Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: - PowerPoint PPT Presentation

Honors Combinatorics CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 8, Tuesday, May 26, 2020 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Inclusion–Exclusion Events: A 1 , . . . , A n ⊆ Ω   �   Input data: P I for I ⊆ [ n ] where P I = P A i         i ∈ I Output: n   �   P A i         i = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Inclusion–Exclusion Events: A 1 , . . . , A n ⊆ Ω   �   Input data: P I for I ⊆ [ n ] where P I = P A i         i ∈ I Output: n   �   P A i         i = 1 How many input data? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Inclusion–Exclusion Events: A 1 , . . . , A n ⊆ Ω   �   Input data: P I for I ⊆ [ n ] where P I = P A i         i ∈ I Output: n   �   P A i         i = 1 2 n − 1 How many input data? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   Events: A 1 , . . . , A n ⊆ Ω P I = P A i         i ∈ I Input data: P I for | I | ≤ k only CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   Events: A 1 , . . . , A n ⊆ Ω P I = P A i         i ∈ I Input data: P I for | I | ≤ k only n   �   Output: approximate value of P A i         i = 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   Events: A 1 , . . . , A n ⊆ Ω P I = P A i         i ∈ I Input data: P I for | I | ≤ k only n   �   Output: approximate value of P A i         i = 1 How many input data? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   Events: A 1 , . . . , A n ⊆ Ω P I = P A i         i ∈ I Input data: P I for | I | ≤ k only n   �   Output: approximate value of P A i         i = 1 k � � n � How many input data? j j = 0 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   Events: A 1 , . . . , A n ⊆ Ω P I = P A i         i ∈ I Input data: P I for | I | ≤ k only n   �   Output: approximate value of P A i         i = 1 k � � � k n � e n � < How many input data? j k j = 0 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   Events: A 1 , . . . , A n ⊆ Ω P I = P A i         i ∈ I Input data: P I for | I | ≤ k only n   �   Output: approximate value of P A i         i = 1 k � � � k n � e n � < How many input data? j k j = 0 Where is the threshold? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   Events: A 1 , . . . , A n ⊆ Ω P I = P A i         i ∈ I Input data: P I for | I | ≤ k only n   �   Output: approximate value of P A i         i = 1 k � � � k n � e n � < How many input data? j k j = 0 Where is the threshold? When k above threshold we get good approximation When k below threshold we can say very little CHAT! CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion   �   A 1 , . . . , A n ⊆ Ω P I = P Events: A i         i ∈ I Input data: P I for | I | ≤ k only n   �   Output: approximate value of P A i         i = 1 k � � � k n � e n � < How many input data? j k j = 0 Where is the threshold? When k above threshold we get good approximation When k below threshold we can say very little √ threshold k ≈ n CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion A 1 , . . . , A n , B 1 , . . . , B n ⊆ Ω events Assumption:       � �       ( ∀ I ⊆ [ m ] , | I | ≤ k )  P A i  = P B i                       i ∈ I i ∈ I n n       Let � �       E ( k , n ) = sup  P A i  − P B i                       i = 1 i = 1 subject to the Assumption. CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion A 1 , . . . , A n , B 1 , . . . , B n ⊆ Ω events Assumption:       � �       ( ∀ I ⊆ [ m ] , | I | ≤ k )  P A i  = P B i                       i ∈ I i ∈ I n n       Let � �       E ( k , n ) = sup  P A i  − P B i                       i = 1 i = 1 subject to the Assumption. Theorem (Nati Linial, Noam Nisan 1990) √ √ � n � e − 2 k / (a) If k = Ω( n ) then E ( k , n ) = O √ � � n (b) For k = O ( n ) then E ( k , n ) = O . k 2 While (a) has later been improved, (b) is best possible. CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion Ingredients: CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion Ingredients: LP Duality CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion Ingredients: LP Duality Approximation theory approximation of functions by polynomials CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Approximate Inclusion–Exclusion Ingredients: LP Duality Approximation theory approximation of functions by polynomials Chebyshev polynomials CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind cos ( 2 θ ) = T 2 ( cos θ ) where T 2 ( x ) = CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind cos ( 2 θ ) = T 2 ( cos θ ) where T 2 ( x ) = 2 x 2 − 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind cos ( 2 θ ) = T 2 ( cos θ ) where T 2 ( x ) = 2 x 2 − 1 cos ( 3 θ ) = T 3 ( cos θ ) where T 3 ( x ) = CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind cos ( 2 θ ) = T 2 ( cos θ ) where T 2 ( x ) = 2 x 2 − 1 cos ( 3 θ ) = T 3 ( cos θ ) where T 3 ( x ) = 4 x 3 − 3 x CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind cos ( 2 θ ) = T 2 ( cos θ ) where T 2 ( x ) = 2 x 2 − 1 cos ( 3 θ ) = T 3 ( cos θ ) where T 3 ( x ) = 4 x 3 − 3 x cos ( 4 θ ) = T 4 ( cos θ ) where T 4 ( x ) = 8 x 4 − 8 x 2 + 1 cos ( 5 θ ) = T 5 ( cos θ ) where T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x cos ( 6 θ ) = T 6 ( cos θ ) where T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind T 0 ( x ) = 1 T 1 ( x ) = x T 2 ( x ) = 2 x 2 − 1 T 3 ( x ) = 4 x 3 − 3 x T 4 ( x ) = 8 x 4 − 8 x 2 + 1 T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind T 0 ( x ) = 1 T 1 ( x ) = x T 2 ( x ) = 2 x 2 − 1 T 3 ( x ) = 4 x 3 − 3 x T 4 ( x ) = 8 x 4 − 8 x 2 + 1 T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 cos ( k θ ) = T k ( cos θ ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind T 0 ( x ) = 1 T 1 ( x ) = x T 2 ( x ) = 2 x 2 − 1 T 3 ( x ) = 4 x 3 − 3 x T 4 ( x ) = 8 x 4 − 8 x 2 + 1 T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 cos ( k θ ) = T k ( cos θ ) Recurrence: T k + 1 ( x ) = 2 x · T k ( x ) − T k − 1 ( x ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Chebyshev polynomials of the first kind T 0 ( x ) = 1 T 1 ( x ) = x T 2 ( x ) = 2 x 2 − 1 T 3 ( x ) = 4 x 3 − 3 x T 4 ( x ) = 8 x 4 − 8 x 2 + 1 T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 cos ( k θ ) = T k ( cos θ ) Recurrence: T k + 1 ( x ) = 2 x · T k ( x ) − T k − 1 ( x ) ∞ 1 − tx T k ( x ) t k = � Generating function: 1 − tx + t 2 k = 0 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics