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55
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.
Enumeration and generation with a string automata representation
 THEORET. COMPUT. SCI.
, 2007
"... In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initiallyconnected deterministic automata (ICDFA’s) with n states over an alphabet of k symbols can be used ..."
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Cited by 16 (9 self)
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In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initiallyconnected deterministic automata (ICDFA’s) with n states over an alphabet of k symbols can be used for counting, exact enumeration, sampling and optimal coding, not only the set of ICDFA’s but, to some extent, the set of regular languages. An exact generation algorithm can be used to partition the set of ICDFA’s in order to parallelize the counting of minimal automata (and thus of regular languages). We present also a uniform random generator for ICDFA’s that uses a table of precalculated values. Based on the same table it is also possible to obtain an optimal coding for ICDFA’s.
Boltzmann oracle for combinatorial systems
 In Algorithms, Trees, Combinatorics and Probabilities
, 2008
"... Boltzmann random generation applies to welldefined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide s ..."
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Cited by 13 (5 self)
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Boltzmann random generation applies to welldefined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide such oracles. In particular, we give a fast oracle based on Newton iteration.
On Properties of Random Dissections and Triangulations
 in Proceedings of the 19th Annual ACMSIAM Symposium on Discrete Algorithms (SODA ’08
"... In the past decades the Gn,p model of random graphs, introduced by Erdős and Rényi in the 60’s, has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in Gn,p appear independently. The independence of the edges allows, for example, to obtain ..."
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Cited by 12 (6 self)
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In the past decades the Gn,p model of random graphs, introduced by Erdős and Rényi in the 60’s, has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in Gn,p appear independently. The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of Gn,p and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. For example, in a random planar graph Rn (a graph drawn uniformly at random from the class of all labeled planar graphs on n vertices) the edges are obviously far from being independent. Consequently, so far basically all results about properties of random graphs with structural side constraints rely on completely different methods, mostly from analytic combinatorics. In this paper we show that recent progress in the construction of socalled Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer (Combinatorics, Probability and Computing 13, 2004) and Fusy (International Conference on Analysis of Algorithms ’05) can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables – to which we can then again apply Chernoff bounds to obtain extremely tight results. We elaborate our ideas by studying random dissections and triangulations of a labeled convex ngon. For both we obtain the degree sequence and the number of induced copies of given fixed graphs. The degree sequence for triangulations was already obtained previously by Gao and Wormald (Combinatorica 23, 2003) using deep methods from analytic combinatorics. We do, however, get better probabilities for the tails of the distributions. 1
Quadratic exactsize and linear approximatesize random sampling of planar graphs
 In Proc. Analysis of Algorithms
, 2005
"... This extended abstract introduces a new algorithm for the 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 judicious use of rejection, a new combinatorial ..."
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Cited by 11 (1 self)
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This extended abstract introduces a new algorithm for the 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 judicious 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. Then, for each generation, the time complexity 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).
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.
CONTROLLED NON UNIFORM RANDOM GENERATION OF DECOMPOSABLE STRUCTURES
"... Controlled non uniform random generation of decomposable structures ..."
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Cited by 8 (5 self)
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Controlled non uniform random generation of decomposable structures
Averagecase analysis of perfect sorting by reversals
, 2009
"... A sequence of reversals that takes a signed permutation to the identity is perfect if at no step a common interval is broken. Determining a parsimonious perfect sequence of reversals that sorts a signed permutation is NPhard. Here we show that, despite this worstcase analysis, with probability one ..."
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Cited by 7 (4 self)
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A sequence of reversals that takes a signed permutation to the identity is perfect if at no step a common interval is broken. Determining a parsimonious perfect sequence of reversals that sorts a signed permutation is NPhard. Here we show that, despite this worstcase analysis, with probability one, sorting can be done in polynomial time. Further, we find asymptotic expressions for the average length and number of reversals in commuting permutations, an interesting subclass of signed permutations. hal00354235, version 1 19 Jan 2009 1
Multidimensional Boltzmann Sampling of Languages
 In DMTCS Proceedings, number 01 in AM
, 2010
"... We address the uniform random generation of words from a contextfree language (over an alphabet of size k), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional extension of Boltzmann samplers. We show that, under mostly strongconnectiv ..."
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Cited by 7 (6 self)
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We address the uniform random generation of words from a contextfree language (over an alphabet of size k), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional extension of Boltzmann samplers. We show that, under mostly strongconnectivity hypotheses, our samplers return a word of size in [(1 − ε)n, (1 + ε)n] and exact frequency in O(n 1+k/2) expected time. Moreover, if we accept tolerance intervals of width in Ω ( √ n) for the number of occurrences of each letters, our samplers perform an approximatesize generation of words in expected O(n) time. We illustrate our approach on the generation of Tetris tessellations with uniform statistics in the different types of tetraminoes.
Maximal biconnected subgraphs of random planar graphs
 In SODA
, 2009
"... Let Pn be the class of simple labeled planar graphs with n vertices, and denote by Pn a graph drawn uniformly at random from this set. Basic properties of Pn were first investigated by Denise, Vasconcellos, and Welsh [7]. Since then, the random planar graph has attracted considerable attention, and ..."
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Cited by 7 (3 self)
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Let Pn be the class of simple labeled planar graphs with n vertices, and denote by Pn a graph drawn uniformly at random from this set. Basic properties of Pn were first investigated by Denise, Vasconcellos, and Welsh [7]. Since then, the random planar graph has attracted considerable attention, and is nowadays an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints. In this paper we study closely the structure of Pn. More precisely, let b(ℓ; Pn) be the number of blocks (i.e. maximal biconnected subgraphs) of Pn that contain exactly ℓ vertices, and let lb(Pn) be the number of vertices in the largest block of Pn. We show that with high probability Pn contains a giant block that includes up to lower order terms cn vertices, where c ≈ 0.959 is an analytically given constant. Moreover, we show that the second largest block contains only ˜ Θ(n 2/3) vertices, and prove sharp concentration results for b(ℓ; Pn), for all 2 ≤ ℓ ≤ n 2/3 (here ˜ Θ(.) stands for “up to logarithmic factors”). In fact, we obtain this result as a consequence of a much more general result that we prove in this paper. Let C be a class of labeled connected graphs, and let Cn be a graph drawn uniformly at random from graphs in C that contain exactly n vertices. Under certain assumptions on C, and depending on the behavior of the singularity of the generating function enumerating the elements of C, Cn belongs with high probability to one of the following three categories, which differ vastly in complexity. Cn either (1) behaves like a random planar graph, i.e. lb(Cn) ∼ cn, for some analytically given c = c(C), and the second largest block is of order n α, where 1> α = α(C), or (2) lb(Cn) = O(log n), i.e., all blocks contain at most logarithmically many vertices, or (3) lb(Cn) = Õ(nα), for some α = α(C) < 1.