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

▶
honors combinatorics
SMART_READER_LITE
LIVE PREVIEW

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

Feb 12, 2023 228 likes •1.03k views

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 InclusionExclusion Events: A 1 , . . . , A n

slide-1
SLIDE 1

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

slide-2
SLIDE 2

Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω Input data: PI for I ⊆ [n] where PI = P      

  • i∈I

Ai       Output: P      

n

  • i=1

Ai      

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-3
SLIDE 3

Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω Input data: PI for I ⊆ [n] where PI = P      

  • i∈I

Ai       Output: P      

n

  • i=1

Ai      

How many input data?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-4
SLIDE 4

Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω Input data: PI for I ⊆ [n] where PI = P      

  • i∈I

Ai       Output: P      

n

  • i=1

Ai      

How many input data? 2n − 1

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-5
SLIDE 5

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-6
SLIDE 6

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only Output: approximate value of P      

n

  • i=1

Ai      

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-7
SLIDE 7

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only Output: approximate value of P      

n

  • i=1

Ai      

How many input data?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-8
SLIDE 8

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only Output: approximate value of P      

n

  • i=1

Ai      

How many input data?

k

  • j=0
  • n

j

  • CMSC-27410=Math-28410∼CMSC-3720

Honors Combinatorics

slide-9
SLIDE 9

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only Output: approximate value of P      

n

  • i=1

Ai      

How many input data?

k

  • j=0
  • n

j

  • <

en k k

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-10
SLIDE 10

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only Output: approximate value of P      

n

  • i=1

Ai      

How many input data?

k

  • j=0
  • n

j

  • <

en k k

Where is the threshold?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-11
SLIDE 11

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only Output: approximate value of P      

n

  • i=1

Ai      

How many input data?

k

  • j=0
  • n

j

  • <

en k k

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

slide-12
SLIDE 12

Approximate Inclusion–Exclusion

Events: A1, . . . , An ⊆ Ω PI = P      

  • i∈I

Ai       Input data: PI for |I| ≤ k only Output: approximate value of P      

n

  • i=1

Ai      

How many input data?

k

  • j=0
  • n

j

  • <

en k k

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

slide-13
SLIDE 13

Approximate Inclusion–Exclusion

A1, . . . , An, B1, . . . , Bn ⊆ Ω events Assumption: (∀I ⊆ [m], |I| ≤ k)      P      

  • i∈I

Ai       = P      

  • i∈I

Bi             Let E(k, n) = sup      P      

n

  • i=1

Ai       − P      

n

  • i=1

Bi             subject to the Assumption.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-14
SLIDE 14

Approximate Inclusion–Exclusion

A1, . . . , An, B1, . . . , Bn ⊆ Ω events Assumption: (∀I ⊆ [m], |I| ≤ k)      P      

  • i∈I

Ai       = P      

  • i∈I

Bi             Let E(k, n) = sup      P      

n

  • i=1

Ai       − P      

n

  • i=1

Bi             subject to the Assumption. Theorem (Nati Linial, Noam Nisan 1990) (a) If k = Ω( √ n) then E(k, n) = O

  • e−2k/

√ n

(b) For k = O( √ n) then E(k, n) = O

  • n

k2

  • .

While (a) has later been improved, (b) is best possible.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-15
SLIDE 15

Approximate Inclusion–Exclusion

Ingredients:

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-16
SLIDE 16

Approximate Inclusion–Exclusion

Ingredients: LP Duality

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-17
SLIDE 17

Approximate Inclusion–Exclusion

Ingredients: LP Duality Approximation theory approximation of functions by polynomials

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-18
SLIDE 18

Approximate Inclusion–Exclusion

Ingredients: LP Duality Approximation theory approximation of functions by polynomials Chebyshev polynomials

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-19
SLIDE 19

Chebyshev polynomials of the first kind

cos(2θ) = T2(cos θ) where T2(x) =

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-20
SLIDE 20

Chebyshev polynomials of the first kind

cos(2θ) = T2(cos θ) where T2(x) = 2x2 − 1

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-21
SLIDE 21

Chebyshev polynomials of the first kind

cos(2θ) = T2(cos θ) where T2(x) = 2x2 − 1 cos(3θ) = T3(cos θ) where T3(x) =

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-22
SLIDE 22

Chebyshev polynomials of the first kind

cos(2θ) = T2(cos θ) where T2(x) = 2x2 − 1 cos(3θ) = T3(cos θ) where T3(x) = 4x3 − 3x

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-23
SLIDE 23

Chebyshev polynomials of the first kind

cos(2θ) = T2(cos θ) where T2(x) = 2x2 − 1 cos(3θ) = T3(cos θ) where T3(x) = 4x3 − 3x cos(4θ) = T4(cos θ) where T4(x) = 8x4 − 8x2 + 1 cos(5θ) = T5(cos θ) where T5(x) = 16x5 − 20x3 + 5x cos(6θ) = T6(cos θ) where T6(x) = 32x6 −48x4 + 18x2 −1

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-24
SLIDE 24

Chebyshev polynomials of the first kind

T0(x) = 1 T1(x) = x T2(x) = 2x2 − 1 T3(x) = 4x3 − 3x T4(x) = 8x4 − 8x2 + 1 T5(x) = 16x5 − 20x3 + 5x T6(x) = 32x6 − 48x4 + 18x2 − 1

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-25
SLIDE 25

Chebyshev polynomials of the first kind

T0(x) = 1 T1(x) = x T2(x) = 2x2 − 1 T3(x) = 4x3 − 3x T4(x) = 8x4 − 8x2 + 1 T5(x) = 16x5 − 20x3 + 5x T6(x) = 32x6 − 48x4 + 18x2 − 1 cos(kθ) = Tk(cos θ)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-26
SLIDE 26

Chebyshev polynomials of the first kind

T0(x) = 1 T1(x) = x T2(x) = 2x2 − 1 T3(x) = 4x3 − 3x T4(x) = 8x4 − 8x2 + 1 T5(x) = 16x5 − 20x3 + 5x T6(x) = 32x6 − 48x4 + 18x2 − 1 cos(kθ) = Tk(cos θ) Recurrence: Tk+1(x) = 2x · Tk(x) − Tk−1(x)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-27
SLIDE 27

Chebyshev polynomials of the first kind

T0(x) = 1 T1(x) = x T2(x) = 2x2 − 1 T3(x) = 4x3 − 3x T4(x) = 8x4 − 8x2 + 1 T5(x) = 16x5 − 20x3 + 5x T6(x) = 32x6 − 48x4 + 18x2 − 1 cos(kθ) = Tk(cos θ) Recurrence: Tk+1(x) = 2x · Tk(x) − Tk−1(x) Generating function:

∞

  • k=0

Tk(x)tk = 1 − tx 1 − tx + t2

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-28
SLIDE 28

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-29
SLIDE 29

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) −1 ≤ x ≤ 1 =⇒ ? ≤ Tk(x) ≤ ? roots: cos(kθ) = 0 iff kθj = (2j − 1)π/2 iff

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-30
SLIDE 30

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) −1 ≤ x ≤ 1 =⇒ −1 ≤ Tk(x) ≤ 1 roots: cos(kθ) = 0 iff kθj = (2j − 1)π/2 iff θj = 2j − 1 2k π (j = 1, . . . , k)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-31
SLIDE 31

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) −1 ≤ x ≤ 1 =⇒ −1 ≤ Tk(x) ≤ 1 roots: cos(kθ) = 0 iff kθj = (2j − 1)π/2 iff θj = 2j − 1 2k π (j = 1, . . . , k) roots: cos(θj) (j = 1, . . . , k)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-32
SLIDE 32

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) −1 ≤ x ≤ 1 =⇒ −1 ≤ Tk(x) ≤ 1 roots: cos(kθ) = 0 iff kθj = (2j − 1)π/2 iff θj = 2j − 1 2k π (j = 1, . . . , k) roots: cos(θj) (j = 1, . . . , k) k roots distributed over (−1, 1) denser at the extremes

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-33
SLIDE 33

Chebyshev polynomials of the first kind

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-34
SLIDE 34

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-35
SLIDE 35

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) 1. Tk(x) has degre k and lead coefficient 2−k+1

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-36
SLIDE 36

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) 1. Tk(x) has degre k and lead coefficient 2−k+1 2. explicit formula outside (−1, 1) interval: Tk(x) = 1 2 ·

  • (x +

√ x2 − 1)k + (x − √ x2 − 1) k

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-37
SLIDE 37

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) 1. Tk(x) has degre k and lead coefficient 2−k+1 2. explicit formula outside (−1, 1) interval: Tk(x) = 1 2 ·

  • (x +

√ x2 − 1)k + (x − √ x2 − 1) k also valid inside (−1, 1) – how come?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-38
SLIDE 38

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) 1. Tk(x) has degre k and lead coefficient 2−k+1 2. explicit formula outside (−1, 1) interval: Tk(x) = 1 2 ·

  • (x +

√ x2 − 1)k + (x − √ x2 − 1) k also valid inside (−1, 1) – how come? 3. −1 ≤ x ≤ 1 =⇒ |Tk(x)| ≤ 1

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-39
SLIDE 39

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) 1. Tk(x) has degre k and lead coefficient 2−k+1 2. explicit formula outside (−1, 1) interval: Tk(x) = 1 2 ·

  • (x +

√ x2 − 1)k + (x − √ x2 − 1) k also valid inside (−1, 1) – how come? 3. −1 ≤ x ≤ 1 =⇒ |Tk(x)| ≤ 1 4. |Tk(x)| = 1 at exactly k + 1 points in [−1, 1]. The sign alternates

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-40
SLIDE 40

Chebyshev polynomials of the first kind cos(kθ) = Tk(cos θ) 1. Tk(x) has degre k and lead coefficient 2−k+1 2. explicit formula outside (−1, 1) interval: Tk(x) = 1 2 ·

  • (x +

√ x2 − 1)k + (x − √ x2 − 1) k also valid inside (−1, 1) – how come? 3. −1 ≤ x ≤ 1 =⇒ |Tk(x)| ≤ 1 4. |Tk(x)| = 1 at exactly k + 1 points in [−1, 1]. The sign alternates 5. derivative |T ′

k(x)| ≤ k2

(−1 ≤ x ≤ 1)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-41
SLIDE 41

Chebyshev polynomials of the first kind

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-42
SLIDE 42

An extremal property of Chebyshev polynomials

Let p(x) be the monic polynomial of degree k (leading term xk) that minimizes max

−1≤x≤1 |p(x)|.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-43
SLIDE 43

An extremal property of Chebyshev polynomials

Let p(x) be the monic polynomial of degree k (leading term xk) that minimizes max

−1≤x≤1 |p(x)|.

Then p(x) = ±21−kTk(x).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-44
SLIDE 44

Chebyshev polynomials of the first kind

Min distance between roots of Tk is Θ(1/k 2)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-45
SLIDE 45

Chebyshev polynomials of the first kind

Min distance between roots of Tk is Θ(1/k 2) will explain change of behavior around k ≈ √ n

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-46
SLIDE 46

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Assumption: (∀I ⊆ [m], |I| ≤ k)      P      

  • i∈I

Ai       = P      

  • i∈I

Bi             Let E(k, n) = sup      P      

n

  • i=1

Ai       − P      

n

  • i=1

Bi             subject to the Assumption.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-47
SLIDE 47

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Assumption: (∀I ⊆ [m], |I| ≤ k)      P      

  • i∈I

Ai       = P      

  • i∈I

Bi             Let E(k, n) = sup      P      

n

  • i=1

Ai       − P      

n

  • i=1

Bi             subject to the Assumption. Theorem (Nati Linial, Noam Nisan 1990) (a) If k = Ω( √ n) then E(k, n) = O

  • e−2k/

√ n

(b) For k = O( √ n) then E(k, n) = O

  • n

k2

  • .

While (a) has later been improved, (b) is best possible.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-48
SLIDE 48

Approximate Inclusion–Exclusion

DEF A j-atom of A = (A1, . . . , An) is a set

  • i∈I

Ai ∩

  • iI

Ai where I ⊆ [n], |I| = j.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-49
SLIDE 49

Approximate Inclusion–Exclusion

DEF A j-atom of A = (A1, . . . , An) is a set

  • i∈I

Ai ∩

  • iI

Ai where I ⊆ [n], |I| = j. DEF A is symmetric if for every j (1 ≤ j ≤ n) all j-atoms have equal probability.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-50
SLIDE 50

Approximate Inclusion–Exclusion

DEF A j-atom of A = (A1, . . . , An) is a set

  • i∈I

Ai ∩

  • iI

Ai where I ⊆ [n], |I| = j. DEF A is symmetric if for every j (1 ≤ j ≤ n) all j-atoms have equal probability. Symmetrization Lemma [easy] E(k, n) remains unchanged if we restrict A and B to be symmetric.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-51
SLIDE 51

Approximate Inclusion–Exclusion

DEF A j-atom of A = (A1, . . . , An) is a set

  • i∈I

Ai ∩

  • iI

Ai where I ⊆ [n], |I| = j. DEF A is symmetric if for every j (1 ≤ j ≤ n) all j-atoms have equal probability. Symmetrization Lemma [easy] E(k, n) remains unchanged if we restrict A and B to be symmetric. Proof: Replace the probability of each j-atom by their average. QED[symmetrization lemma]

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-52
SLIDE 52

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Notation aj := sum of probabilities of the j-atoms of A bj := sum of probabilities of the j-atoms of B

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-53
SLIDE 53

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Notation aj := sum of probabilities of the j-atoms of A bj := sum of probabilities of the j-atoms of B Obs: n

j=1 aj = P(n i=1 Ai)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-54
SLIDE 54

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Notation aj := sum of probabilities of the j-atoms of A bj := sum of probabilities of the j-atoms of B Obs: n

j=1 aj = P(n i=1 Ai)

rj := sum of probabilities of all j-wise intersections (j ≤ k) (this is the same for A and B)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-55
SLIDE 55

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Notation aj := sum of probabilities of the j-atoms of A bj := sum of probabilities of the j-atoms of B Obs: n

j=1 aj = P(n i=1 Ai)

rj := sum of probabilities of all j-wise intersections (j ≤ k) (this is the same for A and B) Ej(x1, . . . , xn) :=

n

  • i=j
  • i

j

  • xi

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-56
SLIDE 56

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Notation aj := sum of probabilities of the j-atoms of A bj := sum of probabilities of the j-atoms of B Obs: n

j=1 aj = P(n i=1 Ai)

rj := sum of probabilities of all j-wise intersections (j ≤ k) (this is the same for A and B) Ej(x1, . . . , xn) :=

n

  • i=j
  • i

j

  • xi

Claim. rj = Ej(a1, . . . , an) (1 ≤ j ≤ k)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-57
SLIDE 57

Approximate Inclusion–Exclusion

A = (A1, . . . , An), B = (B1, . . . , Bn) Notation aj := sum of probabilities of the j-atoms of A bj := sum of probabilities of the j-atoms of B Obs: n

j=1 aj = P(n i=1 Ai)

rj := sum of probabilities of all j-wise intersections (j ≤ k) (this is the same for A and B) Ej(x1, . . . , xn) :=

n

  • i=j
  • i

j

  • xi

Claim. rj = Ej(a1, . . . , an) (1 ≤ j ≤ k) Proof: For j ≤ i, and i-atom contributes to (i

j)

j-intersections.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-58
SLIDE 58

Approximate Inclusion–Exclusion: a LP

Lemma (primal): E(k, n) = max n

i=1 xi subject to the

constraints (1) for 1 ≤ j ≤ k: Ej(x1, . . . , xn) = 0 (2) for S ⊆ [n]: −1 ≤

i∈S xi ≤ 1

Proof: Recall:ai = P(i-atoms of A), bi: same for B xi := ai − bi satisfies constraints; (2) b/c diff of probab (1) b/c Ej( x) = Ej( a − b) = Ej( a) − Ej( b) = rj − rj = 0 This shows max xi ≥ E(k, n).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-59
SLIDE 59

Approximate Inclusion–Exclusion: a LP

Lemma (primal): E(k, n) = max n

i=1 xi subject to the

constraints (1) for 1 ≤ j ≤ k: Ej(x1, . . . , xn) = 0 (2) for S ⊆ [n]: −1 ≤

i∈S xi ≤ 1

Proof continued: Conversely: Assume x satisfies (1), (2). We achieve xi = ai − bi: Let ai := xi if xi > 0 and ai = 0 otherwise. Let bi := −xi if xi < 0 and bi = 0 otherwise. Now we build collection A of events by distributing aj equally among the (n

j) j-atoms. Same for B.

So

i xi = ai − bi = P( Ai) − P( Bi) (Obs)

Also, j-wise intersections for A and B have equal prob. by Claim: this prob is rj/(n

j) and rj = Ej(

a) = Ej( b) because Ej( x) = 0 by constraint (1) QED[Lemma(primal)]

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-60
SLIDE 60

Approximate Inclusion–Exclusion: a LP

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

Lemma (dual) E(k, n) = inff maxi∈[n](1 − fi) where the inf is over f = n

i=1 fixi ∈ span(Ej | j ∈ [k])

satisfying (∀i ∈ [n])(fi ≤ 1).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-61
SLIDE 61

Approximate Inclusion–Exclusion: a LP

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

Lemma (dual) E(k, n) = inff maxi∈[n](1 − fi) where the inf is over f = n

i=1 fixi ∈ span(Ej | j ∈ [k])

satisfying (∀i ∈ [n])(fi ≤ 1).

Not an LP but proof based on dual LP . (Skipping proof.)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-62
SLIDE 62

Approximate Inclusion–Exclusion: a LP

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

Lemma (dual) E(k, n) = inff maxi∈[n](1 − fi) where the inf is over f = n

i=1 fixi ∈ span(Ej | j ∈ [k])

satisfying (∀i ∈ [n])(fi ≤ 1).

Not an LP but proof based on dual LP . (Skipping proof.)

The BIG JUMP Lemma (polynomials) E(k, n) = inf

q max m∈[n](1 − q(m))

where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-63
SLIDE 63

Jumping from linear forms to polynomials

Interpolation: Given h1, . . . , hn ∈ R ∃! polynomial g of deg ≤ n s.t. g(0) = 0 and (∀i ∈ [n])(g(i) = hi).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-64
SLIDE 64

Jumping from linear forms to polynomials

Interpolation: Given h1, . . . , hn ∈ R ∃! polynomial g of deg ≤ n s.t. g(0) = 0 and (∀i ∈ [n])(g(i) = hi). Defining linear map F : linear forms in n variables → polynomials p of deg ≤ n with p(0) = 0 F : n

i=1 cixi → p s.t. p(0) = 0 and p(i) = ci (i ∈ [n])

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

F : Ej → pj s.t. pj(i) = (i

j)

what polynomial pj does this?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-65
SLIDE 65

Jumping from linear forms to polynomials

Interpolation: Given h1, . . . , hn ∈ R ∃! polynomial g of deg ≤ n s.t. g(0) = 0 and (∀i ∈ [n])(g(i) = hi). Defining linear map F : linear forms in n variables → polynomials p of deg ≤ n with p(0) = 0 F : n

i=1 cixi → p s.t. p(0) = 0 and p(i) = ci (i ∈ [n])

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

F : Ej → pj s.t. pj(i) = (i

j)

what polynomial pj does this? E.g. E2 → p2 s.t. p2(i) = i(i − 1)/2 so p2(x) =?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-66
SLIDE 66

Jumping from linear forms to polynomials

Interpolation: Given h1, . . . , hn ∈ R ∃! polynomial g of deg ≤ n s.t. g(0) = 0 and (∀i ∈ [n])(g(i) = hi). Defining linear map F : linear forms in n variables → polynomials p of deg ≤ n with p(0) = 0 F : n

i=1 cixi → p s.t. p(0) = 0 and p(i) = ci (i ∈ [n])

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

F : Ej → pj s.t. pj(i) = (i

j)

what polynomial pj does this? E.g. E2 → p2 s.t. p2(i) = i(i − 1)/2 so p2(x) =? p2(x) = x(x − 1)/2 and generally pj(x) = (x

j )

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-67
SLIDE 67

Jumping from linear forms to polynomials

Interpolation: Given h1, . . . , hn ∈ R ∃! polynomial g of deg ≤ n s.t. g(0) = 0 and (∀i ∈ [n])(g(i) = hi). Defining linear map F : linear forms in n variables → polynomials p of deg ≤ n with p(0) = 0 F : n

i=1 cixi → p s.t. p(0) = 0 and p(i) = ci (i ∈ [n])

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

F : Ej → pj s.t. pj(i) = (i

j)

what polynomial pj does this? E.g. E2 → p2 s.t. p2(i) = i(i − 1)/2 so p2(x) =? p2(x) = x(x − 1)/2 and generally pj(x) = (x

j )

span(p1, . . . , pk) =?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-68
SLIDE 68

Jumping from linear forms to polynomials

Interpolation: Given h1, . . . , hn ∈ R ∃! polynomial g of deg ≤ n s.t. g(0) = 0 and (∀i ∈ [n])(g(i) = hi). Defining linear map F : linear forms in n variables → polynomials p of deg ≤ n with p(0) = 0 F : n

i=1 cixi → p s.t. p(0) = 0 and p(i) = ci (i ∈ [n])

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

F : Ej → pj s.t. pj(i) = (i

j)

what polynomial pj does this? E.g. E2 → p2 s.t. p2(i) = i(i − 1)/2 so p2(x) =? p2(x) = x(x − 1)/2 and generally pj(x) = (x

j )

span(p1, . . . , pk) = all polynomials of deg ≤ k, p(0) = 0

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-69
SLIDE 69

Approximate Inclusion–Exclusion: a LP

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

Lemma (dual) E(k, n) = inff maxi∈[n](1 − fi) where the min is over f = n

i=1 fixi ∈ span(Ej | j ∈ [k]) and

satisfy (∀i ∈ [n])(fi ≤ 1).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-70
SLIDE 70

Approximate Inclusion–Exclusion: a LP

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

Lemma (dual) E(k, n) = inff maxi∈[n](1 − fi) where the min is over f = n

i=1 fixi ∈ span(Ej | j ∈ [k]) and

satisfy (∀i ∈ [n])(fi ≤ 1). Lemma (polynomials) E(k, n) = inf

q max i∈[n] (1 − q(i))

where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-71
SLIDE 71

Approximate Inclusion–Exclusion: a LP

Recall: Ej(x1, . . . , xn) := n

i=j (i j)xi

Lemma (dual) E(k, n) = inff maxi∈[n](1 − fi) where the min is over f = n

i=1 fixi ∈ span(Ej | j ∈ [k]) and

satisfy (∀i ∈ [n])(fi ≤ 1). Lemma (polynomials) E(k, n) = inf

q max i∈[n] (1 − q(i))

where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1). Proof: Apply F operator. QED[Lemma(dual)→ Lemma(polynomials)]

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-72
SLIDE 72

Approximate Inclusion–Exclusion: polynomials

Lemma (polynomials) E(k, n) = inf

q max i∈[n] (1 − q(i))

where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-73
SLIDE 73

Approximate Inclusion–Exclusion: polynomials

Lemma (polynomials) E(k, n) = inf

q max i∈[n] (1 − q(i))

where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1). Switch to D(k, n) := infq maxi |1 − q(i)|

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-74
SLIDE 74

Approximate Inclusion–Exclusion: polynomials

Lemma (polynomials) E(k, n) = inf

q max i∈[n] (1 − q(i))

where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1). Switch to D(k, n) := infq maxi |1 − q(i)| Obs: E(k, n) =

2D(k,n) 1+D(k,n)

New goal: estimate D(k, n)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-75
SLIDE 75

Approximate Inclusion–Exclusion: polynomials

D(k, n) := infq max1≤i≤n |1 − q(i)| where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1). discrete → continuous: D∗(k, n) := infq max1≤x≤n |1 − q(x)| where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀1 ≤ x ≤ n)(q(x) ≤ 1).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-76
SLIDE 76

Approximate Inclusion–Exclusion: polynomials

D(k, n) := infq max1≤i≤n |1 − q(i)| where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀i ∈ [n])(q(i) ≤ 1). discrete → continuous: D∗(k, n) := infq max1≤x≤n |1 − q(x)| where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀1 ≤ x ≤ n)(q(x) ≤ 1). Let p(x) be the monic polynomial of degree k (leading term xk) that minimizes max

−1≤x≤1 |p(x)|.

Then p(x) = ±21−kTk(x) (Chebyshev polynomial). ∴ D∗(k, n) optimized by stretched-shifted Tk(x)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-77
SLIDE 77

Approximate Inclusion–Exclusion: polynomials

D∗(k, n) := infq max1≤x≤n |1 − q(x)| where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀1 ≤ x ≤ n)(q(x) ≤ 1). ∴ D∗(k, n) optimized by stretched-shifted Tk(x)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-78
SLIDE 78

Approximate Inclusion–Exclusion: polynomials

D∗(k, n) := infq max1≤x≤n |1 − q(x)| where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀1 ≤ x ≤ n)(q(x) ≤ 1). ∴ D∗(k, n) optimized by stretched-shifted Tk(x) This almost works for the discrete version, too. Then, for large k, the estimate comes from the explicit formula for the Chebyshev polynomial. For small k we need to look at the derivative: derivative |T ′

k(x)| ≤ k 2

(−1 ≤ x ≤ 1)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

slide-79
SLIDE 79

Approximate Inclusion–Exclusion: polynomials

D∗(k, n) := infq max1≤x≤n |1 − q(x)| where q: polynomial of deg ≤ k satisfying q(0) = 0 and (∀1 ≤ x ≤ n)(q(x) ≤ 1). ∴ D∗(k, n) optimized by stretched-shifted Tk(x) This almost works for the discrete version, too. Then, for large k, the estimate comes from the explicit formula for the Chebyshev polynomial. For small k we need to look at the derivative: derivative |T ′

k(x)| ≤ k 2

(−1 ≤ x ≤ 1) which after stretching translates to ≈ k 2/n, not too steep for k = O( √ n).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics