Matthieu Dien O. Bodini, X. Fontaine, A. Genitrini, H.-K. Hwang - PowerPoint PPT Presentation

Matthieu Dien O. Bodini, X. Fontaine, A. Genitrini, H.-K. Hwang Université Pierre et Marie Curie Ą Laboratoire LIP6 Ą Équipe APR Mardi 8 Mars 2016 1/13

Outline ‚ Introduction ‚ Asymptotic study of the diamonds ‚ Random Generation ‚ Conclusion 2/13

Introduction Motivations ‚ Combinatorial study of concurrents programs (seen as discrete structures) ‚ Quantitative study of the combinatorial explosion phenomena: the large number of possible runs (seen as increasing labellings) Approach: Analytic Combinatorics ‚ symbolic method to modelize (Greene’s “box” operators) ‚ singularity analysis to obtain asymptotics of the number of increasing labellings ‚ based on previous work on increasing trees of [F. Bergeron, P. Flajolet and B. Salvy ’92] 3/13

Combinatorial specifications Skeleton ‚ S “ Z ` Z ¨ G p S q ¨ Z ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ 4/13

Combinatorial specifications Skeleton 1 S “ Z ` Z ¨ G p S q ¨ Z 2 5 4 Increasing labellings 3 6 8 9 7 11 I “ Z ` Z ˝ ‹ G p I q ‹ Z ‚ 12 10 13 14 4/13

Combinatorial specifications Skeleton 1 S “ Z ` Z ¨ G p S q ¨ Z 2 5 4 Increasing labellings 3 6 8 9 7 11 I “ Z ` Z ˝ ‹ G p I q ‹ Z ‚ 12 10 Differential equation 13 I 2 “ G p I q $ & I p 0 q “ 0 14 I 1 p 0 q “ 1 % 4/13

Easy case: non-plane diamonds We start with the differential equation: A 2 p z q “ e A p z q We can solve it: A 1 p z q “ tan z ` sec z The poles are the p 2 k ` 1 2 q π Using the residue theorem we get: a n “ 2 n ` 1 p n ´ 1 q ! `8 1 ÿ p 1 ` 4 j q n . π n j “´8 5/13

Easy case: non-plane diamonds We start with the differential equation: A 2 p z q “ e A p z q We can solve it: A 1 p z q “ tan z ` sec z The poles are the p 2 k ` 1 2 q π Using the residue theorem we get: a n “ 2 n ` 1 p n ´ 1 q ! `8 1 ÿ p 1 ` 4 j q n . π n j “´8 p a n q n ě 1 “ t 1 , 1 , 1 , 2 , 5 , 16 , 61 , 272 , 1385 , 7936 , 50521 , 353792 , . . . u Known in OEIS to count the number of number of increasing unary-binary trees on n vertices. 5/13

Bijection Increasing unary-binary trees Non-plane diamonds T “ Z ` Z ˝ ‹ p T ` Set “ 2 p T qq A “ Z ` Z ˝ ‹ Set p A q ‹ Z ‚ T 2 p z q “ p 1 ` T p z qq ¨ T 1 p z q N 3 p z q “ N 1 p z q ¨ N 2 p z q 1 1 2 2 N 2 “ e N N 1 T 1 1 ` T n Thanks to A. Bacher, G. Collet and C. Mailler (and ALEA Network) 6/13

Elliptic cases Weierstrass’s case F 2 “ P p F q where P is a polynomial of degree 2, then: F p z q “ K ℘ p z ´ ρ ; ω 1 , ω 2 q ż 8 d t with ρ “ and K a constant. b ş t 0 1 ` 2 0 P p v q d v Weierstrass’s elliptic function ℘ is defined periodically over a lattice that contains one double pole in a corner of each cell: ℘ p z ; ω 1 , ω 2 q “ 1 ˆ 1 1 ˙ ÿ z 2 ` p z ` k ω 1 ` l ω 2 q 2 ´ p k ω 1 ` l ω 2 q 2 p k , l qP Z 2 ztp 0 , 0 qu 7/13

Elliptic cases Jacobi’s case F 2 “ P p F q where P is a polynomial of degree 3, then ? let g 2 “ β ´ δ α ´ δ ¨ F ´ 2 α ? 2 β with α , β and δ well chosen then F ´ g 1 p z q “ M p 1 ´ z 2 qp 1 ´ ℓ 2 z 2 q and so a g p z q “ sn p Mz ; ℓ q Jacobi’s elliptic sinus function sn is defined periodically over a lattice that contains two simple poles in each cell and a zero in a corner. 8/13

Elliptic cases: binary and ternary diamonds Weierstrass case: binary diamonds B “ Z ` Z ˝ ‹ p E ` B ‹ B q ‹ Z ‚ B 2 “ 1 ` B 2 b n “ 6 p n ` 1 q ! 1 n Ñ8 6 p n ` 1 q ! ÿ „ ρ n ` 2 ¯ n ` 2 ρ n ` 2 ´ 1 ` k ω 1 ρ ` l ω 2 p k , l qP Z 2 ρ Jacobi’s case: ternary diamonds T “ Z ˝ ‹ p E ` T ‹ T ‹ T q ‹ Z ‚ T 2 “ 1 ` T 3 ? ? 2 n ! 1 1 2 p n ` 1 q ! ÿ t n “ ˘ n ` 1 ´ n Ñ8 6 „ ρ n ` 1 ˘ n ` 1 ρ n ` 1 ` ` 1 ` C k , l 2 ` C k , l p k , l qP Z 2 ? with C k , l “ 3 k 3 2 ` i 2 p k ` 2 l q 9/13

More general cases Asymptotics results ‚ Diamonds of fixed arity ( G P Z r X s and deg p G q “ m ): 2 m ´ 1 n ´ m ´ 3 ˜ a ¸ 2 p m ` 1 q m ´ 1 2 ´ ´ 4 ¯¯ m ´ 1 q ρ ´ n ´ n ´ p m ´ 1 q? b m f n “ n ! 1 ` O m ´ 1 m ´ 1 2 Γ p ‚ Plane general diamonds ( G “ Seq): ˜ ÿ ˙¸ n ! ρ 1 ´ n ˆ p log log n q K P k p log log n q f n “ ` O p log n q K n 2 a p log n q k 2 log n 0 ď k ă K Sequence A032035 in OEIS which also enumerates increasing rooted (2,3)-cacti with n ´ 1 nodes 10/13

Random Generation of the skeletons Boltzmann method ‚ Straightforward use of standard techniques [P. Duchon, P. Flajolet, G. Louchard & G. Schaeffer ’04] ‚ a bit of tricks to draw an object from F from Γ F 2 [O. Bodini, O. Roussel & M. Soria ’12] and [O. Bodini ’10] ñ Boltzmann generator using only uniform random variable to draw object such that F 2 “ φ p F q 11/13

Random Generation of the increasing labellings • • • “ • | B B B • • • 12/13

Random Generation of the increasing labellings diamond ñ increasing labelling ‚ ñ return p 1 q 12/13

Random Generation of the increasing labellings diamond ñ increasing labelling • • • x : “ draw_inc_lbl p B 1 q y : “ draw_inc_lbl p B 2 q B 1 B 2 ñ t : “ shuffle p x , y q | t | “ | x | ` | y | • • return p 1 , t ` 1 , | t | ` 1 q • 12/13

Random Generation of the increasing labellings diamond ñ increasing labelling • • • x : “ draw_inc_lbl p B 1 q y : “ draw_inc_lbl p B 2 q B 1 B 2 ñ t : “ shuffle p x , y q | t | “ | x | ` | y | • • return p 1 , t ` 1 , | t | ` 1 q • Average complexity ‚ The average complexity of draw_inc_lbl in memory writings is O p n ? n q ‚ The average number of random bits needed during the generation is O p n 3 { 2 log n q 12/13

Current work ‚ study of the average of some parameters (width, depth, root’s degree ...) of the increasingly labelled structures ‚ study of a bit more realistic model, from a concurrency point of view: • • • • • FJ “ • | | FJ FJ FJ • • • • FJ • ‚ more efficient algorithms for the random generation of increasing labellings Open question ‚ for the elliptic cases, how to do for showing the periodicity of the solutions directly from the differential equation ? ‚ is this periodic behaviour still present for higher degree of polynomial ? 13/13