AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. What is the - PowerPoint PPT Presentation

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. What is the average number of 0 bits in a random bit string of length n containing no 00? 0 1 a 0 = 0 a 1 = 1/2 1

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. What is the average number of 0 bits in a random bit string of length n containing no 00? 0 1 0 1 0 1 1 1 a 0 = 0 a 1 = 1/2 a 2 = 2/3 1

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. What is the average number of 0 bits in a random bit string of length n containing no 00? 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 1 1 0 a 0 = 0 a 1 = 1/2 1 1 1 a 2 = 2/3 a 3 = 1 1

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. What is the average number of 0 bits in a random bit string of length n containing no 00? 0 1 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 0 1 1 0 1 a 0 = 0 a 1 = 1/2 1 1 1 a 2 = 2/3 1 1 1 0 a 3 = 1 1 1 1 1 a 4 = 10/8 1

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. What is the average number of 0 bits in a random bit string of length n containing no 00? 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 a 0 = 0 a 1 = 1/2 1 1 1 1 1 0 1 1 a 2 = 2/3 1 1 1 0 1 1 0 1 0 a 3 = 1 1 1 1 1 1 1 1 0 1 a 4 = 10/8 1 1 1 1 0 1 1 1 1 1 a 5 =20/13 1

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. Recalling the derivation at left, fill in the boxes at right to prove that the n average number of 0 bits in a random bit string of length n containing no 00 is √ ∼ φ 5 B = E + Z 0 + ( Z 1 + Z 0 × Z 1 ) × B construction B ( z ) = 1 + z + ( z + z 2 ) B ( z ) B ( z, u ) = OGF equation BGF equation 1 + z B ( z ) = OGF BGF 1 − z − z 2 φ n = φ n +2 [ z n ] B ( z ) ∼ 1 + 1 / φ enumeration [ z n ] B u ( z, 1) = cumulated cost √ 1 − ˆ 5 φ / φ (1 − z/ ρ ) α ∼ f ( ρ ) f ( z ) Γ ( α ) ρ − n n α − 1 [ z n ] [ z n ] B u ( z, 1) n solution √ ∼ φ [ z n ] B ( z ) 5 2

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. Recalling the derivation at left, fill in the boxes at right to prove that the n average number of 0 bits in a random bit string of length n containing no 00 is √ ∼ φ 5 B = E + Z 0 + ( Z 1 + Z 0 × Z 1 ) × B construction B ( z ) = 1 + z + ( z + z 2 ) B ( z ) B ( z, u ) = 1 + uz + ( z + uz 2 ) B ( z, u ) OGF equation BGF equation 1 + z B ( z ) = OGF BGF 1 − z − z 2 φ n = φ n +2 [ z n ] B ( z ) ∼ 1 + 1 / φ enumeration [ z n ] B u ( z, 1) = cumulated cost √ 1 − ˆ 5 φ / φ (1 − z/ ρ ) α ∼ f ( ρ ) f ( z ) Γ ( α ) ρ − n n α − 1 [ z n ] [ z n ] B u ( z, 1) n solution √ ∼ φ [ z n ] B ( z ) 5 2

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. Recalling the derivation at left, fill in the boxes at right to prove that the n average number of 0 bits in a random bit string of length n containing no 00 is √ ∼ φ 5 B = E + Z 0 + ( Z 1 + Z 0 × Z 1 ) × B construction B ( z ) = 1 + z + ( z + z 2 ) B ( z ) B ( z, u ) = 1 + uz + ( z + uz 2 ) B ( z, u ) OGF equation BGF equation 1 + z 1 + uz B ( z ) = B ( z, u ) = OGF BGF 1 − z − z 2 1 − z − uz 2 φ n = φ n +2 [ z n ] B ( z ) ∼ 1 + 1 / φ enumeration [ z n ] B u ( z, 1) = cumulated cost √ 1 − ˆ 5 φ / φ (1 − z/ ρ ) α ∼ f ( ρ ) f ( z ) Γ ( α ) ρ − n n α − 1 [ z n ] [ z n ] B u ( z, 1) n solution √ ∼ φ [ z n ] B ( z ) 5 2

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. Recalling the derivation at left, fill in the boxes at right to prove that the n average number of 0 bits in a random bit string of length n containing no 00 is √ ∼ φ 5 B = E + Z 0 + ( Z 1 + Z 0 × Z 1 ) × B construction B ( z ) = 1 + z + ( z + z 2 ) B ( z ) B ( z, u ) = 1 + uz + ( z + uz 2 ) B ( z, u ) OGF equation BGF equation 1 + z 1 + uz B ( z ) = B ( z, u ) = OGF BGF 1 − z − z 2 1 − z − uz 2 φ n = φ n +2 [ z n ] B ( z ) ∼ 1 + 1 / φ z [ z n ] enumeration [ z n ] B u ( z, 1) = cumulated cost √ 1 − ˆ (1 − z − z 2 ) 2 5 φ / φ (1 − z/ ρ ) α ∼ f ( ρ ) f ( z ) Γ ( α ) ρ − n n α − 1 [ z n ] [ z n ] B u ( z, 1) n solution √ ∼ φ [ z n ] B ( z ) 5 2

AC MGFs Q&A 1: 0 bits in 00-free bit strings Q. Recalling the derivation at left, fill in the boxes at right to prove that the n average number of 0 bits in a random bit string of length n containing no 00 is √ ∼ φ 5 B = E + Z 0 + ( Z 1 + Z 0 × Z 1 ) × B construction B ( z ) = 1 + z + ( z + z 2 ) B ( z ) B ( z, u ) = 1 + uz + ( z + uz 2 ) B ( z, u ) OGF equation BGF equation 1 + z 1 + uz B ( z ) = B ( z, u ) = OGF BGF 1 − z − z 2 1 − z − uz 2 φ n = φ n +2 [ z n ] B ( z ) ∼ 1 + 1 / φ z [ z n ] enumeration [ z n ] B u ( z, 1) = cumulated cost √ 1 − ˆ (1 − z − z 2 ) 2 5 φ / φ φ / φ ) 2 n φ n = n φ n +1 1 / φ ∼ (1 − ˆ 5 (1 − z/ ρ ) α ∼ f ( ρ ) f ( z ) Γ ( α ) ρ − n n α − 1 [ z n ] [ z n ] B u ( z, 1) n solution √ ∼ φ [ z n ] B ( z ) 5 2

AC MGFs Q&A 2 Q. Which of the following is true of the average number of 0s in a random bit string of length n excluding any fixed pattern p, as n grows? Approaches 0 Approaches 1 Approaches cn for some fixed constant c Has periodic behavior Asymptotic order of growth depends on the pattern 3

AC MGFs Q&A 2 Q. Which of the following is true of the average number of 0s in a random bit string of length n excluding any fixed pattern p, as n grows? F Approaches 0 Approaches 1 Approaches cn for some fixed constant c Has periodic behavior Asymptotic order of growth depends on the pattern 3

AC MGFs Q&A 2 Q. Which of the following is true of the average number of 0s in a random bit string of length n excluding any fixed pattern p, as n grows? F Approaches 0 Approaches 1 F Approaches cn for some fixed constant c Has periodic behavior Asymptotic order of growth depends on the pattern 3

AC MGFs Q&A 2 Q. Which of the following is true of the average number of 0s in a random bit string of length n excluding any fixed pattern p, as n grows? F Approaches 0 Approaches 1 F Approaches cn for some fixed constant c F Has periodic behavior Asymptotic order of growth depends on the pattern 3

AC MGFs Q&A 2 Q. Which of the following is true of the average number of 0s in a random bit string of length n excluding any fixed pattern p, as n grows? F Approaches 0 Approaches 1 F T Approaches cn for some fixed constant c F Has periodic behavior Asymptotic order of growth depends on the pattern 3

AC MGFs Q&A 2 Q. Which of the following is true of the average number of 0s in a random bit string of length n excluding any fixed pattern p, as n grows? F Approaches 0 Approaches 1 F T Approaches cn for some fixed constant c F Has periodic behavior F Asymptotic order of growth depends on the pattern 3