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33
Boltzmann Samplers For The Random Generation Of Combinatorial Structures
 Combinatorics, Probability and Computing
, 2004
"... 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 combina ..."
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Cited by 67 (2 self)
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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 realarithmetic 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.
Generating Labeled Planar Graphs Uniformly at Random
, 2003
"... We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3con ..."
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Cited by 25 (7 self)
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We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.
Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation
, 2003
"... In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n node ..."
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Cited by 12 (1 self)
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In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive: an enumeration formula, and an asymptotic of 2 3n (log n) ; an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worstcase constant time and neighbors query of a ddegree node in worstcase O(d) time...
Random Sampling from Boltzmann Principles
, 2002
"... 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. ..."
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Cited by 12 (2 self)
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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.
Uniform random sampling of planar graphs in linear time
, 2007
"... Abstract. This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combina ..."
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Cited by 9 (2 self)
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Abstract. This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Giménez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost; and the expected time complexity of generation is quadratic for exactsize uniform sampling and linear for approximatesize sampling. This greatly improves on the best previously known time complexity for exactsize uniform sampling of planar graphs with n vertices, which was a little over O(n 7). This is the extended and revised journal version of a conference paper with the title “Quadratic exactsize and linear approximatesize random generation of planar graphs”, which appeared in the Proceedings of the International Conference on Analysis of Algorithms (AofA’05), 610 June 2005, Barcelona. 1.
Generating outerplanar graphs uniformly at random
 in Combinatorics, Probability, and Computation
, 2003
"... supported by the DFG (GRK 588/1) Abstract. We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block s ..."
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Cited by 9 (6 self)
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supported by the DFG (GRK 588/1) Abstract. We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labeled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs. Next we modify our formulas to count rooted unlabeled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabeled outerplanar graphs uniformly at random in expected polynomial time. random structures, outerplanar graphs, efficient counting, uniform generation 1
A direct decomposition of 3connected planar graphs
 In Proceedings of the 17th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC05
, 2005
"... ABSTRACT. We present a decomposition strategy for cnets, i. e., rooted 3connected planar maps. The decomposition yields an algebraic equation for the number of cnets with a given number of vertices and a given size of the outer face. The decomposition also leads to a deterministic and polynomial ..."
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Cited by 6 (5 self)
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ABSTRACT. We present a decomposition strategy for cnets, i. e., rooted 3connected planar maps. The decomposition yields an algebraic equation for the number of cnets with a given number of vertices and a given size of the outer face. The decomposition also leads to a deterministic and polynomial time algorithm to sample cnets uniformly at random. Using rejection sampling, we can also sample isomorphism types of convex polyhedra, i.e., 3connected planar graphs, uniformly at random. RÉSUMÉ. Nous proposons une stratégie de décomposition pour les cartes pointées 3connexes (créseaux). Cette décomposition permet d’obtenir une équation algébrique pour le nombre de créseaux suivant le nombre de sommets et la taille de la face extèrieure. On en déduit un algorithme de complexité en temps polynomiale pour le tirage aléatoire uniforme des créseaux. En utilisant une méthode à rejet, nous obtenons aussi un algorithme de tirage aléatoire uniforme pour les graphes planaires 3connexes. 1.
Random generation of finitely generated subgroups of a free group
"... We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm ra ..."
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Cited by 5 (3 self)
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We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity O(n) in the RAM model and O(n 2 log 2 n) in the bitcost model.
Sampling unlabeled biconnected planar graphs
 in the Proceedings of the 16th Annual International Symposium on Algorithms and Computation (ISAAC’05), 2005, Springer LNCS 3827, 593 – 603
"... Abstract. We present an expected polynomial time algorithm to generate a 2connected unlabeled planar graph uniformly at random. To do this we first derive recurrence formulas to count the exact number of rooted 2connected planar graphs, based on a decomposition along the connectivity structure. Fo ..."
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Cited by 4 (4 self)
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Abstract. We present an expected polynomial time algorithm to generate a 2connected unlabeled planar graph uniformly at random. To do this we first derive recurrence formulas to count the exact number of rooted 2connected planar graphs, based on a decomposition along the connectivity structure. For 3connected planar graphs we use the fact that they have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sensereversing or a poleexchanging automorphism. We prove a bijection between such symmetric objects and certain colored networks. These colored networks can again be decomposed along their connectivity structure. All the numbers can be evaluated in polynomial time by dynamic programming. To generate 2connected unlabeled planar graphs without a root uniformly at random we apply rejection sampling and obtain an expected polynomial time algorithm. 1
Space of Gene/Species Trees Reconciliations and Parsimonious Models ⋆
"... Abstract. We describe algorithms to study the space of all possible reconciliations between a gene tree and a species tree, that is counting the size of this space, uniformly generate a random reconciliation, and exploring this space in optimal time using combinatorial operators. We also extend thes ..."
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Cited by 4 (2 self)
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Abstract. We describe algorithms to study the space of all possible reconciliations between a gene tree and a species tree, that is counting the size of this space, uniformly generate a random reconciliation, and exploring this space in optimal time using combinatorial operators. We also extend these algorithms for optimal and suboptimal reconciliations according to the three usual combinatorial costs (duplication, loss, and mutation). Applying these algorithms to simulated and real gene family evolutionary scenarios, we observe that the LCA (Last Common Ancestor) based reconciliation is almost always identical to the real one.