Enumeration and uniform sampling of planar structures Mihyun Kang - PowerPoint PPT Presentation

Asymptotic number n t ( n ) z n View a generating function T ( z ) = � n ! as a complex-valued function that is analytic at the origin. Let R be the radius of convergence of T ( z ) . Then n →∞ | θ ( n ) | 1 /n = 1 . [ z n ] T ( z ) = θ ( n ) R − n , lim sup where [Pringsheim’s Theorem] The point z = R is a dominant singularity of T ( z ) , since T ( z ) has non-negative Taylor coefficients.

Asymptotic number n t ( n ) z n View a generating function T ( z ) = � n ! as a complex-valued function that is analytic at the origin. Let R be the radius of convergence of T ( z ) . Then n →∞ | θ ( n ) | 1 /n = 1 . [ z n ] T ( z ) = θ ( n ) R − n , lim sup where [Pringsheim’s Theorem] The point z = R is a dominant singularity of T ( z ) , since T ( z ) has non-negative Taylor coefficients. How to determine • the dominant singularity R and • the subexponential factor θ ( n ) ?

Singularity analysis [ F LAJOLET , S EDGEWICK 07+ ] Let ψ ( u ) be the functional inverse of T ( z ) . (Indeed ψ ( u ) = ue − u for rooted trees.)

Singularity analysis [ F LAJOLET , S EDGEWICK 07+ ] Let ψ ( u ) be the functional inverse of T ( z ) . (Indeed ψ ( u ) = ue − u for rooted trees.) Let r > 0 be the radius of convergence of ψ , and suppose there exists u 0 ∈ (0 , r ) such that ψ � ( u 0 ) = 0 and ψ �� ( u 0 ) � = 0 . ψ ( u ) u 0 Indeed, z 0 = e − 1 and thus t ( n ) = θ ( n ) e n , where lim sup | θ ( n ) | 1 /n = 1 .

Singularity analysis [ F LAJOLET , S EDGEWICK 07+ ] Let ψ ( u ) be the functional inverse of T ( z ) . (Indeed ψ ( u ) = ue − u for rooted trees.) Let r > 0 be the radius of convergence of ψ , and suppose there exists u 0 ∈ (0 , r ) such that ψ � ( u 0 ) = 0 and ψ �� ( u 0 ) � = 0 . T ( z ) z 0 = ψ ( u 0 ) ψ ( u ) u 0 Indeed, z 0 = e − 1 and thus t ( n ) = θ ( n ) e n , where lim sup | θ ( n ) | 1 /n = 1 .

Singularity analysis [ F LAJOLET , S EDGEWICK 07+ ] Let ψ ( u ) be the functional inverse of T ( z ) . (Indeed ψ ( u ) = ue − u for rooted trees.) Let r > 0 be the radius of convergence of ψ , and suppose there exists u 0 ∈ (0 , r ) such that ψ � ( u 0 ) = 0 and ψ �� ( u 0 ) � = 0 . T ( z ) z 0 = ψ ( u 0 ) ψ ( u ) u 0 Indeed, z 0 = e − 1 and thus t ( n ) n ! = θ ( n ) e n , where lim sup | θ ( n ) | 1 /n = 1 .

Local dependency [ F LAJOLET , S EDGEWICK 07+ ] Taylor expansion of z = ψ ( u ) at u 0 is of the form ψ ( u ) = ψ ( u 0 ) + 1 2 ψ �� ( u 0 )( u − u 0 ) 2 + · · · .

Local dependency [ F LAJOLET , S EDGEWICK 07+ ] Taylor expansion of z = ψ ( u ) at u 0 is of the form ψ ( u ) = ψ ( u 0 ) + 1 2 ψ �� ( u 0 )( u − u 0 ) 2 + · · · . It implies a locally quadratic dependency between z and u = T ( z ) : ψ �� ( u 0 )( z − z 0 ) = − 2 ψ ( u 0 ) 2 ( T ( z ) − T ( z 0 )) 2 = ( u − u 0 ) 2 ∼ ψ �� ( u 0 )(1 − z/z 0 )

Local dependency [ F LAJOLET , S EDGEWICK 07+ ] Taylor expansion of z = ψ ( u ) at u 0 is of the form ψ ( u ) = ψ ( u 0 ) + 1 2 ψ �� ( u 0 )( u − u 0 ) 2 + · · · . It implies a locally quadratic dependency between z and u = T ( z ) : ψ �� ( u 0 )( z − z 0 ) = − 2 ψ ( u 0 ) 2 ( T ( z ) − T ( z 0 )) 2 = ( u − u 0 ) 2 ∼ ψ �� ( u 0 )(1 − z/z 0 ) Since T ( z ) is increasing along the positive real axis, we have � − 2 ψ ( u 0 ) /ψ �� ( u 0 ) (1 − z/z 0 ) 1 / 2 T ( z ) − T ( z 0 ) ∼ −

Local dependency [ F LAJOLET , S EDGEWICK 07+ ] Taylor expansion of z = ψ ( u ) at u 0 is of the form ψ ( u ) = ψ ( u 0 ) + 1 2 ψ �� ( u 0 )( u − u 0 ) 2 + · · · . It implies a locally quadratic dependency between z and u = T ( z ) : ψ �� ( u 0 )( z − z 0 ) = − 2 ψ ( u 0 ) 2 ( T ( z ) − T ( z 0 )) 2 = ( u − u 0 ) 2 ∼ ψ �� ( u 0 )(1 − z/z 0 ) Since T ( z ) is increasing along the positive real axis, we have � − 2 ψ ( u 0 ) /ψ �� ( u 0 ) (1 − z/z 0 ) 1 / 2 T ( z ) − T ( z 0 ) ∼ − Using ∆ -analycity of T ( z ) and transfer theorem, we have [ z n ] T ( z ) ∼ − � − 2 ψ ( u 0 ) /ψ �� ( u 0 )[ z n ](1 − z/z 0 ) 1 / 2

Basic scale [ F LAJOLET , S EDGEWICK 07+ ] We have z 0 = e − 1 , u 0 = 1 , ψ ( u ) = ue − u and [ z n ] T ( z ) ∼ − � − 2 ψ ( u 0 ) /ψ �� ( u 0 )[ z n ](1 − z/z 0 ) 1 / 2 .

Basic scale [ F LAJOLET , S EDGEWICK 07+ ] We have z 0 = e − 1 , u 0 = 1 , ψ ( u ) = ue − u and [ z n ] T ( z ) ∼ − � − 2 ψ ( u 0 ) /ψ �� ( u 0 )[ z n ](1 − z/z 0 ) 1 / 2 . R ESCALING RULE / GENERALIZED BINOMIAL THEOREM ∼ n − 3 / 2 � n − 3 / 2 � [ z n ] (1 − z/z 0 ) 1 / 2 = z − n − 2 √ πz − n 0 . 0 n

Basic scale [ F LAJOLET , S EDGEWICK 07+ ] We have z 0 = e − 1 , u 0 = 1 , ψ ( u ) = ue − u and [ z n ] T ( z ) ∼ − � − 2 ψ ( u 0 ) /ψ �� ( u 0 )[ z n ](1 − z/z 0 ) 1 / 2 . R ESCALING RULE / GENERALIZED BINOMIAL THEOREM ∼ n − 3 / 2 � n − 3 / 2 � [ z n ] (1 − z/z 0 ) 1 / 2 = z − n − 2 √ πz − n 0 . 0 n We have that the number of rooted trees on n vertices equals 1 n n − 1 = t ( n ) 2 πn − 3 / 2 e n n ! √ ∼ � n √ 1 2 πn − 3 / 2 e n � n ∼ √ 2 πn (Stirling’s formula) e = n n − 1 (Cayley’s formula)

Block structure of a graph A block of a graph is a maximal connected subgraph without a cutvertex:

Block structure of a graph A block of a graph is a maximal connected subgraph without a cutvertex: • a maximal biconnected subgraph, • an edge (including its ends), or • an isolated vertex

Block structure of a graph A block of a graph is a maximal connected subgraph without a cutvertex: • a maximal biconnected subgraph, • an edge (including its ends), or • an isolated vertex The block structure of a graph is a forest with two types of vertices: the blocks and the cutvertices of the graph.

Blocks of planar structures 2-connected outerplanar graphs:

Blocks of planar structures 2-connected outerplanar graphs: [ B ODIRSKY , G IMÉNEZ , K., N OY 07+ ] ρ . ∼ α n − 5 / 2 ρ n n ! , # outerplanar graphs on n vertices = 7 . 32

Blocks of planar structures 2-connected outerplanar graphs: [ B ODIRSKY , G IMÉNEZ , K., N OY 07+ ] ρ . ∼ α n − 5 / 2 ρ n n ! , # outerplanar graphs on n vertices = 7 . 32 2-connected series-parallel graphs:

Blocks of planar structures 2-connected outerplanar graphs: [ B ODIRSKY , G IMÉNEZ , K., N OY 07+ ] ρ . ∼ α n − 5 / 2 ρ n n ! , # outerplanar graphs on n vertices = 7 . 32 2-connected series-parallel graphs: [ B ODIRSKY , G IMÉNEZ , K., N OY 07+ ] γ . ∼ β n − 5 / 2 γ n n ! , # series-parallel graphs on n vertices = 9 . 07

Labeled planar graphs 2-connected graphs [ T RAKHTENBROT 58; T UTTE 63; W ALSH 82 ]

Labeled planar graphs 2-connected graphs [ T RAKHTENBROT 58; T UTTE 63; W ALSH 82 ] [ B ENDER , G AO , W ORMALD 02 ] The growth constant for biconnected planar graphs: ∼ 26 . 18

Labeled planar graphs 2-connected graphs [ T RAKHTENBROT 58; T UTTE 63; W ALSH 82 ] [ B ENDER , G AO , W ORMALD 02 ] The growth constant for biconnected planar graphs: ∼ 26 . 18 [ B ENDER , R ICHMOND 84 ; B ODIRSKY , G RÖPL , J OHANNSEN , K. 05; F USY , P OULALHON , S CHAEFFER 05 ] The growth constant for 3-connected planar graphs: ∼ 21 . 05

Labeled planar graphs 2-connected graphs [ T RAKHTENBROT 58; T UTTE 63; W ALSH 82 ] [ B ENDER , G AO , W ORMALD 02 ] The growth constant for biconnected planar graphs: ∼ 26 . 18 [ B ENDER , R ICHMOND 84 ; B ODIRSKY , G RÖPL , J OHANNSEN , K. 05; F USY , P OULALHON , S CHAEFFER 05 ] The growth constant for 3-connected planar graphs: ∼ 21 . 05 Uniform sampling algorithm

Labeled planar graphs 2-connected graphs [ T RAKHTENBROT 58; T UTTE 63; W ALSH 82 ] [ B ENDER , G AO , W ORMALD 02 ] The growth constant for biconnected planar graphs: ∼ 26 . 18 [ B ENDER , R ICHMOND 84 ; B ODIRSKY , G RÖPL , J OHANNSEN , K. 05; F USY , P OULALHON , S CHAEFFER 05 ] The growth constant for 3-connected planar graphs: ∼ 21 . 05 Uniform sampling algorithm [ B ODIRSKY , G RÖPL , K. 03; F USY 05 ; G IMÉNEZ , N OY 05 ] Uniform sampling algorithm for planar graphs O ( n 7 ) ; O ( n 2 ) ∼ c n − 7 / 2 27 . 22 n n ! The number of planar graphs is

Scheme Decomposition Recursive Counting Formulas Equations of Generating Functions Singularity Analysis Recursive Method Uniform Generation Asymptotic Number

Scheme Decomposition Recursive Counting Formulas Equations of Generating Functions Singularity Analysis Recursive Method Uniform Generation Asymptotic Number Probabilistic Analysis Typical Properties

Labeled cubic planar graphs [ B ODIRSKY , K., L ÖFFLER , M C D IARMID 07 ] The number of cubic planar graphs on n vertices is asymptotically where ρ . ∼ αn − 7 / 2 ρ n n ! , = 3 . 1325

Labeled cubic planar graphs [ B ODIRSKY , K., L ÖFFLER , M C D IARMID 07 ] The number of cubic planar graphs on n vertices is asymptotically where ρ . ∼ αn − 7 / 2 ρ n n ! , = 3 . 1325 What is the chromatic number of a random cubic planar graph G that is chosen uniformly at random among labeled cubic planar graphs on [ n ] ?

Chromatic number What is the chromatic number of a random cubic planar graph G ? • χ ( G ) ≤ 4 [Four colour theorem] • For any connected graph G that is neither a complete graph nor an odd cycle, χ ( G ) ≤ ∆ ( G ) = 3 [Brooks’ theorem]

Chromatic number What is the chromatic number of a random cubic planar graph G ? • χ ( G ) ≤ 4 [Four colour theorem] • For any connected graph G that is neither a complete graph nor an odd cycle, χ ( G ) ≤ ∆ ( G ) = 3 [Brooks’ theorem] If G contains a component isomorphic to K 4 , then χ ( G ) = 4 . If G contains no isolated K 4 , but at least one triangle, then χ ( G ) = 3 .

Random cubic planar graphs [ B ODIRSKY , K., L ÖFFLER , M C D IARMID 07 ] Let G ( k ) be a random k connected cubic planar graph on n vertices. n

Random cubic planar graphs [ B ODIRSKY , K., L ÖFFLER , M C D IARMID 07 ] Let G ( k ) be a random k connected cubic planar graph on n vertices. n S UBGRAPH CONTAINMENTS Let X n be # isolated K 4 ’s in G (0) and Y n # triangles in G ( k ) n , k > 0 . Then n n →∞ Pr( X n > 0) = 1 − e − ρ 4 4! , lim n →∞ Pr( Y n > 0) = 1 . lim

Random cubic planar graphs [ B ODIRSKY , K., L ÖFFLER , M C D IARMID 07 ] Let G ( k ) be a random k connected cubic planar graph on n vertices. n S UBGRAPH CONTAINMENTS Let X n be # isolated K 4 ’s in G (0) and Y n # triangles in G ( k ) n , k > 0 . Then n n →∞ Pr( X n > 0) = 1 − e − ρ 4 4! , lim n →∞ Pr( Y n > 0) = 1 . lim C HROMATIC NUMBER n →∞ Pr( X n > 0) = 1 − e − ρ 4 n →∞ Pr( χ ( G (0) lim n ) = 4) = lim 4! n →∞ Pr( X n = 0 , Y n > 0) = e − ρ 4 4! . n →∞ Pr( χ ( G (0) lim n ) = 3) = lim = 0 . 9995 . lim n →∞ Pr( χ ( G ( k ) For k = 1 , 2 , 3 , n ) = 3) = lim n →∞ Pr( Y n > 0) = 1 .

Labeled planar structures The number of planar structures on n vertices is asymp. ∼ α n − β γ n n ! . Let G n be a random planar structure on n vertices. Then as n → ∞ , • the expected number of edges in G n is ∼ µn , • G n is connected with probability tending to a constant p con , and • χ ( G n ) is three with probability tending to a constant p χ . Running time of uniform sampler (recursive method): ˜ O ( n k ) Classes β γ µ p con p χ k Trees 5 / 2 2 . 71 1 1 0 4 Outerplanar graphs 5 / 2 7 . 32 1 . 56 0 . 861 1 4 Series-parallel graphs ? ? 5 / 2 9 . 07 1 . 61 0 . 889 Planar graphs ? 7 / 2 27 . 2 2 . 21 0 . 963 7 Cubic planar graphs 7 / 2 3 . 13 1 . 50 ≥ 0 . 998 0 . 999 6

Unlabeled planar structures Difficulty with unlabeled planar structures is symmetry:

Unlabeled planar structures Difficulty with unlabeled planar structures is symmetry: • recursive method: decomposition along symmetry • Pólya Theory: cycle indices

Unlabeled planar structures Difficulty with unlabeled planar structures is symmetry: • recursive method: decomposition along symmetry • Pólya theory: symmetry vs orbits of automorphism group of a graph • Boltzmann sampler: composition operation, cycle-pointing Uniform sampling Asymptotic number Bodirsky, K. 06 Outerplanar graphs Cubic planar graphs Bodirsky, Groepl, K. 04+ 2 − con planar graphs Bodirsky, Groepl, K. 05 Planar graphs

Unlabeled planar structures Difficulty with unlabeled planar structures is symmetry: • recursive method: decomposition along symmetry • Pólya theory: symmetry vs orbits of automorphism group of a graph • Boltzmann sampler: composition operation, cycle-pointing Uniform sampling Asymptotic number cn^{ − 5/2}7.5^n Bodirsky, K. 06 Outerplanar graphs Bodirsky, Fusy, K., Vigerske 07+ Cubic planar graphs Bodirsky, Groepl, K. 04+ 2 − con planar graphs Bodirsky, Groepl, K. 05 Planar graphs

Unlabeled planar structures Difficulty with unlabeled planar structures is symmetry: • recursive method: decomposition along symmetry • Pólya theory: symmetry vs orbits of automorphism group of a graph • Boltzmann sampler: composition operation, cycle-pointing Uniform sampling Asymptotic number cn^{ − 5/2}7.5^n Bodirsky, K. 06 Outerplanar graphs Bodirsky, Fusy, K., Vigerske 07 Bodirsky, Fusy, K., Vigerske 07+ Cubic planar graphs Bodirsky, Groepl, K. 04+ 2 − con planar graphs Bodirsky, Groepl, K. 05 Planar graphs

Unlabeled planar structures Difficulty with unlabeled planar structures is symmetry: • recursive method: decomposition along symmetry • Pólya theory: symmetry vs orbits of automorphism group of a graph • Boltzmann sampler: composition operation, cycle-pointing Uniform sampling Asymptotic number cn^{ − 5/2}7.5^n Bodirsky, K. 06 Outerplanar graphs Bodirsky, Fusy, K., Vigerske 07 Bodirsky, Fusy, K., Vigerske 07+ ? Cubic planar graphs Bodirsky, Groepl, K. 04+ 2 − con planar graphs ? Bodirsky, Groepl, K. 05 ? ? Planar graphs

Outline • Decomposition • Recursive method • Singularity analysis • Probabilistic analysis

Outline • Decomposition • Recursive method • Singularity analysis • Probabilistic analysis • Gaussian matrix integral

Gaussian matrix integral [ W ICK 50 ] Let M = ( M ij ) be an N × N Hermitian matrix (i.e., M ij = M ji ) and dM = � � i dM ii i<j d Re( M ij ) d Im( M ij ) the standard Haar measure.

Gaussian matrix integral [ W ICK 50 ] Let M = ( M ij ) be an N × N Hermitian matrix (i.e., M ij = M ji ) and dM = � � i dM ii i<j d Re( M ij ) d Im( M ij ) the standard Haar measure. The Gaussian matrix integral is defined by f ( M ) e − N Tr( M 2 2 ) dM � < f > = , e − N Tr( M 2 2 ) dM � where the integration is over N × N Hermitian matrices.

Gaussian matrix integral [ W ICK 50 ] Let M = ( M ij ) be an N × N Hermitian matrix (i.e., M ij = M ji ) and dM = � � i dM ii i<j d Re( M ij ) d Im( M ij ) the standard Haar measure. The Gaussian matrix integral is defined by f ( M ) e − N Tr( M 2 2 ) dM � < f > = , e − N Tr( M 2 2 ) dM � where the integration is over N × N Hermitian matrices. Using the source integral < e Tr( MS ) > , we obtain Tr( S 2) S =0 = δ il δ jk ∂ ∂ ∂ ∂ < e Tr( MS ) > ˛ ˛ < M ij M kl > = S =0 = e . ˛ 2 N ˛ ∂S ji ∂S lk ˛ ∂S ji ∂S lk ˛ N

Pictorial interpretation [ B RÉZIN , I TZYKSON , P ARISI , Z UBER 78; Z VONKIN 97; DI F RANCESCO 04] δ il δ jk Pictorial interpretation from : < M ij M kl > = N i M ij j i l, l = i 1 < M ij M kl > = j k, k = j N

Enumeration and uniform sampling of planar structures Mihyun Kang - PowerPoint PPT Presentation

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