On Boltzmann samplers and properties of combinatorial structures joint work with Nicla Bernasconi & Kostas Panagiotou TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

Random Graphs Paul Erdős, Alfred Rényi On the evolution of random graphs Publ. Math. Inst. Int. Hungar. Acad. Sci., 1960 Given a set of vertices. Decide for each potential edge randomly whether edge is present in graph. edge probability p → random graph G n,p Key property: Independence of edges.

Random Partial Orders Various models were proposed and studied, see e.g.

Random Planar Graphs Colin McDiarmid, AS, Dominic Welsh Random planar graphs Journal of Combinatorial Theory, Series B, 2005  c, C: 0 < c < Prob[P n connected ] < C < 1 P n := set of all planar graphs on n (labelled) vertices P n := graph drawn randomly from P n ( → random planar graph)

Connectedness – Proof Idea Direct approach: Counting... Prob[P n connected ] = Later... [ Giménez, Noy, ] # connected planar graphs on n vertices # planar graphs on n vertices

Connectedness – Proof Idea Combinatorial approach: Markov chain... start with empty graph: G := (V, ϕ ) repeat forever: –pick two vertices u,v from V uniformly at random –if {u,v} in G: delete edge {u,v} –if {u,v} not in G and G + {u,v} planar: insert edge {u,v} –otherwise: do nothing Stationary Distribution: uniform distribution on set of all planar graphs

Connectedness – Proof Idea Combinatorial approach: Markov chain... Stationary Distribution: uniform distribution on set of all planar graphs # edges in graph ≈ c 1 n # edges that can be added ≈ c 2 n # vertices of degree one ≈ α 1 n crude counting

Questions Combinatorics: Number of (planar) graphs on n vertices Properties of random (planar) graph P n : Expected number of edges Degree distribution Expected number of substructures (of a given type)... Algorithmic: Generate a random (planar) graph P n Aim of this talk: describe some recent progress towards the development of methods for answering this kind of questions.

Counting Philippe Flajolet, Robert Sedgewick Analytic Combinatorics Cambridge University Press, to appear Generating functions: G ≈ class of objects, G n ≈ objects in G of size n G(x) = Σ n≥0 | G n | x n (unlabelled objects) G(x) = Σ n≥0 x n (labelled objects) |Gn||Gn|

Generating Functions G ≈ class of (unlabelled) objects C ≈ class of connected objects in G G(x), C(x) ≈ corresponding generating functions G(x) = Σ k≥0 = e C(x) Observation: G(x) and C(x) have same radius of convergence... and thus the same growth constant C(x) k G(x) = Σ n≥0 | G n | x n

Counting Planar Graphs Tutte connected planar maps Bender, Gao, Wormald connected planar graphs Giménez, Noy connected planar graphs, general planar graphs

Giménez, Noy P ≈ class of (labelled) planar graphs | P n | = p · n −7/2 · γ n · n! where p ≈ · 10 −6 γ ≈ and for C ≈ class of connected planar graphs | C n | = c · n −7/2 · γ n · n! where c ≈ · 10 −6

Generation of Random Objects Duchon, Flajolet, Louchard, Schaeffer Boltzmann Samplers for the Random Generation of Combinatorial Structures Combinatorics, Probability and Computing, 2004 Éric Fusy Quadratic exact-size and linear approximate-size random sampling of planar graphs Analysis of Algorithms AofA’05

Boltzmann Sampler Observations: If we condition on | ΓD (x)| = n, then ΓD (x) is a uniform sampler. Expected size of the output depends on the parameter x: E [ | ΓD (x)| ] = An algorithm ΓD (x) that generates an element D  D is called Boltzmann Sampler iff x D´(x) D(x)

From Decompositions to Samplers Duchon, Flajolet, Louchard, Schaeffer, 2004

Boltzmann Sampler ΓD (x) D (  1,  2, …,  i, …) Input: sequence of  i ‘s that are independent and identically distributed Output: object D  D s.t. size of x determines expected size of ΓD(x) Idea: establish a relation between properties of the sequence (  1,  2, …,  i, …) and properties of ΓD (x).

Random Dissections D n := set of all dissections on n (labelled) vertices D n := graph drawn randomly from D n ( → random dissection)

rooted (unlabelled) dissection labelled dissection 2 ↔ (n-1)! Observation: It suffices to consider rooted unlabelled structures.

D =  D D D D D   … Generating Function D  class of all edge-rooted dissections Bodirsky, Giménez, Kang, Noy, = c  n -3/2 δ n

Boltzmann Sampler Aim: procedure ΓD (x) that generates an element D  D such that recall: Idea: - first generate face containing the root-edge - then call procedure recursively for each edge of cycle D =  D D D D D   …

Boltzmann Sampler ΓD (x) D (  1,  2, …,  i, …) Input: sequence of  i ‘s that are independent and distributed s.t. Output: random dissection D s.t.

Sampler - How does it work? path of length α 1 -1 path of length α 2 -2 path of length α k -2 What can we deduce about the degrees ? root edge path of length Σ i (α i -2) α k+1 = 2 D2D2 DiDi D Σ (αi-2) D1D1 path of length α 1 -2 degree = 1 + min{ k : α k+1 =2 }

Sampler - How does it work? degree = min{ k : α k =2 } root edge α k+1 = 2 D2D2 DiDi D Σ (αi-1) D1D1 Observations: # of used α i ‘s = # of edges of D 1 + #{ i : α i = 2 } = # of vertices of D α 1,α 2,…,α k,2,α k+2, …,2, ….,2, …, α |E(D)|-1, 2 # blocks = # vertices - 1 pre-degree

Sampler - How does it work? root edge D2D2 DiDi D Σ (αi-1) D1D1 Key observation: # pre-deg‘( D, k) = # blocks‘ of size k deg(root, D ) = size of first block do not count root vertices do not count first block

Sampler - Summary ΓD (x) D (  1,  2, …,  i, …) If |V(D)| = n then ΓD (x) used  3n of the  i ‘s. # pre-deg ( D ;k) = number of blocks of size k among the first n blocks.

A Sampler for Dissections of Size n for i = 1 to n 3 do: if |V( D i )| = n then return D i D   (  1 1,  1 2, …,  1 3n ) Easy probabilistic arguments/Chernoff: Prob[ algorithm returns D   ]  1 – e -  (n)  k: Prob[  i s.t. (  i 1,  i 2, …,  i 3n ) contains  (1  ) c k n blocks of size k among first n blocks]  e -  (n) Easy probabilistic arguments/Chernoff: Prob[ algorithm returns D   ]  1 – e -  (n)  k: Prob[  i s.t. (  i 1,  i 2, …,  i 3n ) contains  (1  ) c k n blocks of size k among first n blocks]  e -  (n) (  2 1,  2 2, …,  2 3n ) (  n 3 1,  n 3 2, …,  n 3 3n ) ΓD(ρD)ΓD(ρD) Di  Di   (  i 1,  i 2, …,  i 3n ) c k ~ geometric distribution

A Sampler for Dissections of Size n D   (  1 1,  1 2, …,  1 3n ) Properties: Prob[ algorithm returns D   ]  1 – e -  (n)  k: Prob[ #pre-deg( D, k)  (1  ) c k n]  e -  (n)  k: Prob[ #post-deg( D, k)  (1  ) c k n]  e -  (n)  k, : Prob[ #pre-deg( D, k)  (1  ) c k n & #post-deg( D, )  (1  ) c n]  e -  (n)  k: Prob[ #deg( D, k)  (1  ) d k n]  e -  (n) Properties: Prob[ algorithm returns D   ]  1 – e -  (n)  k: Prob[ #pre-deg( D, k)  (1  ) c k n]  e -  (n)  k: Prob[ #post-deg( D, k)  (1  ) c k n]  e -  (n)  k, : Prob[ #pre-deg( D, k)  (1  ) c k n & #post-deg( D, )  (1  ) c n]  e -  (n)  k: Prob[ #deg( D, k)  (1  ) d k n]  e -  (n) (  2 1,  2 2, …,  2 3n ) (  n 3 1,  n 3 2, …,  n 3 3n ) ΓD‘ (ρ D ) k = k(n) is also possible Δ(D n ) = Θ(log n) Similar results for subdisections

Extension: Outerplanar Graphs biconnected outerplanar graphs (dissections) connected outerplanar graphs outerplanar graphs McDiarmid, St., Welsh (2005): degree sequence of a random planar graph ≈ degree sequence of random planar connected graph

Let C  be the class of labelled rooted connected (outerplanar) graphs: Z is the class consisting of one single element, B  is the class of rooted biconnected (outerplanar) graphs, and B ‘ the derivated class From Biconnected to Connected

A Sampler for Rooted Connected Outerplanar Graphs Γ C  (x) C ( λ 1, λ 2, …, λ i, …) (B 1, B 2, …, B i, …) List of parameters indep. distributed according to Po( B‘ (C  (x)) ). List of vertex rooted dissections according to Γ B ‘(x). Similarly as before: If we run Γ C  (ρ C ) n 3 times then one of the graphs will have size exactly n C with probability 1 – e -  (n). Need to study properties of one run only …

A Sampler for Rooted Connected Outerplanar Graphs Intuitively :  Every vertex is born with a certain degree  It then receives a certain number of new neighbors  Every vertex is born with a certain degree  It then receives a certain number of new neighbors d = P [ born with degree ] p k- = P [ receive k- more neighbors later ]  inner vertex of dissection  Poisson many copies of a root of a dissection  inner vertex of dissection  Poisson many copies of a root of a dissection ( )

Summary and Outlook dissections and triangulations of a polygon connected outerplanar graphs → general scheme to lift properties from 2-connected graphs to 1-connected graphs 1 outerplanar graphs → same as 1-connected due to McDiarmid, St, Welsh Degree sequence/substructures of random … 2 3 Extensions: series-parallel graphs Work in progress: general scheme, other graph classes, … [Drmota, Giménez, Noy 2008+]