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Boltzmann Samplers, Pólya Theory, and Cycle Pointing [article]

Manuel Bodirsky and Éric Fusy and Mihyun Kang and Stefan Vigerske
2011 arXiv   pre-print
We extend Polya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on P\'olya operators.  ...  All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results.  ...  Omid Amini, Olivier Bodini, Philippe Flajolet, and Pierre Leroux are greatly thanked for fruitful discussions and suggestions. Further, we thank two anonymous referees for their helpful comments.  ... 
arXiv:1003.4546v2 fatcat:x6cznmq7ezdhdhwaajtjbubpmi

Boltzmann Samplers, Pólya Theory, and Cycle Pointing

Manuel Bodirsky, Éric Fusy, Mihyun Kang, Stefan Vigerske
2011 SIAM journal on computing (Print)  
We extend Pólya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on Pólya operators.  ...  All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results.  ...  Omid Amini, Olivier Bodini, Philippe Flajolet, and Pierre Leroux are greatly thanked for fruitful discussions and suggestions. Further, we thank two anonymous referees for their helpful comments.  ... 
doi:10.1137/100790082 fatcat:yjbet7b2obditkq352p2xirphm

Enumeration and Random Generation of Unlabeled Classes of Graphs: A Practical Study of Cycle Pointing and the Dissymmetry Theorem [article]

Alexander Iriza
2015 arXiv   pre-print
Finally, we apply cycle pointing to enumerate and implement samplers for the classes of distance-hereditary graphs and three-leaf power graphs.  ...  In this work, we extend the power of the dissymmetry theorem by showing that it in fact provides a Boltzmann sampler for the class in question.  ...  Just as cycle index sums provide transfer theorems for the substitution and cycle-pointed substitution operators, Pólya-Boltzmann samplers allow us to build Boltzmann samplers for classes specified with  ... 
arXiv:1511.06037v1 fatcat:s3fyjp7n3rggrdjo46ayovxsgu

Boltzmann Sampling of Unlabelled Structures

Philippe Flajolet, Éric Fusy, Carine Pivoteau
2007 2007 Proceedings of the Fourth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
, series-parallel circuits, term trees and acyclic molecules obeying a variety of constraints, and so on.  ...  Boltzmann models from statistical physics combined with methods from analytic combinatorics give rise to efficient algorithms for the random generation of unlabelled objects.  ...  For multisets, it can be observed that the algorithm emulates Γ MSet[A](x) conditioned upon the value of k (alternatively, use basic Burnside-Pólya theory [15, 22]). Similarly for cycles.  ... 
doi:10.1137/1.9781611972979.5 dblp:conf/analco/FlajoletFP07 fatcat:odvvz5kljfherghn2crwlwuvuy

A note on the scaling limits of random Pólya trees

Bernhard Gittenberger, Emma Yu Jin, Michael Wallner
2017 2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
Their proof used the framework of a Boltzmann sampler and deviation inequalities.  ...  Panagiotou and Stufler (arXiv:1502.07180v2 ) recently proved one important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size n consists of a conditioned  ...  Let (T, c) be a cycle-pointed structure considered up to symmetry where T is a Pólya tree and c is a cycle of an automorphism σ ∈ Aut(T ).  ... 
doi:10.1137/1.9781611974775.8 dblp:conf/analco/GittenbergerJW17 fatcat:ly37lt3gnfgq5ck6hqillzaove

A note on the scaling limits of random Pólya trees [article]

Bernhard Gittenberger, Emma Yu Jin, Michael Wallner
2016 arXiv   pre-print
Their proof used the framework of a Boltzmann sampler and deviation inequalities.  ...  Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size n consists of a conditioned  ...  Acknowledgements: This work was supported by the SFB project F50-03 "Combinatorics of Tree-Like Structures and Enriched Trees". We also thank the three referees for their feedback.  ... 
arXiv:1606.08769v3 fatcat:uvrdojk5pnaudlxknzzllcjv74

The continuum random tree is the scaling limit of unlabelled unrooted trees [article]

Benedikt Stufler
2016 arXiv   pre-print
We prove that the uniform unlabelled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable rescaling to the Brownian continuum random  ...  The corresponding pointed cycle index sum is given bȳ (Pólya-)Boltzmann samplers Boltzmann samplers were introduced in [18, 19, 20] and generalized to Pólya-Boltzmann samplers in [11] .  ...  Pólya-Boltzmann samplers for cycle-pointed species In the following, we suppose that F is a cycle pointed species and that s 1 , t 1 , s 2 , t 2 , . . . are non-negative real numbers such that 0 <Z F (  ... 
arXiv:1412.6333v4 fatcat:gfvkrj5vmbdvjf6ghnilf53zky

Boltzmann Samplers for Colored Combinatorial Objects [article]

Olivier Bodini
2009 arXiv   pre-print
In this paper, we give a general framework for the Boltzmann generation of colored objects belonging to combinatorial constructible classes.  ...  We propose an intuitive notion called profiled objects which allows the sampling of size-colored objects (and also of k-colored objects) although the corresponding class cannot be described by an analytic  ...  This example points out that symmetries increase the difficulty to build an uniform sampler. That can be also observed by comparing Boltzmann sampler codes for labelled and unlabelled structures.  ... 
arXiv:0911.2801v1 fatcat:blvctm3synbqrewdvct7dtlcxi

On the shape of random Pólya structures [article]

Bernhard Gittenberger, Emma Yu Jin, Michael Wallner
2018 arXiv   pre-print
Their proof used the framework of a Boltzmann sampler and deviation inequalities.  ...  Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size n consists of a conditioned critical Galton-Watson  ...  Combinatorial Species and Tree-Like Structures. Cambridge, 1998. [3] M. Bodirsky, É. Fusy, M. Kang and S. Vigerske. Boltzmann samplers, Pólya theory, and cycle pointing. SIAM J.  ... 
arXiv:1707.02144v2 fatcat:dzdnyxl535guvpcg7qxiptftiu

Polynomial tuning of multiparametric combinatorial samplers [chapter]

Maciej Bendkowski, Olivier Bodini, Sergey Dovgal
2018 2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA  ...  Finally, we illustrate the efficiency of our approach using several applications of rational, algebraic and Pólya structures including polyomino tilings with prescribed tile frequencies, planar trees with  ...  for plane partitions [BFP10] or the cycle pointing operator for Pólya structures [Bod+11] .  ... 
doi:10.1137/1.9781611975062.9 dblp:conf/analco/BendkowskiBD18 fatcat:jhnvqea5arbr7ksxau5cddxzoq

Polynomial tuning of multiparametric combinatorial samplers [article]

Maciej Bendkowski, Olivier Bodini, Sergey Dovgal
2017 arXiv   pre-print
Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA  ...  with a given specific node degree distribution, and weighted partitions.  ...  for plane partitions [BFP10] or the cycle pointing operator for Pólya structures [Bod+11] .  ... 
arXiv:1708.01212v2 fatcat:iqtyt3fxnfb2vc77gchesc7k7m

Boltzmann samplers for first-order differential specifications

Olivier Bodini, Olivier Roussel, Michèle Soria
2012 Discrete Applied Mathematics  
This paper proposes an efficient Boltzmann sampler for ordered structures defined by first-order differential specifications.  ...  In the framework of analytic combinatorics, Boltzmann models give rise to efficient algorithms for the random generation of combinatorial objects.  ...  Acknowledgments The authors are very grateful to the referees for their perceptive and encouraging comments.  ... 
doi:10.1016/j.dam.2012.05.022 fatcat:2iw5thfgtvfj7hazjqbfkcjf5a

Tuning as convex optimisation: a polynomial tuner for multi-parametric combinatorial samplers [article]

Maciej Bendkowski, Olivier Bodini, Sergey Dovgal
2021 arXiv   pre-print
Combinatorial samplers are algorithmic schemes devised for the approximate- and exact-size generation of large random combinatorial structures, such as context-free words, various tree-like data structures  ...  We show how our method can be adapted to a broad range of less typical combinatorial constructions, including symmetric polynomials, labelled sets and cycles with cardinality lower bounds, simple increasing  ...  We are especially grateful to Cedric Chauve, Yann Ponty, and Sebastian Will for showing us numerous applications of Boltzmann sampling in biology, and to Sergey Tarasov for asking a question about computational  ... 
arXiv:2002.12771v2 fatcat:xucfxmndz5hnnp4e2cv4lbnl3m

How to generate random lambda terms? [article]

Maciej Bendkowski
2020 arXiv   pre-print
We discuss methods of exact- and approximate-size generation, as well as methods of achieving size-uniform and non-uniform outcome distributions.  ...  We survey several methods of generating large random lambda-terms, focusing on their closed and simply-typed variants.  ...  We would like to thank Pierre Lescanne and Sergey Dovgal for encouraging us to write this survey and their valuable remarks during its formation.  ... 
arXiv:2005.08856v2 fatcat:qefstenqtnb7pilumaueb7nsva

Random Enriched Trees with Applications to Random Graphs

Benedikt Stufler
2018 Electronic Journal of Combinatorics  
We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry.  ...  For both models we establish a Gromov–Hausdorff scaling limit, a Benjamini–Schramm limit, and a local weak limit that describes the asymptotic shape near the fixed root.  ...  Acknowledgement I thank Jetlir Duraj, Markus Heydenreich, Grégory Miermont, and Vitali Wachtel for related discussions.  ... 
doi:10.37236/7328 fatcat:jzcvjfvmdvgvrdhcdekkus7fdq
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