Singularity Analysis: A Perspective Philippe Flajolet ( Inria , - PowerPoint PPT Presentation

MSRI, Berkeley, June 2004 Singularity Analysis: A Perspective Philippe Flajolet ( Inria , France)

Analysis of Algorithms ↓ Average-Case, Probabilistic ↓ Properties of Random Structures? • Counting and asymptotics n ! ∼ n n e − n √ 2 πn Z x 1 e − t 2 / 2 dt. D • Asymptotic laws Ω n → √ (e.g., Monkey and typewriter!) 2 π −∞ — Probabilistic, stochastic — Analytic Combinatorics: Generating Functions

1. Introduction “Symbolic” Methods Rota-Stanley; Foata-Schutzenberger; Joyal and uqam group; Jackson-Goulden, &c; F.; ca 1980 ± . F-Salvy-Zimmermann 1991 ❀ Computer Algebra . Basic combinatorial constructions admit of direct translations as operators over generating functions (GF’s) .

C : class of comb. structures; C n : # objects of size n ↓ ↓ ↓  �  C n z n C ( z ) := (counting) � z n  � C ( z ) := C n n !  � C n,k z n u k  C ( z, u ) := � C n,k u k z n (params)  � C ( z, u ) := n ! Ordinary GF’s for unlabelled structures. Exponential GF’s for labelled structures.

“Dictionaries” = Constructions viewed as Operators over GF’s. Constr. Operations Union + + × × Product (1 − f ) − 1 (1 − f ) − 1 S equence e f M ultiSet P´ olya Exp. log(1 − f ) − 1 C ycle P´ olya Log. (unlab.) (lab.) f ( z ) + 1 2 f ( z 2 ) + · · · ` ´ Exp( f ) := exp 1 Log( f ) := log 1 − f ( z ) + · · · Books: Goulden-Jackson, Bergeron-LL, Stanley, F-Sedgewick ⇒ How to extract coeff., especially, asymptotically? = ?

“Complex–analytic Structures” Interpret: ♥ Counting GF as analytic transformation of C ; ♥ Comb. Construction as analytic functional. Singularities are crucial to asymptotic prop’s! (cf. analytic number theory, complex analysis, etc) Asymptotic counting via Singularity Analysis (S.A.) Asymptotic laws via Perturbation + S.A.

1 1 dz Z f ( z ) = (1 − z − z 2) − 1. ℑ f ( z ) , 1 − z − z 2 zn +1 2 iπ Refs: F–Odlyzko, SIAM A&DM, 1990 ≪ FO82 on tree height; Odlyzko’s 1995 survey in Handbook of Combinatorics + Banderier, Fill, J. Gao, Gonnet, Gourdon, Kapur, G. Labelle, Laforest, T. Lafforgue, Noy, Odlyzko, Panario, Poblete, Pouyanne, Prodinger, Puech, Richmond, Robson, Salvy, Schaeffer, Sipala, Soria, Steyaert, Szpankowski, B. Vall´ ee, Viola .

♠ Location of singularity at z = ρ : coeff. [ z n ] f ( z ) = ρ − n · coeff. [ z n ] f ( ρz ) ♠ Nature of singularity at z = 1: 1 − → n + 1 ∼ n (1 − z ) 2 1 1 H n ≡ 1 1 + ... + 1 1 − z log − → ∼ log n n 1 − z 1 − → 1 ∼ 1 1 − z ! 1 1 2 n 1 √ 1 − z − → ∼ √ πn 2 2 n n 8 > > ρ − n < Location of sing’s : Exponential factor > > : Nature of sing’s : “Polynomial” factor ϑ ( n )

Generating Function ❀ Coefficients Solving a “Tauberian” problem R eal–Tauberian Darboux-P´ olya Singularity An. 0 1 (large = ⇒ large) (smooth = ⇒ small) (Full mappings) Combinatorial constructions ❀ Analytic Functionals ⇒ Analytic continuation prevails for comb. GF’s =

2. Basic Singularity Analysis Theorem 1. Basic scale translates: σ α,β ( z ) := (1 − z ) − α � � β 1 1 z log 1 − z n α − 1 [ z n ] σ α,β Γ( α ) (log n ) β . = ⇒ ∼ n →∞ Proof . Cauchy’s coefficient integral, f ( z ) = (1 − z ) − α Z 1 dz [ z n ] f ( z ) = f ( z ) z n +1 2 iπ γ ( z = 1 + t ↓ ↓ ↓ ↓ n ) „ « − α Z 1 − t e − t dt 2 iπ n n H n α − 1 × 1 Γ( α ) .

“Camembert” Theorem 2. O –transfers: Under continuation in a ∆ -domain, [ z n ] f ( z ) = O ([ z n ] σ α,β ( z )) . ⇒ f ( z ) = O ( σ α,β ( z )) = Proof :

  f ( z ) = λσ ( z ) + µτ ( z ) + ... + O ( ω ( z ))   Usage: ⇒ =    f n = λσ n + µτ n + ... + O ( ω n ) . Similarly: o -transfer. • Dominant singularity at ρ gives factor ρ − n . • Finitely many singularities work fine

Example 1 . 2-regular graphs [Comtet] (Originally by Darboux-P´ olya.) „ 1 « G = M 2 C ≥ 3 ( Z ) „ 1 « 2 − z 2 1 − z − z 1 b G ( z ) = exp 2 log 4 e − 3 / 4 b G ( z ) ∼ √ 1 − z z → 1 e − 3 / 4 G n ∼ √ πn . n ! n →∞ ✷ > equivalent(exp(-z/2-z^2/4)/sqrt(1-z),z,n,4); # By SALVY 1/2 3/2 5/2 exp(-3/4) (1/n) exp(-3/4) (1/n) exp(-3/4) (1/n) ------------------ - 5/8 ------------------ + 1/128 ------------------ 1/2 1/2 1/2 Pi Pi Pi

Example 2 . Richness index of trees [F-Sipala-Steyaert,90] = Number of different terminal subtrees. Catalan case: ! “p ” X √ K ( z ) = 1 1 2 k 1 − 4 z − 4 z k +1 − 1 − 4 z 2 z k + 1 k k ≥ 0 1 ≈ √ Z log Z , Z := 1 − 4 z K ( z ) z → 1 / 4 r n 8 log 2 √ log n, Mean index n →∞ C ∼ C ≡ . π = Compact tree representations as dag s = Common Subexpression Pb. ✷

Extensions ♥ Slowly varying = ⇒ slowly varying: Log-log = ⇒ Log-Log, . . . ♥ Full asymptotic expansions ♥ Uniformity of coefficient extraction [ z n ] { F u ( z ) } u ∈ Ω = ❀ later!. ♥ Some cases with natural boundary [Fl-Gourdon-Panario-Pouyanne] Example 3 . Distinct Degree Factorization [DDF] in Polynomial Fact ❀ Greene–Knuth: „ « ∞ Y 1 + z k [ z n ] . k k =1 Hybrid w/ Darboux: e − γ + e − γ + · · · + ⋆ ( − 1) n + ⋆ ω n ✷ n 3 + · · · n 3 n Cf. Hardy-Ramanujan’s partition analysis “without contrast”.

3. Closure Properties Function of S.A.–type = amenable to singularity analysis • is continuable in a ∆-domain, • admits singular expansion in scale { σ α,β } . Theorem 3. Generalized polylogarithms � (log n ) k n − α z n Li α,k := are of S.A.-type. Proof . Cauchy-Lindel¨ of representations Z 1 / 2+ i ∞ X ϕ ( n )( − z ) n = − 1 π ϕ ( s ) z s sin πs ds. 2 iπ 1 / 2 − i ∞ + Mellin transform techniques (Ford, Wong, F.).

Example 4 . Entropy of Bernoulli distribution X ` n H n := − p k (1 − p ) n − k π n,k log π n,k , ´ π n,k ≡ k X k log( k !) z k = (1 − z ) − 1 Li 0 , 1 ( z ) involves p 1 2 log n + 1 2 πp (1 − p ) + · · · . 2 + log Redundancy, coding, information th.; Jacquet-Szpankowski via Analytic ✷ dePoissonization. • Elements like log n, √ n in combinatorial sums

Theorem 4. Functions of S.A.-type are closed under integration and differentiation. Proof . Adapt from Olver, Henrici, etc. Theorem 5. Functions of S.A.-type are closed under Hadamard product � ( f n g n ) z n . f ( z ) ⊙ g ( z ) := n Proof . Start from Hadamard’s formula Z ” dt “ w 1 f ( z ) ⊙ g ( z ) = f ( t ) g t . 2 iπ t γ + adapt Hankel contours [H., Jungen, R. Wilson ❀ Fill-F-Kapur]

Example 5 . Divide-and -conquer recurrences X f n = t n + π n,k ( f k + f n − k ) Sing( f ( z )) = Φ(Sing( t ( z ))) Asympt[ f n ] = Ψ(Sing( t )) . E.g., Catalan statistics: need P ` 2 n ´ log n · z n . n Useful in random tree applications [Fill-F-Kapur, 2004 + , Fill-Kapur] // Neininger-Hwang et al. ≪ Knuth-Pittel. Moments ↔ contraction method ✷ [R¨ osler-R¨ uschendorf-Neininger] K * n ? n−K

4. Functional Equations • Rational functions. Linear system Q ≥ 0 [ z ] implies polar singularities: X [ z n ] f ( z ) ≈ ω n n k , ω ∈ Q , k ∈ Z ≥ 0 . + irreducibility: Perron-Frobenius = ⇒ simple dom. pole . • Word problems from regular language models; • Transfer matrices [Bender-Richmond]: dimer in strip, knights, etc. ❀ Vall´ ee’s generalization to dynamical sources via transfer operators. • Algebraic functions, by Puiseux expansions ( Z p/q ) ≪ S.A. or Darboux! X X [ z n ] f ( z ) ≈ ω n n p/q , ω ∈ Q , p/q ∈ Q , Asymptotics of coeff. is decidable [Chabaud-F-Salvy]. • Word problems from context-free models ; • Trees ; Geom. configurations (non-crossing graphs, polygonal triangs.); Planar Maps [Tutte...]; Walks [Banderier Bousquet-M., Schaeffer], . . .

(1 − √ 1 − 4 z ) / (2 z ) Square-root singularity is “ universal ” for many recursive classes = controlled “failure” of Implicit Function Theorem Z ∝ Y 2 Entails coeff. asymptotic ≈ ω n n − 3 / 2 with critical exponent − 3 / 2. E.g., unbalanced 2–3 trees (Meir-Moon): f = zφ ( f ) , φ ( u ) = 1 + u 2 + u 3 . P´ olya’s combinatorial chemistry programme: f ( z ) = z Exp( f ( z )) ≡ ze f ( z )+ 1 2 f ( z 2 )+ 1 3 f ( z 3 )+ ··· Starting with P´ olya 1937; Otter 1949; Harary-Robinson et al. 1970’s; Meir-Moon 1978; Bender/Meir-Moon; Drmota-Lalley-Woods thm. 1990 +

• “Holonomic” functions. Defined as solutions of linear ODE’s with coeffs in C ( z ) [Zeilberger] ≡ D -finite. L [ f ( z )] = 0 , L ∈ C ( z )[ ∂ z ] . • Stanley, Zeilberger, Gessel: Young tableaux and permutation statistics; regular graphs, constrained matrices, etc. Fuchsian case (or “regular” singularity) ( Z β log k Z ): � [ z n ] f ( z ) ≈ ω n n β (log n ) k , ω, β ∈ Q , k ∈ Z ≥ 0 . S.A. applies automatically to classical classification. Asymptotics of coeff is decidable — general case: modulo oracle for connection problem; — strictly positive case: “usually” OKay.

QTrees: Example 6 . Quadtrees—Partial Match [FGPR’92] Divide-and-conquer recurrence with coeff. in Q ( n ) Fuchsian equation of order d (dimension) for GF √ Q ( d =2) ≍ n ( 17 − 3) / 2 . n ✷ E.g., d = 2: Hypergeom 2 F 1 with algebraic arguments. Extended by Hwang et al. Cf also Hwang’s Cauchy ODE cases. Panholzer-Prodinger+Martinez, . . .