Pointing, asymptotics, and random generation in unlabelled classes ´ Eric Fusy LIX, ´ Ecole Polytechnique (Paris) Joint work with Manuel Bodirsky, Mihyun Kang and Stefan Vigerske . – p.1/37
Motivations Automatic methods for • Enumeration (exact/asymptotic) • Random generation (cf [Flajolet,F,Pivoteau’07]) in the unlabelled setting. . – p.2/37
Motivations Automatic methods for • Enumeration (exact/asymptotic) • Random generation (cf [Flajolet,F,Pivoteau’07]) in the unlabelled setting. References: • Short version in SODA’07 • Long version almost finished, written in the framework of “Combinatorial species” , cf [Bergeron, Labelle, Leroux’98] . – p.2/37
Labelled/Unlabelled structures • labelled class C = ∪ n C n 3 Labeled graph of size 5 1 5 4 2 � 1 n ! c n x n , with c n = | C n | EGF : C ( x ) = n • Unlabelled class � C = ∪ n � C n Unlabeled tree of size 7 � c n = | � c n x n , with � OGF : C ( x ) = � C n | n . – p.3/37
Plan • Decomposition strategy for labelled structures • Pointing + recursive decomp. + gen. functions • Examples: trees, planar graphs... . – p.4/37
Plan • Decomposition strategy for labelled structures • Pointing + recursive decomp. + gen. functions • Examples: trees, planar graphs... • We adapt the method to the unlabelled setting • Difficulties due to symmetries • Solution: unbiased pointing + P´ olya theory . – p.4/37
Plan • Decomposition strategy for labelled structures • Pointing + recursive decomp. + gen. functions • Examples: trees, planar graphs... • We adapt the method to the unlabelled setting • Difficulties due to symmetries • Solution: unbiased pointing + P´ olya theory • Application to asymptotic enumeration . – p.4/37
Plan • Decomposition strategy for labelled structures • Pointing + recursive decomp. + gen. functions • Examples: trees, planar graphs... • We adapt the method to the unlabelled setting • Difficulties due to symmetries • Solution: unbiased pointing + P´ olya theory • Application to asymptotic enumeration • Application to random generation: ⇒ Boltzmann samplers without rejection . – p.4/37
Decomposition strategy for labelled structures . – p.5/37
Dictionary for EGF • labelled class C = ∪ n C n � 1 n ! c n x n , with c n = | C n | EGF : C ( x ) = n • Simple computation rule for each construction: Disjoint union C = A + B C ( x ) = A ( x ) + B ( x ) Cartesion product C = A × B C ( x ) = A ( x ) · B ( x ) Set C = Set( A ) C ( x ) = exp( A ( x )) Substitution C = A ◦ B C ( x ) = A ( B ( x )) . – p.6/37
Dictionary for EGF • labelled class C = ∪ n C n � 1 n ! c n x n , with c n = | C n | EGF : C ( x ) = n • Simple computation rule for each construction: Disjoint union C = A + B C ( x ) = A ( x ) + B ( x ) Cartesion product C = A × B C ( x ) = A ( x ) · B ( x ) Set C = Set( A ) C ( x ) = exp( A ( x )) Substitution C = A ◦ B C ( x ) = A ( B ( x )) • Remark. Substitution rule implies Set rule since the EGF of the class Set is exp( z ) (same for cycle, set, unoriented sequence, etc...) . – p.6/37
Decomposition strategy for trees • Goal: find t n the number of (unrooted) trees of size n • Important tool: pointing: A �→ A • Let r n be the number of rooted trees of size n 1 1 1 1 1 3 3 3 3 3 2 2 2 2 2 4 4 4 4 4 r n = n · t n ⇒ Counting trees reduces to counting rooted trees. . – p.7/37
Rooted trees are decomposable • The class R of rooted trees satisfies the decomposition 8 8 3 3 1 1 7 7 1 1 1 1 5 1 3 5 1 3 6 6 1 0 1 0 4 9 4 9 1 2 1 2 2 2 R = Z × Set( R ) ⇒ R ( x ) = x exp( R ( x )) . – p.8/37
Rooted trees are decomposable • The class R of rooted trees satisfies the decomposition 8 8 3 3 1 1 7 7 1 1 1 1 5 1 3 5 1 3 6 6 1 0 1 0 4 9 4 9 1 2 1 2 2 2 R = Z × Set( R ) ⇒ R ( x ) = x exp( R ( x )) • Lagrange inversion formula inverse of R ( x ) is R ( − 1) ( y ) = y exp( − y ) ) ⇒ Rooted trees: r n = n n − 1 ⇒ Trees: c n = n n − 2 . – p.8/37
Counting labelled trees: summary • Decomposition of rooted trees R ( x ) = x exp( R ( x )) yields r n = n n − 1 from Lagrange inversion formula • Pointing relation: t n = r n /n : R ( x ) = xT ′ ( x ) yields t n = n n − 2 Same method applies for many classes (planar graphs) . – p.9/37
Adaptation to the unlabelled setting . – p.10/37
Unlabelled setting • Unlabelled structures=labelled structures up to relabeling 2 2 1 2 3 1 ← 2 2 ← 3 ∈ 3 ← 5 5 5 3 4 ← 1 5 ← 4 5 4 4 1 1 . – p.11/37
Unlabelled setting • Unlabelled structures=labelled structures up to relabeling 2 2 1 2 3 1 ← 2 2 ← 3 ∈ 3 ← 5 5 5 3 4 ← 1 5 ← 4 5 4 4 1 1 • Examples: 1 1 1 2 2 3 3 1 1 2 1 2 2 3 3 2 3 3 1 2 3 labeled objects (instead of 3! = 6 ) 3 2 1 2 1 3 2 3 1 3 1 2 1 1 2 2 3 3 6 labeled objects (no symmetry) . – p.11/37
Unlabelled setting • Unlabelled structures=labelled structures up to relabeling 2 2 1 2 3 1 ← 2 2 ← 3 ∈ 3 ← 5 5 5 3 4 ← 1 5 ← 4 5 4 4 1 1 • Examples: 1 1 1 2 2 3 3 1 1 2 1 2 2 3 3 2 3 3 1 2 3 labeled objects (instead of 3! = 6 ) 3 2 1 2 1 3 2 3 1 3 1 2 1 1 2 2 3 3 6 labeled objects (no symmetry) • Unlabelled struct. size n → at most n ! labelled structures. ⇒ (EGF) A ( x ) � � ⇒ 1 n ! a label . ≤ a unlabel . A ( x ) (OGF). n n . – p.11/37
Symmetries Let A be a labelled class, • A symmetry of size n on A is a pair ( σ ∈ S n , A ∈ A n ) such that A is fixed by the action of σ . σ = (1 6 5 8)(4 7 3 9)(2) 4 9 4 9 9 3 1 8 1 8 8 5 2 2 2 6 5 6 5 1 6 7 7 3 3 4 7 (rotation by π/ 2) . – p.12/37
Burnside’s lemma Given A a labelled class (species of structures) let � A = A / isomorphisms , Sym( A ) = { Symmetries from A } • Burnside’s lemma ⇒ Sym( A ) n ≃ n ! × � A n 1 3 2 1 3 2 2 1 3 2 3 1 3 2 3 2 1 1 2 1 2 3 3 1 1 3 1 3 1 2 1 2 2 3 2 3 1 1 1 1 1 1 2 3 2 3 3 2 3 2 3 3 2 2 . – p.13/37
Burnside’s lemma Given A a labelled class (species of structures) let � A = A / isomorphisms , Sym( A ) = { Symmetries from A } • Burnside’s lemma ⇒ Sym( A ) n ≃ n ! × � A n 1 3 2 1 3 2 2 1 3 2 3 1 3 2 3 2 1 1 2 1 2 3 3 1 1 3 1 3 1 2 1 2 2 3 2 3 1 1 1 1 1 1 2 3 2 3 3 2 3 2 3 3 2 2 • Hence EGF of Sym( A ) = � A ( x ) (OGF) . – p.13/37
Cycle index sum Let A be a labelled class, Sym( A ) the symmetry class. • Refined weight for ( σ, A ) ∈ Sym( A ) 1 n ! s c 1 ( σ ) s c 2 ( σ ) · · · s c n ( σ ) W ( σ, A ) := n 1 2 where c i ( σ ) = # { cycles length i in σ } • Cycle index sum of A (cf P´ olya) is the multivariate series � Z A ( s 1 , s 2 , . . . ) = W ( σ,A ) σ · A = A � � 1 s c 1 1 . . . s c n = n # (Fix σ ) n ! n ≥ 1 σ ∈ S n • OGF of � A = EGF of Sym( A ) = Z A ( x, x 2 , x 3 , . . . ) . – p.14/37
Examples of cycle index sums 1 2 1 3 3 2 2 1 3 1 2 3 3 3 2 2 1 1 1 1 6 s 3 6 s 3 6 s 3 1 6 s 3 1 1 6 s 3 1 6 s 3 Z = + + + + + 1 1 1 1 1 1 s 3 = 1 . – p.15/37
Examples of cycle index sums 1 2 1 3 3 2 2 1 3 1 2 3 3 3 2 2 1 1 1 1 6 s 3 6 s 3 1 6 s 3 6 s 3 1 1 6 s 3 1 6 s 3 Z = + + + + + 1 1 1 1 1 1 s 3 = 1 2 2 3 3 1 1 3 1 3 1 2 1 2 1 2 3 2 3 1 6 s 3 1 1 1 1 1 Z = 6 s 3 6 s 3 + + + + + 6 s 1 s 2 6 s 1 s 2 6 s 1 s 2 1 1 1 1 2 s 3 1 + 1 = 2 s 1 s 2 . – p.15/37
Examples of cycle index sums 1 2 1 3 3 2 2 1 3 1 2 3 3 3 2 2 1 1 1 1 6 s 3 6 s 3 1 6 s 3 1 6 s 3 1 6 s 3 1 6 s 3 Z = + + + + + 1 1 1 1 1 1 s 3 = 1 2 2 3 3 1 1 3 1 3 1 2 1 2 1 2 3 2 3 1 6 s 3 1 1 1 1 1 Z = 6 s 3 6 s 3 + + + + + 6 s 1 s 2 6 s 1 s 2 6 s 1 s 2 1 1 1 1 2 s 3 1 + 1 = 2 s 1 s 2 1 1 1 1 1 1 3 3 2 3 2 3 3 2 2 3 2 2 1 6 s 3 1 1 1 Z = 1 1 + + + + + 6 s 3 6 s 1 s 2 6 s 1 s 2 6 s 3 6 s 1 s 2 1 . – p.15/37 1 1 3 =
Dictionary for OGF • Unlabelled class � c n = Card( � C = ∪ n C n / S n � C n ) � � c n x n OGF : C ( x ) = � n ≥ 0 • Dictionary (computation rules): C ( x ) = � � A ( x ) + � Disjoint union C = A + B B ( x ) C ( x ) = � � A ( x ) � Product C = A × B B ( x ) �� � � k � 1 A ( x k ) Set C = Set( A ) C ( x ) = exp k ≥ 1 C ( x ) � = � � A ( � Substitution C = A ◦ B B ( x )) . – p.16/37
Dictionary for OGF • Unlabelled class � c n = Card( � C = ∪ n C n / S n � C n ) � � c n x n OGF : C ( x ) = � n ≥ 0 • Dictionary (computation rules): C ( x ) = � � A ( x ) + � Disjoint union C = A + B B ( x ) C ( x ) = � � A ( x ) � Product C = A × B B ( x ) �� � � k � 1 A ( x k ) Set C = Set( A ) C ( x ) = exp k ≥ 1 C ( x ) = Z A ( � � B ( x ) , � B ( x 2 ) , . . . ) Substitution C = A ◦ B . – p.16/37
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