## Uniform random sampling of planar graphs in linear time (2007)

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Citations: | 9 - 2 self |

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@MISC{Fusy07uniformrandom,

author = {Éric Fusy},

title = {Uniform random sampling of planar graphs in linear time},

year = {2007}

}

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### Abstract

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 exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size 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 exact-size and linear approximate-size random generation of planar graphs”, which appeared in the Proceedings of the International Conference on Analysis of Algorithms (AofA’05), 6-10 June 2005, Barcelona. 1.

### Citations

773 |
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Citation Context ...addition, A0 is irreducible, i.e., the dependency graph induced by the nonzero coefficients of A0 is strongly connected. A well known result of Markov chain theory ensures that (I − A0) is invertible =-=[22]-=-. Hence, (I − A) is invertible for z close to z0, and (I − A) −1 converges to the matrix (I − A0) −1 . Moreover, the components of L are of the form ( L = a, b, c, d · Λ −→ G3 ′ (z, w) + e · Λ −→ ) G3... |

380 | The factors of graphs
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Citation Context ...Their idea is to apply the recursive method of sampling to a well known combinatorial decomposition of planar graphs according to successive levels of connectivity, which has been formalised by Tutte =-=[33]-=-. Precisely, the decomposition yields some recurrences satisfied by the coefficients counting planar graphs as well as subfamilies (connected, 2-connected, 3-connected), which in turn yield an explici... |

317 |
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Citation Context ...(x, y0) = P(x) + cα · (x0 − x) α + o((x0 − x) α ), where cα is a constant, P(x) is rational with no poles in the disk |z| ≤ x0, and where the expansion holds in a so-called ∆-neighbourhood of x0, see =-=[14, 13]-=-. In the special case α = 1/2, the class is said to have square-root singularities. 5 In an earlier version of the article and in the conference version [16], we derived 3 times—as prescribed by [11]—... |

292 | Graphical Enumeration - Harary, Palmer - 1973 |

176 |
Combinatorial Species and Tree-like Structures
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Citation Context ...nvolves both (labelled) vertices and (unlabelled) edges. The constructions needed to formulate the decomposition of planar graphs are classical ones in combinatorics: Sum, Product, Set, Substitutions =-=[3, 14]-=-. In Section 3.2, for each of the constructions, we describe a sampling rule, so that Boltzmann samplers can be assembled for any class that admits a decomposition in terms of these constructions. Mor... |

143 |
A census of planar maps
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Citation Context ...y Schaeffer in his thesis [29], and many other families of rooted maps have been counted in this way [17, 27, 28, 7]. The advantage of bijective constructions over recursive methods for counting maps =-=[32]-=- is that the bijections yield efficient —linear-time— generators for maps, as random sampling of maps is reduced to the much easier task of random sampling of trees, see [30]. The method has been rece... |

137 |
Combinatorial algorithms
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Citation Context ...g algorithm. More importantly, the rate of convergence to the uniform distribution is unknown. A second approach for uniform random generation is the recursive method introduced by Nijenhuis and Wilf =-=[25]-=- and formalised by Flajolet, Van Cutsem and Zimmermann [15]. The recursive method is a general framework for the random generation of combinatorial classes admitting a recursive decomposition. For suc... |

105 | Cutsem. A calculus for the random generation of labelled combinatorial structures - Flajolet, Zimmermann, et al. - 1994 |

69 |
Conjugaison d’arbres et cartes combinatoires aléatoires
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Citation Context ...heorem of Whitney [35], 3-connected planar graphs have a unique embedding (up to reflection), so they are equivalent to 3-connected planar maps. Following the general approach introduced by Schaeffer =-=[29]-=-, a bijection has been described by the author, Poulalhon, and Schaeffer [18] to enumerate 3-connected maps [18] from binary trees, which yields an explicit Boltzmann sampler for (rooted) 3-connected ... |

67 | Boltzmann samplers for the random generation of combinatorial structures
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(Show Context)
Citation Context ...ion into a random generator using the framework of Boltzmann samplers, instead of the recursive method. Boltzmann samplers have been recently developed by Duchon, Flajolet, Louchard, and Schaeffer in =-=[11]-=- as a powerful framework for the random generation of decomposable combinatorial structures. The idea of Boltzmann sampling is to gain efficiency by relaxing the constraint of exact-size sampling. As ... |

45 | Random planar graphs
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(Show Context)
Citation Context ... the implementation to connected graphs, which are conveniently manipulated using the half-edge data structure.) However, from the works of Giménez and Noy [20] and previous work by MacDiarmid et al. =-=[23]-=-, a random planar graph consists of a huge connected component, plus other components whose total expected size is O(1). Thus, statistical properties like those stated in Conjecture 23 should be the s... |

39 | Asymptotic enumeration and limit laws of planar graphs
- Giménez, Noy
(Show Context)
Citation Context ...graphs that are labelled, i.e., the n vertices bear distinct labels in [1..n], and simple, i.e., with no loop nor multiple edges. Statistical properties of planar graphs have been intensively studied =-=[6, 19, 20]-=-. Very recently, Giménez and Noy [20] have solved exactly the difficult problem of the asymptotic enumeration of labelled planar graphs. They also provide exact analytic expressions for the asymptotic... |

37 | Optimal coding and sampling of triangulations
- Poulalhon, Schaeffer
(Show Context)
Citation Context ... satisfy simple context-free decomposition grammars. Such constructions have first been described by Schaeffer in his thesis [29], and many other families of rooted maps have been counted in this way =-=[17, 27, 28, 7]-=-. The advantage of bijective constructions over recursive methods for counting maps [32] is that the bijections yield efficient —linear-time— generators for maps, as random sampling of maps is reduced... |

35 | The number of labeled 2-connected planar graphs
- Bender, Gao, et al.
(Show Context)
Citation Context ...eat flexibility to design Boltzmann samplers, since it makes it possible to adjust the distributions of the samplers. Lemma 5 (Rejection). Given a combinatorial class C, let W : C ↦→ R + and p : C ↦→ =-=[0, 1]-=- be two functions, called weight-function and rejection-function, respectively. Assume that W is summable, i.e., ∑ γ∈C W(γ) is finite. Let A be a random generator for C that draws each object γ ∈ C wi... |

34 | Planar maps as labeled mobiles
- Bouttier, Francesco, et al.
- 2004
(Show Context)
Citation Context ... satisfy simple context-free decomposition grammars. Such constructions have first been described by Schaeffer in his thesis [29], and many other families of rooted maps have been counted in this way =-=[17, 27, 28, 7]-=-. The advantage of bijective constructions over recursive methods for counting maps [32] is that the bijections yield efficient —linear-time— generators for maps, as random sampling of maps is reduced... |

34 |
The enumeration of c-nets via quadrangulations
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(Show Context)
Citation Context ... rise to an edge of µ connecting the two (opposite) black vertices of f, see Figure 6(c)-(d). The map µ is naturally rooted so as to have the same root-vertex as κ. Theorem 9 (Mullin and Schellenberg =-=[24]-=-). The primal-map construction is a bijection between rooted irreducible quadrangulations with n black vertices and m faces, and rooted 3connected maps with n vertices and m edges3 . In other words, t... |

33 | The random planar graph
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Citation Context ...s planarity testing, embedding algorithms, procedures for finding geometric cuts, and so on. Denise, Vasconcellos, and Welsh have proposed a first algorithm for the random generation of planar graphs =-=[8]-=-, by defining a Markov chain on the set Gn of labelled planar graphs with n vertices. At each step, two different vertices v and v ′ are chosen at random. If they are adjacent, the edge (v, v ′ ) is d... |

32 |
Counting labelled three-connected and homeomorphically irreducible twoconnected graphs
- Walsh
- 1982
(Show Context)
Citation Context ...ur vertices (the 2-connected ones are surrounded). Below each graph is indicated the number of distinct labellings. components, which has been formalised by Trakhtenbrot [31] (and later used by Walsh =-=[34]-=- to count 2-connected planar graphs and by Bender, Gao, Wormald to obtain asymptotic enumeration [1]). Finally, connected planar graphs are generated from 2-connected ones by using the well-known deco... |

30 | Uniform Random Generation of Decomposable Structures Using Floating-Point Arithmetic
- Denise, Zimmermann
- 1997
(Show Context)
Citation Context ...an auxiliary memory of size O(n 5 log n). Once the tables have been computed, the complexity of each generation is O(n 3 ). A more recent optimisation of the recursive method by Denise and Zimmermann =-=[9]-=- —based on controlled real arithmetics— should be applicable; it would improve the time complexity somewhat, but the storage complexity would still be large. In this article, we introduce a new random... |

28 |
Random sampling of large planar maps and convex polyhedra
- Schaeffer
- 1999
(Show Context)
Citation Context ...methods for counting maps [32] is that the bijections yield efficient —linear-time— generators for maps, as random sampling of maps is reduced to the much easier task of random sampling of trees, see =-=[30]-=-. The method has been recently applied to the family of 3-connected maps, which is of interest here. Precisely, as described in [18], there is a bijection between binary trees and irreducible dissecti... |

25 | Generating Labelled Planar Graphs Uniformly at Random
- Bodirsky, Gröpl, et al.
- 2003
(Show Context)
Citation Context ...tion. As a consequence, this method requires a preprocessing step where large tables of large coefficients are calculated using the recursive relations they satisfy. Bodirsky et al. have described in =-=[5]-=- the first polynomial-time random sampler for planar graphs. Their idea is to apply the recursive method of sampling to a well known combinatorial decomposition of planar graphs according to successiv... |

23 |
Towards a theory of non–repeating contact schemes (russian
- Trakhtenbrot
- 1958
(Show Context)
Citation Context ... planar graphs with at most four vertices (the 2-connected ones are surrounded). Below each graph is indicated the number of distinct labellings. components, which has been formalised by Trakhtenbrot =-=[31]-=- (and later used by Walsh [34] to count 2-connected planar graphs and by Bender, Gao, Wormald to obtain asymptotic enumeration [1]). Finally, connected planar graphs are generated from 2-connected one... |

22 | 2007. “Boltzmann sampling of unlabelled structures
- Flajolet, Fusy, et al.
(Show Context)
Citation Context ...bution Px. The authors of [11] give sampling rules associated to classical combinatorial constructions, such as Sum, Product, and Set. (For the unlabelled setting, we refer to the more recent article =-=[12]-=-, and to [4] for the specific case of plane partitions.) In order to translate the combinatorial decomposition of planar graphs into a Boltzmann sampler, we need to extend the framework of Boltzmann s... |

19 | G.: Planar graphs, via wellorderly maps and trees
- Bonichon, Gavoille, et al.
- 2006
(Show Context)
Citation Context ...graphs that are labelled, i.e., the n vertices bear distinct labels in [1..n], and simple, i.e., with no loop nor multiple edges. Statistical properties of planar graphs have been intensively studied =-=[6, 19, 20]-=-. Very recently, Giménez and Noy [20] have solved exactly the difficult problem of the asymptotic enumeration of labelled planar graphs. They also provide exact analytic expressions for the asymptotic... |

19 | Dissections and trees, with applications to optimal mesh encoding and to random sampling
- Fusy, Poulalhon, et al.
- 2005
(Show Context)
Citation Context ...rview The algorithm we describe relies mainly on two ingredients. The first one is a recent correspondence, called the closure-mapping, between binary trees and (edge-rooted) 3connected planar graphs =-=[18]-=-, which makes it possible to obtain a Boltzmann sampler for 3-connected planar graphs. The second one is a decomposition formalised by Tutte [33], which ensures that any planar graph can be decomposed... |

16 | Degree distribution in random planar graphs
- Drmota, Giménez, et al.
(Show Context)
Citation Context ...where d is the maximal degree of γ and, for 1 ≤ k ≤ d, Z (k) (γ) is the proportion of vertices of γ that have degree k. Let us mention some progress on this conjecture. It has recently been proved in =-=[10]-=- that the expected values E(Z (k) n ) converge as n → ∞ to constants π (k) that are computable and satisfy ∑ k π(k) = 1. Hence, what remains to be shown regarding the conjecture is the concentration p... |

13 | Boltzmann oracle for combinatorial systems
- Pivoteau, Salvy, et al.
(Show Context)
Citation Context ...er log(n) k (again, this is yet to be proved rigorously). The following informal statement summarizes the discussion; making a theorem of it is the subject of ongoing research (see the recent article =-=[26]-=-): Fact. With high probability, the auxiliary memory necessary to generate planar graphs of size n is of order O(log(n)) and the preprocessing time complexity is of order O(log(n) k ) for some low int... |

12 | Transversal structures on triangulations, with application to straight-line drawing., in Graph Drawing(Proc
- Fusy
- 2007
(Show Context)
Citation Context ... satisfy simple context-free decomposition grammars. Such constructions have first been described by Schaeffer in his thesis [29], and many other families of rooted maps have been counted in this way =-=[17, 27, 28, 7]-=-. The advantage of bijective constructions over recursive methods for counting maps [32] is that the bijections yield efficient —linear-time— generators for maps, as random sampling of maps is reduced... |

11 |
A bijection for triangulations of a polygon with interior points and multiple edges
- Poulalhon, Schaeffer
- 2003
(Show Context)
Citation Context |

11 | An unbiased pointing operator for unlabelled structures, with applications to counting and sampling - Bodirsky, Fusy, et al. - 2007 |

10 |
The asymptotic enumeration of rooted convex polyhedra
- Bender, Richmond
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(Show Context)
Citation Context ...as K. Proof. The class J is equal to 3 ⋆ ZL ⋆ ZU ⋆ I, which is isomorphic to 3 ⋆ ZL ⋆ ZU ⋆ K, so J has the same singular points and singularity type as K. □ Lemma 35 (rooted 3-connected planar graphs =-=[2]-=-). The class −→ G3 of edge-rooted 3-connected planar graphs is 3/2-singular; and the class −→ G3 of U-derived edge-rooted 3-connected planar graphs is 1/2-singular. These classes have the same singula... |

9 |
Analytic Combinatorics. Preliminary version available at http://pauillac.inria.fr/algo/flajolet/Publications/books.html
- Flajolet, Sedgewick
(Show Context)
Citation Context ...nvolves both (labelled) vertices and (unlabelled) edges. The constructions needed to formulate the decomposition of planar graphs are classical ones in combinatorics: Sum, Product, Set, Substitutions =-=[3, 14]-=-. In Section 3.2, for each of the constructions, we describe a sampling rule, so that Boltzmann samplers can be assembled for any class that admits a decomposition in terms of these constructions. Mor... |

7 | Analytic combinatorics. Available at http://algo.inria.fr/flajolet/Publications/book051001.pdf. Preliminary version of the forthcoming book - Flajolet, Sedgewick |

6 |
Quadratic exact size and linear approximate size random generation of planar graphs
- Fusy
- 2005
(Show Context)
Citation Context ...a so-called ∆-neighbourhood of x0, see [14, 13]. In the special case α = 1/2, the class is said to have square-root singularities. 5 In an earlier version of the article and in the conference version =-=[16]-=-, we derived 3 times—as prescribed by [11]—in order to get a singularity type (1 − x/ρ) −1/2 (efficient targetted samplers are obtained when taking x = ρ(1 − 1/(2n))). We have recently discovered that... |

5 | Random planar graphs with a fixed number of edges - Gerke, McDiarmid, et al. - 2005 |

4 | Random sampling of plane partitions
- Bodini, Fusy, et al.
(Show Context)
Citation Context ...e authors of [11] give sampling rules associated to classical combinatorial constructions, such as Sum, Product, and Set. (For the unlabelled setting, we refer to the more recent article [12], and to =-=[4]-=- for the specific case of plane partitions.) In order to translate the combinatorial decomposition of planar graphs into a Boltzmann sampler, we need to extend the framework of Boltzmann samplers to t... |

2 |
Algorithms project, INRIA Rocquencourt 78153 Le Chesnay Cedex, France E-mail address: eric.fusy@inria.fr
- Math
- 1933
(Show Context)
Citation Context ...oltzmann samplers Our algorithm starts with the generation of 3-connected planar graphs, which have the nice feature that they are combinatorially tractable. Indeed, according to a theorem of Whitney =-=[35]-=-, 3-connected planar graphs have a unique embedding (up to reflection), so they are equivalent to 3-connected planar maps. Following the general approach introduced by Schaeffer [29], a bijection has ... |