5 applications of rational and meromorphic asymptotics
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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 5. Applications of Rational and Meromorphic Asymptotics http://ac.cs.princeton.edu Analytic combinatorics overview specification A. SYMBOLIC METHOD 1. OGFs 2. EGFs GF


  1. 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 5. Applications of Rational and Meromorphic Asymptotics http://ac.cs.princeton.edu

  2. ⬅ 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

  3. Bottom line from last lecture Speci fi cation Symbolic transfer GF equation Analytic transfer for meromorphic GFs: f ( z )/ g ( z ) ~ c β N • Compute the dominant pole α (smallest real with g ( z ) = 0). • Compute the residue h − 1 = − f ( α )/ g' ( α ). Analytic transfer Not order 1 if g '( α ) = 0. Adjust to (slightly) more • Constant c is h − 1 / α . complicated order M case. • Exponential growth factor β is 1 / α Asymptotics This lecture: Numerous applications 3

  4. 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 5. Applications of Rational and Meromorphic Asymptotics Analytic •Bitstrings Combinatorics •Other familiar examples Philippe Flajolet and •Compositions Robert Sedgewick OF •Supercritical sequence schema CAMBRIDGE http://ac.cs.princeton.edu II.5a.RMapps.Bitstrings

  5. Warmup: Bitstrings How many bitstrings of length N ? 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 counting sequence OGF 0 1 1 0 0 0 0 1 0 � 0 1 1 1 0 0 1 0 1 1 � � = � � � − � � 1 1 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 B 0 = 1 B 1 = 2 1 0 1 0 � 1 1 0 � � � � = ( � � ) � = � � B 2 = 4 1 0 1 1 1 1 1 � − � � � ≥ � � ≥ � 1 1 0 0 B 3 = 8 1 1 0 1 1 1 1 0 1 1 1 1 B 4 = 16 5

  6. Warmup: Bitstrings B , the class of all bitstrings Speci fi cation B = E + ( Z 0 + Z 1 ) × B Symbolic transfer � � ( � ) = GF equation � − � � Dominant singularity: pole at α = � / � � � � = − � ( � ) � � ( � ) = � Analytic transfer Residue: � � � Coefficient of z N : ∼ � − � � � = � � α α [ � � ] � ( � ) = � � Asymptotics 6

  7. Example 1: Bitstrings with restrictions on consecutive 0s How many bitstrings of length N have no two consecutive 0s ? 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 1 1 0 1 T 0 = 1 T 1 = 2 1 1 1 1 1 0 1 1 T 2 = 3 1 1 1 0 1 1 0 1 0 T 3 = 5 1 1 1 1 1 1 1 0 1 1 1 1 1 0 T 4 = 8 1 1 1 1 1 T 5 =13 7

  8. Example 1: Bitstrings with restrictions on consecutive 0s B 00 , the class of all bitstrings having no 00 Speci fi cation B 00 = E + Z 0 + ( Z 0 + Z 0 × Z 1 ) × B 00 √ � − � ˆ φ = � √ � + � φ = Symbolic transfer � � + � ˆ ˆ � �� ( � ) = φ φ Dominant singularity: pole at GF equation � − � − � � Residue: � � � = − � (ˆ = � + ˆ φ ) φ � � (ˆ � + � ˆ φ ) φ Analytic transfer Coefficient of z N : ∼ � − � � � � + ˆ � � φ φ � φ � = φ + � ˆ ˆ ˆ ˆ φ φ [ � � ] � �� ( � ) = φ � φ � φ ˆ φ = � √ � Asymptotics � = � . ����� φ � = φ + � β � . ∼ � � β � ���� � � � . = � . ����� 8

  9. Example 1: Bitstrings with restrictions on consecutive 0s B 4 , the class of all bitstrings having no 0 4 Speci fi cation B 4 = Z <4 ( E + Z 1 B 4 ) Symbolic transfer � � ( � ) = ( � + � + � � + � � )( � + �� � ( � )) � + � + � � + � � Dominant singularity: pole at α GF equation = � − � − � � − � � − � � � + α + α � + α � Residue: � � � = − � ( � ) � � ( � ) = α + � α + � α � + � α � � � Analytic transfer [ � � ] � � ( � ) ∼ � − � � � Coefficient of z N : α α � = � . ���� β � . [ � � ] � � ( � ) ∼ � � β � ���� Asymptotics � � � . = � . ���� 9

  10. Example 1: Bitstrings with restrictions on consecutive 0s � + � � − � − � � � + � + � � � − � − � � − � � � + � + � � + � � � − � − � � − � � − � � � + � + � � + � � + � � � − � − � � − � � − � � − � � 10

  11. Example 1: Bitstrings with restrictions on consecutive 0s � + � + � � + � � + � � + � � + � � + � � + � � + � � � − � − � � − � � − � � − � � − � � − � � − � � − � � − � �� 11

  12. Information on consecutive 0s in GFs for strings [from AC Part I Lecture 5] � | � | = { # ����������������������� � ������� � � } � � � � ( � ) = � � � ∈ � � � ≥ � = � + � + � � + . . . + � � − � � − � � = � − � − � � − . . . � � � − � � + � � + � { # ����������������������� � ��������������� � �� } / � � � � � � � � ( � / � ) = � � ≥ � { # ����������������������� � ��������������� � �� } / � � � � ( � / � ) = � � ≥ � � �� { ��� � ������������������������������������������ � �� } = � ≥ � �� { ����������������������� � � �� > � } = �������������������������������� � � � = � ≥ � Theorem. Probability that an N -bit random bitstring has no 0 M : [ � � ] � � ( � / � ) ∼ � � ( β � / � ) � Theorem. Expected wait time for the first 0 M in a random bitstring: � � ( � / � ) = � � + � − � 12

  13. Autocorrelation [from AC Part I Lecture 5] The probability that an N -bit random bitstring does not contain 0000 is ~1.0917 × . 96328 N The expected wait time for the first occurrence of 0000 in a random bitstring is 30. 0001 occurs much Q. Do the same results hold for 0001? earlier than 0000 A. NO! 10111110100101001100111 0001 001111101101101 0000 0111100001 Observation. Consider first occurrence of 000. • 0000 and 0001 equally likely, BUT • mismatch for 0000 means 0001, so need to wait four more bits • mismatch for 0001 means 0000, so next bit could give a match. Q. What is the probability that an N -bit random bitstring does not contain 0001? Q. What is the expected wait time for the first occurrence of 0001 in a random bitstring? 13

  14. Constructions for strings without specified patterns [from AC Part I Lecture 5] Cast of characters: p — a pattern p 101001010 S p — binary strings that do not contain p S p 10111110101101001100110000011111 T p — binary strings that end in p T p 10111110101101001100110101001010 and have no other occurrence of p First construction • S p and T p are disjoint • the empty string is in S p • adding a bit to a string in S p gives a string in S p or T p � � + � � = � + � � × { � � + � � } 14

  15. Constructions for bitstrings without specified patterns [from AC Part I Lecture 5] Every pattern has an autocorrelation polynomial • slide the pattern to the left over itself. • for each match of i trailing bits with the leading bits include a term z |p| − i 101001010 � � 101001010 101001010 101001010 101001010 101001010 � � 101001010 101001010 � � 101001010 101001010 � ��������� ( � ) = � + � � + � � autocorrelation polynomial 15

  16. Constructions for bitstrings without specified patterns [from AC Part I Lecture 5] Second construction • for each 1 bit in the autocorrelation of any string in T p add a “tail” • result is a string in S p followed by the pattern p 101001010 a string in T p 10111110101101001100110101001010 first tail is null 10111110101101001100110101001010 strings in S p 10111110101101001100110101001010 01010 10111110101101001100110101001010 1001010 � � × { � } = � � × � { � � } � � � = � 16

  17. Bitstrings without specified patterns [from AC Part I Lecture 5] How many N -bit strings do not contain a specified pattern p ? � | � | � � ( � ) = � OGFs S p — the class of binary strings with no p Classes � ∈ � � � | � | � � ( � ) = � T p — the class of binary strings that end in p and have no other occurence � ∈ � � � � + � � = � + � � × { � � + � � } � � × { � } = � � × � { � � } Constructions � � � = � � � ( � ) � � = � � ( � ) � � ( � ) � � ( � ) + � � ( � ) = � + � �� � ( � ) OGF equations � � ( � ) � � ( � ) = Solution � � + ( � − � � ) � � ( � ) β � ����������������������� � � + ( � − � � ) � � ( � ) � [ � � ] � � ( � ) ∼ � � β � ����� Extract cofficients � � � = ���������������������������� 17

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