7. Applications of Singularity Analysis http://ac.cs.princeton.edu - PowerPoint PPT Presentation

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 7. Applications of Singularity Analysis http://ac.cs.princeton.edu

⬅ Analytic combinatorics overview specification A. SYMBOLIC METHOD 1. OGFs 2. EGFs GF equation 3. MGFs SYMBOLIC METHOD B. COMPLEX ASYMPTOTICS asymptotic 4. Rational & Meromorphic estimate 5. Applications of R&M COMPLEX ASYMPTOTICS 6. Singularity Analysis desired 7. Applications of SA result ! 8. Saddle point 2

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 7. Applications of Singularity Analysis Analytic •Simple varieties of trees Combinatorics •Labelled sets •Mappings Philippe Flajolet and Robert Sedgewick OF •Tree-like classes CAMBRIDGE http://ac.cs.princeton.edu II.7a.SAapps.Sets

Transfer theorem for invertible tree classes [from Lecture 6] Theorem. If a simple variety of trees F = Z [ × or ★ ] SEQ Φ ( F ) applications applications � ( � ) = � φ ( � ( � )) is -invertible where the GF satisfies and λ φ ( λ ) = λφ � ( λ ) is the positive real root of then general trees � φ � ( λ ) � � � � / � [ � � ] � ( � ) ∼ � � φ �� ( λ ) / φ ( λ ) � binary trees � ( � ) ∼ λ − � � φ ( λ ) / φ �� ( λ ) � � − � φ � ( λ ) and unary-binary trees Important note: Singularity analysis gives both • Coefficient asymptotics. • Asymptotic estimate of GF near dominant singularity. Cayley trees [and many [and many, many more...] 4

Example 1: Rooted ordered trees Q. How many trees with N nodes? G 1 = 1 G 2 = 1 G 3 = 2 G 4 = 5 G 5 =14 5

Example 1: Rooted ordered trees G , the class of rooted ordered trees Speci fi cation G = Z × SEQ( G )) Symbolic transfer � � ( � ) = GF equation � − � ( � ) simple variety � Analytic transfer of trees φ ( � ) = λ = � / � � − � � λ φ ( λ ) = � � � − λ = ( � − λ ) � φ � ( � ) = ( � − � ) � φ � ( λ ) = � � φ �� ( λ ) = �� � φ �� ( � ) = � √ � � � � � / � ( � − � ) � � � ∼ Asymptotics 6

Example 2: Binary trees How many binary trees with N nodes? T 1 = 1 T 2 = 2 T 3 = 5 T 4 = 14 7

Example 2: Binary trees B , the class of binary trees Speci fi cation B = ● × ( E + B ) × ( E + B ) Symbolic transfer B = ● + ● × SEQ 0,2 ( B ) Expecting ? Stay tuned. � ( � ) = � ( � + � ( � )) � GF equation simple variety Analytic transfer of trees λ = � φ ( � ) = ( � + � ) � φ ( λ ) = � ( � + λ ) � = � λ ( � + λ ) φ � ( � ) = � ( � + � ) φ � ( λ ) = � φ �� ( � ) = � � φ �� ( λ ) = � [ � � ] � ( � ) ∼ √ � � � � � / � Asymptotics 8

Example 3: Unary-binary trees Q. How many unary-binary trees with N nodes? M 1 = 1 M 2 = 1 M 3 = 2 degrees of all nodes 0, 1, or 2 M 4 = 4 M 5 =9 9

Example 3: Unary-binary trees M , the class of all unary-binary trees Speci fi cation M = Z × SEQ 0,1,2 ( M ) Symbolic transfer � ( � ) = � ( � + � ( � ) + � ( � ) � ) GF equation simple variety Analytic transfer of trees λ = � φ ( � ) = � + � + � � � + λ + λ � = λ + � λ φ ( λ ) = � φ � ( � ) = � + � � φ � ( λ ) = � φ �� ( � ) = � φ �� ( λ ) = � � Asymptotics � � � − � / � � � ∼ � � / � � 10

Example 4: Cayley trees Q. How many different labelled rooted unordered trees of size N ? 1 1 2 2 3 3 24 ways to label 2 3 1 3 1 2 1 2 3 2 3 1 2 1 1 12 ways to label 2 1 1 2 3 T 1 = 1 T 2 = 2 2 3 1 3 1 2 24 ways to label T 3 = 9 1 way to label 4 ways to label 2 ways 6 ways to label to label 3 ways to label T 4 = 64 A. N N − 1 . (See EGF lecture.) 11

Example 4: Cayley trees (exact, from EGF lecture) Example 2 5 7 Class C , the class of labelled rooted unordered trees 6 6 2 1 1 2 2 5 1 � | � | � � � � � ( � ) = � � 1 | � | ! ≡ EGF 8 � ! � ∈ C � ≥ � 4 3 � = � ⋆ ( ��� ( � )) Construction "a tree is a root connected to a set of trees" � ( � ) = �� � ( � ) EGF equation � [ � � ] � ( � ) = � Extract coefficients � � � [ � � − � ] � by Lagrange inversion � / � � with f ( u ) = u / e u � [ � � − � ] ��� = � � − � = � � ! � � = � ![ � � ] � ( � ) = � � − � ✓ 12

Example 4: Cayley trees C , the class of all labelled rooted 10 13 unordered trees 12 9 7 Speci fi cation 3 17 15 C = Z ★ SET ( C ) 18 6 1 8 2 5 14 Symbolic transfer 16 4 11 � ( � ) = �� � ( � ) GF equation simple variety Analytic transfer of trees λ = � φ ( � ) = � � φ ( λ ) = � φ � ( � ) = � � � λ = λ � λ φ � ( λ ) = � � φ �� ( � ) = � � [ � � ] � ( � ) = � � � − � / � φ �� ( λ ) = � Asymptotics � � √ 13

Aside: Stirling’s formula via Cayley tree enumeration Exact, via Lagrange inversion Approximate, via singularity analysis � � � � − � ∼ � ! √ � �� � � � √ � � � ! ∼ � �� Theorem. Stirling's formula � 14

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 7. Applications of Singularity Analysis Analytic •Simple varieties of trees Combinatorics •Labelled sets •Mappings Philippe Flajolet and Robert Sedgewick OF •Tree-like classes CAMBRIDGE http://ac.cs.princeton.edu II.7a.SAapps.Sets

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 7. Applications of Singularity Analysis Analytic •Simple varieties of trees Combinatorics •Labelled sets •Mappings Philippe Flajolet and Robert Sedgewick OF •Tree-like classes CAMBRIDGE http://ac.cs.princeton.edu II.7a.SAapps.Sets

Transfer theorem for exp-log labelled set classes [from Lecture 6] Theorem. Asymptotics of exp-log labelled sets . Suppose that a labelled set class F = SET Φ ( G ) is exp-log( α , β , ρ ) � � � α � ( � ) ∼ α log � ( � ) ∼ � β � � − � / ρ + β with . Then � − � / ρ � � and � β � � [ � � ] � ( � ) ∼ � � − α Γ ( α ) ρ Corollary. The expected number of G -components in a random F -object of size N is ~ α ln N. and is concentrated there 17

Example 5: Cycles in permutations Q. How many permutations of N elements? Q. How many cycles in a random permutation of N elements? P 1 = 1 avg. # cycles: 1 P 2 = 2 avg. # cycles: 1.5 P 3 = 6 avg. # cycles: 1.8333 P 4 =24 18 avg. # cycles: 2.08333

Example 5: Cycles in permutations P , the class of all permutations Speci fi cation P = SET(CYC( Z )) Symbolic transfer � � ( � ) = exp(ln � − � ) GF equation exp-log � � � − � = α log � − � / ρ + β ln Analytic transfer ��� α = � , β = � , ��� ρ = � [ � � ] � ( � ) ∼ � # permutations: ∼ � ! Asymptotics ∼ ln � avg # cycles: 19

Example 6: Cycles in derangements Q. How many derangements of N elements? Q. How many cycles in a random derangement of N elements? D 1 = 0 avg. # cycles: 0 D 2 = 1 avg. # cycles: 1 D 3 = 2 avg. # cycles: 1 D 4 =9 20 avg. # cycles: 1.33333

Example 6: Cycles in derangements D , the class of all derangements Speci fi cation D = SET(CYC >0 ( Z )) Symbolic transfer � � ( � ) = exp(ln � − � − � ) GF equation � � exp-log � − � − � = α log � − � / ρ + β ln Analytic transfer ��� α = � , β = − � , ��� ρ = � [ � � ] � ( � ) ∼ � − � # derangements: ∼ � ! / � Asymptotics ∼ ln � avg # cycles: 21

Example 6: Cycles in generalized derangements D , the class of all permutations having no cycles of length w 1 , w 2 , ... w t Speci fi cation D = SET(CYC ≠ w i ( Z )) Symbolic transfer � − � − � � � − . . . − � � � � � ( � ) = exp(ln ) GF equation � � � � � � � − � − � = α log � − � / ρ + β ln ��� α = � , β = − � − . . . − � exp-log � � � � Analytic transfer [ � � ] � ( � ) = exp( − � − . . . − � ��� ρ = � ) � � � � Asymptotics ∼ � ! / � � / � � + ... + � / � � # derangements: ∼ ln � avg # cycles: 22

Example 7: 2-regular graphs undirected graphs with Q. How many labelled 2-regular graphs of N elements? all nodes degree 2 Q. How many components in a random 2-regular graph of N elements? 1 way to label 3 ways to label 12 ways to label R 3 = 1 R 4 = 3 R 5 = 12 60 ways to label 360 ways to label 1-2 1 3 1-3 2-4 4 2 3-4 1-2 1-2 1 1 2 1-4 1-3 2-3 2-3 10 ways to label 2 3 105 ways to label 3 4 3-4 R 6 = 70 R 7 = 465 1-3 1 3 1-4 avg. # components: avg. # components: 2-3 2 4 2-4 (1 ⋅ 60 + 2 ⋅ 10)/70 ≐ 1.143 (1 ⋅ 360 + 2 ⋅ 105)/465 ≐ 1.226 23