R won't work in polynomial time since such words have an exponentially sma... |

56 |
Ordered cycle lengths in a random permutation
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(Show Context)
Citation Context ...eedom and order (for blocks). Example 6. Cycles in permutations. This corresponds to P = P(C1 (Z)) and is obtained by a (PoissonLog) process. This example is loosely related to the Shepp{Lloyd model [27] that generates permutations by ordered cycle lengths. Example 7. Assemblies ofslaments in a liquid. We may model these as sets of sequences, F = P(S1 (Z)). The EGF is exp z 1 z . The random ge... |

42 |
Finite range random walk on free groups and homogeneous trees. The Annals of Probability
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(Show Context)
Citation Context ...s matrix is Perron-Frobenius by \irreducibility". By the analytic version of the implicit function theorem, is determined by () = 1. A local analysis (essentially of the Drmota{Lalley{ Woods typ=-=e [5, 18,-=- 32]) establishes that each F j has a singularity at that is of the square-root type in the complex plane. Thus the coecients are asymptotically n n 3=2 ; see [24]. The cost of Boltzmann sampling i... |

38 |
Statistical Mechanics, 2nd ed
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(Show Context)
Citation Context ...of success). Precisely, the size of the resulting object under a Boltzmann model is a random variable denoted throughout by N , whose law is quantied by the following lemma. (See, e.g., Huang's book [=-=16-=-] 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 satises (1) E x (N) = x C 0 (x... |

31 | Uniform random generation of decomposable structures using floating-point arithmetic, Theoret
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(Show Context)
Citation Context ...tes within computer algebra systems, since then arbitrary precision is available. One could use adaptive precision, interval arithmetics, or randomized rounding, as suggested by Denise and Zimmermann =-=[3]-=- in a related context. Roughly put, say we tolerate a discrepancy of abouts= 10 16 from the uniform distribution under the Boltzmann model. Say we aim at sizes of at most S 0 = 10 9 for random generat... |

29 |
Computation of generating functions for biological molecules
- Howell, S, et al.
(Show Context)
Citation Context ...y-binary trees by V = Z(1 + V + V 2 ), etc. Example 3. Secondary structures. This example is inspired by Waterman et al., themselves motivated by the problem of enumerating secondary RNA structures [=-=15]. -=-Tosx ideas, consider rooted binary trees where edges contain 2 or 3 atoms and leaves (\loops") contain 4 or 5 atoms. A specication is S = (Z 4 + Z 5 ) + (Z 2 +Z 3 ) 2 (S S). A Bernoulli switch w... |

28 |
Random sampling of large planar maps and convex polyhedra
- Schaeffer
- 1999
(Show Context)
Citation Context ...ia.fr. Version of Nov. 11, 2001. Supported in part by the ALCOM-FT Project IST-1999-14186 of the E.U. 1 2 P. DUCHON, P. FLAJOLET, G. LOUCHARD, G. SCHAEFFER types of trees, polyominos, and planar maps =-=[6, 21, 26]. Th-=-ere are also similarities with the technique of \shifting the mean" (see Greene and Knuth's book [13, p. 78{ 80]) and we have beneted from discussions with Alain Denise on these aspects. Finally,... |

21 | Systems of functional equations
- Drmota
- 1997
(Show Context)
Citation Context ...s matrix is Perron-Frobenius by \irreducibility". By the analytic version of the implicit function theorem, is determined by () = 1. A local analysis (essentially of the Drmota{Lalley{ Woods typ=-=e [5, 18,-=- 32]) establishes that each F j has a singularity at that is of the square-root type in the complex plane. Thus the coecients are asymptotically n n 3=2 ; see [24]. The cost of Boltzmann sampling i... |

10 |
Largest component in random combinatorial structures
- Gourdon
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(Show Context)
Citation Context ...plies (write O = S(Q) and O(z) = (1 Q(z)) 1 ). The root of Q(z) = 1 is easily found to 50D: := 0:5761487691 . The objects of Q needed are with high probability of size at most O(log n), see [12], so that they can be generated by whichever subexponential method is convenient (e.g., Combstruct). The overall (theoretical and practical) complexity is O(n) with very low implementation constants. ... |

10 | Probabilistic analysis of column-convex and directed diagonally-convex animals, Random Struct. Algorithms 11
- Louchard
- 1996
(Show Context)
Citation Context ...x) at entails the promised characteristics by virtue of common properties of meromorphic functions. This theorem provides a global setting for a variety of ad hoc algorithms developed by Louchard [1=-=9, 20, 22] in the context-=- of ecient generation of certain brands (directed, convex) of random planar diagrams known as \animals" and \polyominos ". Here, it applies for instance to \words without long runs" and... |

9 | Large deviations for integer partitions
- Dembo, Vershik, et al.
- 2000
(Show Context)
Citation Context ...ajolet, Zimmermann, and Van Cutsem [10] served as a key conceptual guide. Ideas from a statistical physics point of view on combinatorics, of which great use was made by Vershik and his collaborators =-=[2, 31]-=-, then provided important intuition regarding the new class of algorithms for random generation that is presented here. Another important ingredient is the collection of rejection algorithms developed... |

9 | Object grammars and random generation
- Dutour, Fédou
- 1998
(Show Context)
Citation Context ...etical and practical) complexity is O(n) with very low implementation constants. Random generation well in the range of millions is now easy thanks to the singular Boltzmann generator. (Dutour et al. =-=[7-=-] employed the recursive method, but it is limited to sizes in the order of hundreds, perhaps thousands.) Continuations. We have work in progress on: generation of unlabelled multisets and cycles; ge... |

8 |
Probabilistic analysis of some (un)directed animals
- Louchard
- 1996
(Show Context)
Citation Context ...x) at entails the promised characteristics by virtue of common properties of meromorphic functions. This theorem provides a global setting for a variety of ad hoc algorithms developed by Louchard [1=-=9, 20, 22] in the context-=- of ecient generation of certain brands (directed, convex) of random planar diagrams known as \animals" and \polyominos ". Here, it applies for instance to \words without long runs" and... |

7 |
The random generation of directed animals
- Barcucci, Pinzani, et al.
- 1994
(Show Context)
Citation Context ...s rise to a random generator C given A and B: function gC(x : real); {generates C = A + B} Uses generators gA(x), gB(x), and values of the OGFs A(x), B(x). pA := A(x)/(A(x)+B(x)); draw U uniformly in =-=[0,-=-1]; if U Cartesian Product. Write C = A B if C is the set of ordered pairs from A and B, and size on C is inherited additively from A; B. For generating functions (3) C(x) = A(x) B(x) since C(x) = X... |

6 |
Méthodes d’analyse pour les constructions combinatoires et les algorithmes, Doctorat ès sciences, Université de Paris–Sud
- Soria-Cousineau
- 1990
(Show Context)
Citation Context ... Boltzmann generator for C produces a random C object of size n +O(1) in one trial, with high probability. Proof. The notion and properties of supercriticality in this context are borrowed from Soria =-=[29]-=-. The adapted algorithm does simply as follows: repeat: draw in A according to A(rho) until total size >= n; 10 P. DUCHON, P. FLAJOLET, G. LOUCHARD, G. SCHAEFFER (Literally taken, the singular Boltzma... |

4 | Asymptotic properties of some underdiagonal walks generation algorithms
- Louchard
- 1999
(Show Context)
Citation Context ...ia.fr. Version of Nov. 11, 2001. Supported in part by the ALCOM-FT Project IST-1999-14186 of the E.U. 1 2 P. DUCHON, P. FLAJOLET, G. LOUCHARD, G. SCHAEFFER types of trees, polyominos, and planar maps =-=[6, 21, 26]. Th-=-ere are also similarities with the technique of \shifting the mean" (see Greene and Knuth's book [13, p. 78{ 80]) and we have beneted from discussions with Alain Denise on these aspects. Finally,... |

3 |
Analytic combinatorics, 2001, Book in preparation; see also
- Flajolet, Sedgewick
(Show Context)
Citation Context ...e of \shifting the mean" (see Greene and Knuth's book [13, p. 78{ 80]) and we have beneted from discussions with Alain Denise on these aspects. Finally, the principles of analytic combinatorics (=-=see [-=-9]) provide essential clues for deciding situations in which the algorithms are likely to be ecient. We consider a class C of combinatorial objects of sorts, with j j the size function from C to Z0 .... |

3 |
Coloring rules for trees, and probabilities of monadic second order sentences. Random Structures and Algorithms
- Woods
- 1997
(Show Context)
Citation Context ...s matrix is Perron-Frobenius by \irreducibility". By the analytic version of the implicit function theorem, is determined by () = 1. A local analysis (essentially of the Drmota{Lalley{ Woods typ=-=e [5, 18,-=- 32]) establishes that each F j has a singularity at that is of the square-root type in the complex plane. Thus the coecients are asymptotically n n 3=2 ; see [24]. The cost of Boltzmann sampling i... |

2 |
Relaxed random generation of trees. Algorithms Seminar
- Duchon
- 2001
(Show Context)
Citation Context ...ia.fr. Version of Nov. 11, 2001. Supported in part by the ALCOM-FT Project IST-1999-14186 of the E.U. 1 2 P. DUCHON, P. FLAJOLET, G. LOUCHARD, G. SCHAEFFER types of trees, polyominos, and planar maps =-=[6, 21, 26]. Th-=-ere are also similarities with the technique of \shifting the mean" (see Greene and Knuth's book [13, p. 78{ 80]) and we have beneted from discussions with Alain Denise on these aspects. Finally,... |

2 |
The editor's corner: n coins in a fountain. American Matematical Monthly 95
- Odlyzko, Wilf
- 1988
(Show Context)
Citation Context ...ounding should suce for all practical purposes. Thus, the exact arithmetic model assumed over the reals is not so unrealistic! Example 8. Coin fountains (O). These were enumerated by Odlyzko and Wilf =-=[-=-25]. They correspond to Dyck paths (aka Bernoulli excursions) taken according to area (disregarding length). The OGF is the continued fraction O(z) = 1 (1 z (1 z 2 (1 z 3 ( )))): At top level, the s... |

1 | Boltzmann samplers for random combinatorial generation - Duchon, Flajolet, et al. - 2002 |