5. Analytic Combinatorics

Transcription

1 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics

2 Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Features: Analysis begins with formal combinatorial constructions. The generating function is the central object of study. Transfer theorems can immediately provide results from formal descriptions. Results extend, in principle, to any desired precision on the standard scale. Variations on fundamental constructions are easily handled. combinatorial constructions symbolic transfer theorem generating function equation analytic transfer theorem coefficient asymptotics the symbolic method 2

3 Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Ex: How many binary trees with N nodes? T = E + Z T T combinatorial construction () = + () GF equation coefficient asymptotics 3

4 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5a.AC.Symbolic

5 The symbolic method is an approach for translating combinatorial constructions to GF equations Define a class of combinatorial objects. Define a notion of size. Define a GF whose coefficients count objects of the same size. Define operations suitable for constructive definitions of objects. Develop translations from constructions to operations on GFs. Examples A, B, Z b A(z) A B A(z)B(z) Formal basis: A combinatorial class is a set of objects and a size function. An atom is an object of size. An neutral object is an atom of size 0. A combinatorial construction uses the union, product, and sequence operations to define a class in terms of atoms and other classes. Building blocks notation denotes contains Z E Φ atomic class neutral class empty class an atom neutral object nothing 5

6 Unlabelled class example : natural numbers Def. A natural number is a set (or a sequence) of atoms. counting sequence OGF = I = I2 = I3 = I4 = I5 = = unary notation 6

7 Unlabelled class example 2: bitstrings Def. A bitstring is a sequence of 0 or bits. 0 B0 = B = B2 = B3 = counting sequence OGF = = () = B4 = 6 7

8 Unlabelled class example 3: binary trees Def. A binary tree is empty or a sequence of a node and two binary trees counting sequence OGF T = T2 = 2 = + ( ) Catalan numbers (see Lecture 3) () = + () T3 = 5 T4 = 4 8

9 Combinatorial constructions for unlabelled classes construction notation semantics disjoint union A + B disjoint copies of objects from A and B Cartesian product A B ordered pairs of copies of objects, one from A and one from B A and B are combinatorial classes of unlabelled objects sequence SEQ ( A ) sequences of objects from A Ex. ( ) ( ) = Ex 2. SEQ( ) =... Ex 3. = "unlabelled"?? Stay tuned. 9

10 The symbolic method for unlabelled classes (transfer theorem) Theorem. Let A and B be combinatorial classes of unlabelled objects with OGFs A(z) and B(z). Then construction notation semantics OGF disjoint union A + B disjoint copies of objects from A and B ()+() Cartesian product A B ordered pairs of copies of objects, one from A and one from B ()() sequence SEQ ( A ) sequences of objects from A () 0

11 Proofs of transfers are immediate from GF counting A + B = + = ()+() + A B = + = = ()() SEQ ( A ) () ɛ ()+() + () + () +...= ()

12 Symbolic method: binary trees How many binary trees with N nodes? Class T, the class of all binary trees Atoms Size t, the number of internal nodes in t OGF Construction = + OGF equation () = = () = + () type class size GF external node 0 internal node z a binary tree is an external node or an internal node connected to two binary trees or [ ]() = + see Lecture 3 and stay tuned. 2

13 Symbolic method: binary trees How many binary trees with N external nodes? Class T, the class of all binary trees Atoms type class size GF Size OGF OGF equation, t the number of external nodes in t () = t Construction = + () = + () external node z internal node 0 a binary tree is an external node or an internal node connected to two binary trees or () =() [ ] () =[ ]() = same as # binary trees with N internal nodes 3

14 Symbolic method: binary strings Warmup: How many binary strings with N bits? Class B, the class of all binary strings Atoms type class size GF Size b, the number of bits in b 0 bit z OGF () = = bit z Construction = ( + ) OGF equation () = a binary string is a sequence of 0 bits and bits [ ]() = 4

15 Symbolic method: binary strings (alternate) Warmup: How many binary strings with N bits? Class B, the class of all binary strings Atoms type class size GF Size b, the number of bits in b 0 bit z OGF () = = bit z Construction OGF equation Solution = +( + ) () = + () () = a binary string is empty or a bit followed by a binary string [ ]() = 5

16 Symbolic method: binary strings with restrictions Ex. How many N-bit binary strings have no two consecutive 0s? 0 T0 = T = T2 = T3 = T4 = T5 =3 Stay tuned for general treatment (Chapter 8) 6

17 Symbolic method: binary strings with restrictions Ex. How many N-bit binary strings have no two consecutive 0s? Class B00, the class of binary strings with no 00 Atoms type class size GF Size OGF b, the number of bits in b () = 0 bit z bit z Construction OGF equation = + +( + ) () = + +( + ) () a binary string with no 00 is either empty or 0 or it is or 0 followed by a binary string with no 00 solution () = + [ ] () = + + = +, 2, 5, 8, 3,... 7

18 Symbolic method: many, many examples to follow How many... with...? Class Atoms type class size GF Size OGF Construction a... is either... or... and... OGF equation solution 8

19 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5a.AC.Symbolic

20 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5b.AC.Labelled

21 Labelled combinatorial classes have objects composed of N atoms, labelled with the integers through N. Ex. Different unlabelled objects Ex. Different labelled objects

22 Labelled class example : urns Def. An urn is a set of labelled atoms counting sequence = EGF U = U2 = U3 = U4 =! = 22

23 Labelled class example 2: permutations Def. A permutation is a sequence of labelled atoms P = 2 2 P2 = P3 = counting sequence EGF =!!! = = P4 = 6 23

24 Labelled class example 3: cycles Def. A cycle is a cyclic sequence of labelled atoms C = 2 C2 = C3 = counting sequence EGF =( )! ln ( )!! = =ln 2 3 C4 = 6 24

25 Star product operation Analog to Cartesian product requires relabelling in all consistent ways. 25 Ex. Ex = =

26 Combinatorial constructions for labelled classes construction notation semantics disjoint union A + B disjoint copies of objects from A and B labelled product A B ordered pairs of copies of objects, one from A and one from B A and B are combinatorial classes of labelled objects sequence SEQ ( A ) sequences of objects from A set SET ( A ) sets of objects from A cycle CYC ( A ) cyclic sequences of objects from A 26

27 The symbolic method for labelled classes (transfer theorem) Theorem. Let A and B be combinatorial classes of labelled objects with EGFs A(z) and B(z). Then construction notation semantics EGF disjoint union A + B disjoint copies of objects from A and B ()+() labelled product A B ordered pairs of copies of objects, one from A and one from B ()() SEQk ( A ) k- sequences of objects from A () sequence SEQ ( A ) sequences of objects from A () set SETk ( A ) SET ( A ) k-sets of objects from A sets of objects from A () /! () CYCk ( A ) k-cycles of objects from A () / cycle CYC ( A ) cycles of objects from A ln () 27

28 The symbolic method for labelled classes: basic constructions class construction EGF counting sequence construction notation EGF disjoint union A + B ()+() urns U = SET ( Z ) () = = labelled product A B ()() SEQk ( A ) () cycles C = CYC ( Z ) P = SEQ ( Z ) permutations P = E + Z P () =ln () = =( =! )! sequence set cycle SEQ ( A ) SETk ( A ) SET ( A ) CYCk ( A ) CYC ( A ) () () /! () () / ln () 28

29 Proofs of transfers are immediate from GF counting A + B +! =! +! = ()+() A B A B! = A B + + ( + )! = A! B! = ()() Notation. We write A 2 for A A, A 3 for A A A, etc. 29

30 Proofs of transfers are immediate from GF counting () = {# }! = {# }! =!{# }! () = {# }! ()! = {# }! class construction EGF k-sequence SEQk( A ) sequence SEQk( A ) = SEQ0( A ) + SEQ( A ) + SEQ2( A ) +... k-cycle CYCk( A ) cycle CYCk( A ) = CYC0( A ) + CYC( A ) + CYC2( A ) +... k-set SETk( A ) set SETk( A ) = SET0( A ) + SET( A ) + SET2( A ) ()+() + () +...= + () + ()! + () + ()! () () + () ()! + ()! +...=ln () () +...= () 30

31 Labelled class example 4: sets of cycles Q. How many sets of cycles of labelled atoms? P * = P * 2 = 2 P * 3 = 6 P * 4 = 24 3

32 Symbolic method: sets of cycles How many sets of cycles of length N? Class P*, the class of all sets of cycles of atoms Atom type class size GF Size p, the number of atoms in p labelled atom Z z EGF () =! =! Construction OGF equation Counting sequence = (()) () =exp ( ln ) = =![ ] () =! 32

33 Aside: A combinatorial bijection A permutation is a set of cycles. Standard representation Set of cycles representation

34 Derangements N people go to the opera and leave their hats on a shelf in the cloakroom. When leaving, they each grab a hat at random. Q. What is the probability that nobody gets their own hat? Definition. A derangement is a permutation with no singleton cycles 34

35 Derangements (various versions) A group of N people go to the opera and leave their hats in the cloakroom. When leaving, they each grab a hat at random. Q. What is the probability that nobody gets their own hat? A professor returns exams to N students by passing them out at random. Q. What is the probability that nobody gets their own exam? A group of N sailors go ashore for revelry that leads to a state of inebriation. When returning, they each end up sleeping in a random cabin. Q. What is the probability that nobody sleeps in their own cabin? A group of N students who live in single rooms go to a party that leads to a state of inebriation. When returning, they each end up in a random room. Q. What is the probability that nobody ends up in their own room? 35

36 Derangements are permutations with no singleton cycles. D = 0 D2 = D3 = 2 D4 = 9 36

37 Symbolic method: derangements How many derangements of length N? Class D, the class of all derangements Atom type class size GF Size p, the number of atoms in p labelled atom Z z EGF () =! =! Construction = ( > ()) =exp ( OGF equation () = /+ /+ /+... ln ) = Expansion [ ( ) ]() =!! probability that a random permutation is a derangement simple convolution Derangements are permutations with no singleton cycles" see Asymptotics lecture Alternate derivation () = () = 37

38 Derangements A group of N students who live in single rooms go to a party that leads to a state of inebriation. When returning, they each end up in a random room. Q. What is the probability that nobody ends up in their own room? A.. =. 38

39 Derangements A group of N graduating seniors each throw their hats in the air and each catch a random hat. Q. What is the probability that nobody gets their own hat back? A.. =. 39

40 Generalized derangements In the hats-in-the-air scenario, a student can get her hat back by "following the cycle" Q. What is the probability that all cycles are of length > M? 40

41 Symbolic method: generalized derangements How many permutations of length N have no cycles of length M? Class DM, the class of all generalized derangements Atom type class size GF labelled atom Z z Size d, the number of atoms in d EGF () =! =! Construction OGF equation = ( > ()) () = =... =exp ( ln /... / ) Expansion =?? M-way convolution (stay tuned) 4

42 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5b.AC.Labelled

43 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5c.AC.Asymptotics

44 Generating coefficient asymptotics are often immediately derived via general "analytic" transfer theorems. Example. Taylor's theorem Theorem. If f (z) has N derivatives, then [ z N ]f (z) = f (N ) (0)/N! Example 2. Rational functions transfer theorem (see "Asymptotics" lecture) Theorem. If f (z) and g (z) are polynomials, then [ ] () () = β(/β) (β) β see Asymptotics lecture for general case where /β is the largest root of g (provided that it has multiplicity ). Example 3. Radius-of-convergence transfer theorem [see next slide] Most are based on complex asymptotics. Stay tuned for Part 2 44

45 Radius-of-convergence transfer theorem Theorem. If f (z) has radius of convergence > with f () 0, then ( ) [ () + α ] ( ) α () for any real α 0,, 2,... () Γ(α) α Gamma function (generalized factorial ) Γ() = Γ(α + ) =αγ(α) Γ( + ) =! convolution, f + f fn ~ f () standard asymptotics with generalized binomial coefficient Γ() = Γ(/) = Corollary. If f (z) has radius of convergence >ρ with f (ρ) 0, then [ () ] ( /ρ) α (ρ) Γ(α) ρ α for any real α 0,, 2,... 45

46 Radius-of-convergence transfer theorem: applications Corollary. If f (z) has radius of convergence >ρ with f (ρ) 0, then [ () ] ( /ρ) α (ρ) Γ(α) ρ α for any real α 0,, 2,... Ex : Catalan () = ( ) = / = / () = / ( /) = (/) = [ ]() Ex 2: Derangements () = /... / [! ] () = = () = /... / 46

47 Transfer theorems based on complex asymptotics provide universal laws of sweeping generality Example: Context-free constructions A system of combinatorial constructions < G > = (< G >, < G >,..., < G t >) < G > = (< G >, < G >,..., < G t >)... < G t > = (< G >, < G >,..., < G t >) that reduces to a single GF equation () =( (), (),..., ()) symbolic method Grobner basis elimination Drmota-Lalley-Woods theorem transfers to a system of GF equations () = ( (), (),..., ()) () = ( (), (),..., ())... () = ( (), (),..., ()) that has an explicit () solution singularity analysis Stay tuned for many more (in Part 2). that transfers to a simple asymptotic form!! 47

48 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5c.AC.Asymptotics

49 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5d.AC.Perspective

50 Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Ex: How many binary trees with N nodes? T = E + Z T T combinatorial construction () = ( ) GF coefficient asymptotics 50

51 Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Ex: How many binary trees with N nodes? T = E + Z T T combinatorial construction () = + () GF equation Note: With complex asymptotics, we can transfer directly from GF equation (no need to solve it). See Part 2. coefficient asymptotics 5

52 Old vs. New: Two ways to count binary trees Old Recurrence GF Expand GF Asymptotics New T = E + Z T T () = + () 52

53 Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Ex: How many generalized derangements? = ( > ()) /... /! combinatorial construction GF equation coefficient asymptotics 53

54 A standard paradigm for analytic combinatorics Fundamental constructs elementary or trivial confirm intuition Compound constructs many possibilities classical combinatorial objects expose underlying structure Variations unlimited possibilities not easily analyzed otherwise

55 Combinatorial parameters are handled as two counting problems via cumulated costs. Ex: How many leaves in a random binary tree?. Count trees T = E + Z T T () = ( ) 2. Count leaves in all trees T = E + Z T T (, ) = Symbolic method works for BGFs (see text) 3. Divide 55

56 Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Features: Analysis begins with formal combinatorial constructions. The generating function is the central object of study. Transfer theorems can immediately provide results from formal descriptions. Results extend, in principle, to any desired precision on the standard scale. Variations on fundamental constructions are easily handled. combinatorial constructions symbolic transfer theorem generating function equation analytic transfer theorem coefficient asymptotics the symbolic method 56

57 Stay tuned for many applications of analytic combinatorics and applications to the analysis of algorithms Trees Permutations Mappings Bitstrings

58 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics OF The symbolic method Labelled objects Coefficient asymptotics Perspective 5d.AC.Perspective

59 Exercise 5. Practice with counting bitstrings. AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS S E C O N D E D I T I O N R O B E R T S E D G E W I C K PHILIPPE FLAJOLET 59

60 Exercise 5.3 Practice with counting trees. AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS S E C O N D E D I T I O N R O B E R T S E D G E W I C K PHILIPPE FLAJOLET 60

61 Exercise 5.7 Practice with counting permutations. AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS S E C O N D E D I T I O N R O B E R T S E D G E W I C K PHILIPPE FLAJOLET 6

62 Exercises 5.5 and 5.6 Practice with tree parameters. AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS S E C O N D E D I T I O N R O B E R T S E D G E W I C K PHILIPPE FLAJOLET 62

63 Assignments for next lecture. Read pages in text. AN INTRODUCTION TO THE ANALYSIS OF ALGORITHMS S E C O N D E D I T I O N R O B E R T S E D G E W I C K PHILIPPE FLAJOLET 2. Write up solutions to Exercises 5., 5.3, 5.7, 5.5, and

64 ANALYTIC COMBINATORICS P A R T O N E 5. Analytic Combinatorics