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

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

Apr 29, 2023 546 likes •1.54k views

Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 8, Thursday, May 28, 2020 CMSC-27410=Math-28410 CMSC-3720 Honors Combinatorics Variance of sum of random variables Var ( X ) = E

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Honors Combinatorics

CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 8, Thursday, May 28, 2020

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

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Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

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

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Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) =

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

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Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) = 0

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

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Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) = 0 Cov(X, X) =

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

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Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) = 0 Cov(X, X) = Var(X)

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

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SLIDE 7

Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) = 0 Cov(X, X) = Var(X) Y = n

i=1 Xi

Var(Y) =

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

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Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) = 0 Cov(X, X) = Var(X) Y = n

i=1 Xi

Var(Y) =

i

  • j Cov(Xi, Xj) =

i Var(Xi) + ij Cov(Xi, Xj)

If the Xi are independent then Var(Y) =

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

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SLIDE 9

Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) = 0 Cov(X, X) = Var(X) Y = n

i=1 Xi

Var(Y) =

i

  • j Cov(Xi, Xj) =

i Var(Xi) + ij Cov(Xi, Xj)

If the Xi are independent then Var(Y) =

i Var(Xi)

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

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Variance of sum of random variables

variance Var(X) = E((X − E(X))2) = E(X 2) − (E(X))2 Var(X) ≥ 0 standard deviation σ(X) =

  • Var(X)

covariance Cov(X, Y) = E(XY) − E(X)E(Y) If X, Y independent then Cov(X, Y) = 0 Cov(X, X) = Var(X) Y = n

i=1 Xi

Var(Y) =

i

  • j Cov(Xi, Xj) =

i Var(Xi) + ij Cov(Xi, Xj)

If the Xi are pairwise independent then Var(Y) =

i Var(Xi)

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) =

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) =

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) =

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

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

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SLIDE 17

Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

standard deviation of Y:

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

standard deviation of Y: √ Var Y = √ n

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

standard deviation of Y: √ Var Y = √ n If k = O( √ n) and k ≡ n (mod 2) then P(Y = k) = Θ(

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

standard deviation of Y: √ Var Y = √ n If k = O( √ n) and k ≡ n (mod 2) then P(Y = k) = Θ(1/ √ n)

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

standard deviation of Y: √ Var Y = √ n If k = O( √ n) and k ≡ n (mod 2) then P(Y = k) = Θ(1/ √ n) If k = Θ( √ n) then P(|Y| ≤ k) =

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

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Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

standard deviation of Y: √ Var Y = √ n If k = O( √ n) and k ≡ n (mod 2) then P(Y = k) = Θ(1/ √ n) If k = Θ( √ n) then P(|Y| ≤ k) = Θ(1)

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

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SLIDE 23

Variance of sum of random variables

X = ±1 with prob (1/2, 1/2) E(X) = 0 Var(X) = E(X 2) − (E(X))2 = 1 − 0 = 1 Xi = ±1 with prob (1/2, 1/2) independent (i ∈ [n]) Y := n

i=1 Xi

Var(Y) = n

i=1 Var(Xi) = n

standard deviation of Y: √ Var Y = √ n If k = O( √ n) and k ≡ n (mod 2) then P(Y = k) = Θ(1/ √ n) If k = Θ( √ n) then P(|Y| ≤ k) = Θ(1) Central Limit Theorem: for k = O( √ n) P(|Y| ≤ k) ≈ 1 2π √ n k

−k

e−t2/(2n)dt

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

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Strong concentration of sum of independent random variables

Central Limit Theorem: for k = O( √ n) P(|Y| ≤ k) ≈ 1 2π √ n k

−k

e−t2/(2n)dt tail hard to estimate – too small against the error we may conjecture P(|Y| ≥ k) ≈ e−k2/(2n)

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

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Strong concentration of sum of independent random variables

Central Limit Theorem: for k = O( √ n) P(|Y| ≤ k) ≈ 1 2π √ n k

−k

e−t2/(2n)dt tail hard to estimate – too small against the error we may conjecture P(|Y| ≥ k) ≈ e−k2/(2n) Theorem (Chernoff bound) X1, . . . , Xn independent random variables, |Xi| ≤ 1 Y = n

i=1 Xi

=⇒ (∀a ∈ R)       P       

  • i

Xi

  • ≥ a

       < 2e−a2/(2n)       

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

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Discrepancy

H = (V, E) hypergraph f : V → {1, −1} 2-coloring (not “legal”)

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

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Discrepancy

H = (V, E) hypergraph f : V → {1, −1} 2-coloring (not “legal”) discrepancy of E ⊆ V under f: disc(E, f) = |

v∈E f(v)|

discrepancy of f on H: disc(H, f) = maxE∈E disc(E, f) discrepancy of H: disc(H) = min

f

disc(H, f) = min

f

max

E∈E disc(E, f)

Goal: keep it small

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

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Discrepancy

H = (V, E) hypergraph f : V → {1, −1} 2-coloring (not “legal”) discrepancy of E ⊆ V under f: disc(E, f) = |

v∈E f(v)|

discrepancy of f on H: disc(H, f) = maxE∈E disc(E, f) discrepancy of H: disc(H) = min

f

disc(H, f) = min

f

max

E∈E disc(E, f)

Goal: keep it small Question: can f be nearly balanced simultaneously on all edges?

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

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Discrepancy

f : V → {1, −1} 2-coloring (not “legal”)

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

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SLIDE 30

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW?

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

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SLIDE 31

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f

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

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SLIDE 32

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Proof. P(disc(E, f) > t) < 2e−t2/2|E| ≤ 2e−t2/(2n)

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

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SLIDE 33

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Proof. P(disc(E, f) > t) < 2e−t2/2|E| ≤ 2e−t2/(2n) P((∃E ∈ E)(disc(E, f) > t)) < 2m · e−t2/(2n)

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

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SLIDE 34

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Proof. P(disc(E, f) > t) < 2e−t2/2|E| ≤ 2e−t2/(2n) P((∃E ∈ E)(disc(E, f) > t)) < 2m · e−t2/(2n)

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

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SLIDE 35

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Proof. P(disc(E, f) > t) < 2e−t2/2|E| ≤ 2e−t2/(2n) P((∃E ∈ E)(disc(E, f) > t)) < 2m · e−t2/(2n) if 2m · e−t2/(2n) ≤ 1 then disc(H) ≤ t

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

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SLIDE 36

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Proof. P(disc(E, f) > t) < 2e−t2/2|E| ≤ 2e−t2/(2n) P((∃E ∈ E)(disc(E, f) > t)) < 2m · e−t2/(2n) if 2m · e−t2/(2n) ≤ 1 then disc(H) ≤ t ln(2m) − t2/(2n) ≤ 0 i.e. t ≤

  • 2m ln(2m)

QED

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

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SLIDE 37

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

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

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SLIDE 38

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Major goal: Beat random!

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

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SLIDE 39

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Major goal: Beat random! (a) quantitatively: reduce discrepancy (math)

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

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SLIDE 40

Discrepancy

f : V → {1, −1} 2-coloring (not “legal”) discrepancy of H: disc(H) = min

f

max

E∈E

  • v∈E

f(v)

  • Goal: keep it small

HOW? Try random f Theorem disc(H) <

  • 2n ln(2m)

Major goal: Beat random! (a) quantitatively: reduce discrepancy (math) (b) qualitatively: find explicit coloring (TCS)

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

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SLIDE 41

Discrepancy: beat random

Theorem disc(H) <

  • 2n ln(2m)

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

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SLIDE 42

Discrepancy: beat random

Theorem disc(H) <

  • 2n ln(2m)

Theorem (Beck–Fiala 1981) disc(H) < 2 degmax

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

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SLIDE 43

Discrepancy: beat random

Theorem disc(H) <

  • 2n ln(2m)

Theorem (Beck–Fiala 1981) disc(H) < 2 degmax Theorem (“Standard deviation bound” Spencer 1985) If m ≥ n then disc(H) = O(

  • n ln(2m/n))

In particular, if m = O(n) then disc(H) = O( √ n)

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

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Beck–Fiala Theorem

f : V → {1, −1} 2-coloring discrepancy of H: disc(H) = minf maxE∈E

  • v∈E f(v)
  • CMSC-27410=Math-28410∼CMSC-3720

Honors Combinatorics

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SLIDE 45

Beck–Fiala Theorem

f : V → {1, −1} 2-coloring discrepancy of H: disc(H) = minf maxE∈E

  • v∈E f(v)
  • Theorem (Beck–Fiala 1981)

disc(H) < 2 degmax

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

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SLIDE 46

Beck–Fiala Theorem

f : V → {1, −1} 2-coloring discrepancy of H: disc(H) = minf maxE∈E

  • v∈E f(v)
  • Theorem (Beck–Fiala 1981)

disc(H) < 2 degmax Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges}

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

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SLIDE 47

Beck–Fiala Theorem

f : V → {1, −1} 2-coloring discrepancy of H: disc(H) = minf maxE∈E

  • v∈E f(v)
  • Theorem (Beck–Fiala 1981)

disc(H) < 2 degmax Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing |U| in each round. Never update f(v) for v U.

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

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SLIDE 48

Beck–Fiala Theorem

f : V → {1, −1} 2-coloring discrepancy of H: disc(H) = minf maxE∈E

  • v∈E f(v)
  • Theorem (Beck–Fiala 1981)

disc(H) < 2 degmax Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing |U| in each round. Never update f(v) for v U. Claim (∀f)(U ∅ =⇒ |U| > |F |) Proof If F ∅ count incidences ∈ U × F d · |U| ≥ # incidences > d · |F | qed[Claim]

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

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SLIDE 49

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |)

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

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SLIDE 50

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

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

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SLIDE 51

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

|F | equations in |U| unknowns

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

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SLIDE 52

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

|F | equations in |U| unknowns x = (x1, . . . , x|U|) ∈ RU

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

slide-53
SLIDE 53

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

|F | equations in |U| unknowns x = (x1, . . . , x|U|) ∈ RU ∴ ∃ affine line of solutions

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

slide-54
SLIDE 54

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

|F | equations in |U| unknowns x = (x1, . . . , x|U|) ∈ RU ∴ ∃ affine line of solutions line hits boundary of box (∀i)(|xi| ≤ 1) (∀i)(|xi| ≤ 1)&(∃j)(|xj| = 1)

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

slide-55
SLIDE 55

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

|F | equations in |U| unknowns x = (x1, . . . , x|U|) ∈ RU ∴ ∃ affine line of solutions line hits boundary of box (∀i)(|xi| ≤ 1) (∀i)(|xi| ≤ 1)&(∃j)(|xj| = 1) update f(vi) := xi (vi ∈ U)

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

slide-56
SLIDE 56

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

|F | equations in |U| unknowns x = (x1, . . . , x|U|) ∈ RU ∴ ∃ affine line of solutions line hits boundary of box (∀i)(|xi| ≤ 1) (∀i)(|xi| ≤ 1)&(∃j)(|xj| = 1) update f(vi) := xi (vi ∈ U) progress:

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

slide-57
SLIDE 57

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Claim (∀f)(U ∅ =⇒ |U| > |F |) Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

|F | equations in |U| unknowns x = (x1, . . . , x|U|) ∈ RU ∴ ∃ affine line of solutions line hits boundary of box (∀i)(|xi| ≤ 1) (∀i)(|xi| ≤ 1)&(∃j)(|xj| = 1) update f(vi) := xi (vi ∈ U) progress: j removed from U

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

slide-58
SLIDE 58

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

End of process: U = F = ∅

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

slide-59
SLIDE 59

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

End of process: U = F = ∅ Need to verify: (∀E ∈ E)(|

v∈E f(v)| < 2d)

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

slide-60
SLIDE 60

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

End of process: U = F = ∅ Need to verify: (∀E ∈ E)(|

v∈E f(v)| < 2d)

time t: round when E became stable

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

slide-61
SLIDE 61

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

End of process: U = F = ∅ Need to verify: (∀E ∈ E)(|

v∈E f(v)| < 2d)

time t: round when E became stable ft: function f after round t E was still balanced under ft:

  • v∈E ft(v) = 0

# undecided vertices in E after round t?

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

slide-62
SLIDE 62

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

End of process: U = F = ∅ Need to verify: (∀E ∈ E)(|

v∈E f(v)| < 2d)

time t: round when E became stable ft: function f after round t E was still balanced under ft:

  • v∈E ft(v) = 0

# undecided vertices in E after round t? ≤ d

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

slide-63
SLIDE 63

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

End of process: U = F = ∅ Need to verify: (∀E ∈ E)(|

v∈E f(v)| < 2d)

time t: round when E became stable ft: function f after round t E was still balanced under ft:

  • v∈E ft(v) = 0

# undecided vertices in E after round t? ≤ d |ft(v)| < 1, |f(v)| = 1 =⇒ |f(v) − ft(v)| < 2

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

slide-64
SLIDE 64

Beck–Fiala Theorem

Beck–Fiala Thm. (∃f)(∀E ∈ E)(disc(E, f) < 2 degmax) Proof. d := degmax. f : V → [−1, 1]. Vertex v undecided if f(v) {±1}. U = {undec vertices} Edge E unstable if |E ∩ U| > d. F = {unstab edges} Initialize f ≡ 0. Update f, reducing # undecided vertices in each round. Never update decided vertex. Loop invariant: E ∈ F =⇒ E balanced:

v∈E f(v) = 0

End of process: U = F = ∅ Need to verify: (∀E ∈ E)(|

v∈E f(v)| < 2d)

time t: round when E became stable ft: function f after round t E was still balanced under ft:

  • v∈E ft(v) = 0

# undecided vertices in E after round t? ≤ d |ft(v)| < 1, |f(v)| = 1 =⇒ |f(v) − ft(v)| < 2 total imbalance at finish < 2d QED[Beck-Fiala]

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

slide-65
SLIDE 65

Beck–Fiala Theorem

Theorem (József Beck–Tibor Fiala 1981) disc(H) < 2 degmax Source of proof presented:

Bernard Chazelle: The Discrepancy Method

Cambridge University Press, 2000

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

slide-66
SLIDE 66

Beck–Fiala Conjecture

Theorem (Beck–Fiala 1981) disc(H) < 2 degmax

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

slide-67
SLIDE 67

Beck–Fiala Conjecture

Theorem (Beck–Fiala 1981) disc(H) < 2 degmax Conjecture (Beck–Fiala 1981) disc(H) = O(

  • degmax)

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

slide-68
SLIDE 68

Beck–Fiala Conjecture

Theorem (Beck–Fiala 1981) disc(H) < 2 degmax Conjecture (Beck–Fiala 1981) disc(H) = O(

  • degmax)

STILL OPEN

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

slide-69
SLIDE 69

Beck–Fiala Conjecture

Theorem (Beck–Fiala 1981) disc(H) < 2 degmax Conjecture (Beck–Fiala 1981) disc(H) = O(

  • degmax)

STILL OPEN Theorem (Wojciech Banaszczyk 1998) disc(H) = O(

  • degmax · ln m)

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

slide-70
SLIDE 70

Beck–Fiala Conjecture

Theorem (Beck–Fiala 1981) disc(H) < 2 degmax Conjecture (Beck–Fiala 1981) disc(H) = O(

  • degmax)

STILL OPEN Theorem (Wojciech Banaszczyk 1998) disc(H) = O(

  • degmax · ln m)

Banaszczyk: “Balancing vectors and Gaussian measures

  • f n-dimensional convex bodies.”

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

slide-71
SLIDE 71

Beck–Fiala Conjecture

Conjecture (József Beck–Tibor Fiala 1981) disc(H) = O(

  • degmax)

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

slide-72
SLIDE 72

Beck–Fiala Conjecture

Conjecture (József Beck–Tibor Fiala 1981) disc(H) = O(

  • degmax)

x = (x1, . . . , xn) x2 = n

i=1 x2 i

x∞ = maxn

i=1 |xi|

unit ball in Rn: Bn := {u ∈ Rn : u2 ≤ 1} m(u1, . . . , un) := min{

i ǫiui∞ : ǫi = ±1}

Conjecture (János Komlós 1980s) (∃K ∈ R)(∀u1, . . . , un ∈ Bm)(m(u1, . . . , un) ≤ K)

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

slide-73
SLIDE 73

Beck–Fiala Conjecture

Conjecture (József Beck–Tibor Fiala 1981) disc(H) = O(

  • degmax)

x = (x1, . . . , xn) x2 = n

i=1 x2 i

x∞ = maxn

i=1 |xi|

unit ball in Rn: Bn := {u ∈ Rn : u2 ≤ 1} m(u1, . . . , un) := min{

i ǫiui∞ : ǫi = ±1}

Conjecture (János Komlós 1980s) (∃K ∈ R)(∀u1, . . . , un ∈ Bm)(m(u1, . . . , un) ≤ K) Komlós’s conjecture =⇒ the Beck–Fiala conjecture

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

slide-74
SLIDE 74

Beck–Fiala Conjecture

Conjecture (József Beck–Tibor Fiala 1981) disc(H) = O(

  • degmax)

x = (x1, . . . , xn) x2 = n

i=1 x2 i

x∞ = maxn

i=1 |xi|

unit ball in Rn: Bn := {u ∈ Rn : u2 ≤ 1} m(u1, . . . , un) := min{

i ǫiui∞ : ǫi = ±1}

Conjecture (János Komlós 1980s) (∃K ∈ R)(∀u1, . . . , un ∈ Bm)(m(u1, . . . , un) ≤ K) Komlós’s conjecture =⇒ the Beck–Fiala conjecture Exercise (∀u1, . . . , un ∈ Bn)(m(u1, . . . , un) ≤ √ n)

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

slide-75
SLIDE 75

Komlós Conj vs. Beck–Fiala Thm

x = (x1, . . . , xn) Norms: x1 = n

i=1 |xi|

x2 = n

i=1 x2 i

x∞ = maxn

i=1 |xi|

m(u1, . . . , un) := min{

i ǫiui∞ : ǫi = ±1}

Beck–Fiala theorem rephrased Theorem (Beck–Fiala 1981) If u1, . . . , un ∈ Rm, ui1 ≤ 1 then m(u1, . . . , un) < 2

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

slide-76
SLIDE 76

Komlós Conj vs. Beck–Fiala Thm

x = (x1, . . . , xn) Norms: x1 = n

i=1 |xi|

x2 = n

i=1 x2 i

x∞ = maxn

i=1 |xi|

m(u1, . . . , un) := min{

i ǫiui∞ : ǫi = ±1}

Beck–Fiala theorem rephrased Theorem (Beck–Fiala 1981) If u1, . . . , un ∈ Rm, ui1 ≤ 1 then m(u1, . . . , un) < 2 Komlós’s conjecture restated Conjecture (János Komlós 1980s) ∃K ∈ R s.t. if u1, . . . , un ∈ Rm, ui2 ≤ 1 then m(u1, . . . , un) < K

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

slide-77
SLIDE 77

Komlós Conj vs. Beck–Fiala Thm

x = (x1, . . . , xn) Norms: x1 = n

i=1 |xi|

x2 = n

i=1 x2 i

x∞ = maxn

i=1 |xi|

m(u1, . . . , un) := min{

i ǫiui∞ : ǫi = ±1}

Beck–Fiala theorem rephrased Theorem (Beck–Fiala 1981) If u1, . . . , un ∈ Rm, ui1 ≤ 1 then m(u1, . . . , un) < 2 Komlós’s conjecture restated Conjecture (János Komlós 1980s) ∃K ∈ R s.t. if u1, . . . , un ∈ Rm, ui2 ≤ 1 then m(u1, . . . , un) < K The difference: assumption in ℓ1-norm (Beck–Fiala Thm) assumption in ℓ2-norm (Komlós Conj)

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

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SLIDE 78

The “standard deviation bound”

Theorem (Joel H. Spencer 1985) If m ≥ n then disc(H) = O(

  • n ln(2m/n))

In particular, if m = O(n) then disc(H) = O( √ n)

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

slide-79
SLIDE 79

The “standard deviation bound”

Theorem (Joel H. Spencer 1985) If m ≥ n then disc(H) = O(

  • n ln(2m/n))

In particular, if m = O(n) then disc(H) = O( √ n) method: Beck 1981: “pigeonhole on discrepancy vector”

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

slide-80
SLIDE 80

The “standard deviation bound”

Theorem (Joel H. Spencer 1985) If m ≥ n then disc(H) = O(

  • n ln(2m/n))

In particular, if m = O(n) then disc(H) = O( √ n) method: Beck 1981: “pigeonhole on discrepancy vector” partial coloring f : V → {1, 0, −1} “uncolored”: f(v) = 0 disc(H, f) = maxE∈E |

v∈E f(v)|

Lemma If m ≥ n then ∃ partial coloring f s.t. disc(H, f) = O(

  • n ln(2m/n)) and

|f −1(0)| ≤ 0.9n (at most 90% of vertices is uncolored)

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

slide-81
SLIDE 81

The “standard deviation bound”

Theorem (Joel H. Spencer 1985) If m ≥ n then disc(H) = O(

  • n ln(2m/n))

In particular, if m = O(n) then disc(H) = O( √ n) method: Beck 1981: “pigeonhole on discrepancy vector” partial coloring f : V → {1, 0, −1} “uncolored”: f(v) = 0 disc(H, f) = maxE∈E |

v∈E f(v)|

Lemma If m ≥ n then ∃ partial coloring f s.t. disc(H, f) = O(

  • n ln(2m/n)) and

|f −1(0)| ≤ 0.9n (at most 90% of vertices is uncolored) Exercise: Lemma =⇒ Theorem

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

slide-82
SLIDE 82

The “standard deviation bound”

partial coloring f : V → {1, 0, −1} “uncolored”: f(v) = 0 Lemma If m ≥ n then ∃ partial coloring f s.t. disc(H, f) = O(

  • n ln(2m/n)) and

|f −1(0)| ≤ 0.9n (at most 90% of vertices is uncolored)

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

slide-83
SLIDE 83

The “standard deviation bound”

partial coloring f : V → {1, 0, −1} “uncolored”: f(v) = 0 Lemma If m ≥ n then ∃ partial coloring f s.t. disc(H, f) = O(

  • n ln(2m/n)) and

|f −1(0)| ≤ 0.9n (at most 90% of vertices is uncolored) Proof: Let f : V → {±1} be a 2-coloring. Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy: d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm

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

slide-84
SLIDE 84

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy of 2-coloring f : V → {±1} d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm

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

slide-85
SLIDE 85

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy of 2-coloring f : V → {±1} d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Technique (Beck 1981): many 2-colorings give the same discrepancy vector → use Pigeonhole

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

slide-86
SLIDE 86

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy of 2-coloring f : V → {±1} d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Technique (Beck 1981): many 2-colorings give the same discrepancy vector → use Pigeonhole Lemma (concentration of discrepancy vectors) (∃ℓ ∈ Zm)(P(D(f) = ℓ) > 2−n/5) Proof: Chernoff bound and entropy argument “QED”[Lemma]

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

slide-87
SLIDE 87

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy: d(E, f) = σ(E, f) c

  • n ln(2m/n)

discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Lemma (concentration of discrepancy vectors) (∃ℓ ∈ Zm)(P(D(f) = ℓ) > 2−n/5) In other words, (∃ℓ ∈ Zm)(|D−1(ℓ)| > 24n/5)

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

slide-88
SLIDE 88

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy: d(E, f) = σ(E, f) c

  • n ln(2m/n)

discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Lemma (concentration of discrepancy vectors) (∃ℓ ∈ Zm)(P(D(f) = ℓ) > 2−n/5) In other words, (∃ℓ ∈ Zm)(|D−1(ℓ)| > 24n/5) D−1(ℓ) = {f | D(f) = ℓ}

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

slide-89
SLIDE 89

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy of 2-coloring f : V → {±1} d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Lemma: (∃ℓ ∈ Zm)(|D−1(ℓ)|) > 24n/5

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

slide-90
SLIDE 90

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy of 2-coloring f : V → {±1} d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Lemma: (∃ℓ ∈ Zm)(|D−1(ℓ)|) > 24n/5 Pick f1 ∈ D−1(ℓ) and let f ′ = (f − f1)/2 for f ∈ D−1(ℓ). f ′: partial coloring disc(f ′) < (c/2)

  • n ln(2m/n)

b/c σ(f ′) = (1/2)(σ(f) − σ(f1))

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

slide-91
SLIDE 91

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy of 2-coloring f : V → {±1} d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Lemma: (∃ℓ ∈ Zm)(|D−1(ℓ)|) > 24n/5 Pick f1 ∈ D−1(ℓ) and let f ′ = (f − f1)/2 for f ∈ D−1(ℓ). f ′: partial coloring disc(f ′) < (c/2)

  • n ln(2m/n)

b/c σ(f ′) = (1/2)(σ(f) − σ(f1)) as desired

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

slide-92
SLIDE 92

The “standard deviation bound”

Signed discrepancy: σ(E, f) =

v∈E f(v)

Normalized, rounded, signed discrepancy of 2-coloring f : V → {±1} d(E, f) =         σ(E, f) c

  • n ln(2m/n)

        Let E = {E1, . . . , Em} discrepancy vector: D(f) = (d(E1, f), . . . , d(Em, f)) ∈ Zm Lemma: (∃ℓ ∈ Zm)(|D−1(ℓ)|) > 24n/5 Pick f1 ∈ D−1(ℓ) and let f ′ = (f − f1)/2 for f ∈ D−1(ℓ). f ′: partial coloring disc(f ′) < (c/2)

  • n ln(2m/n)

b/c σ(f ′) = (1/2)(σ(f) − σ(f1)) as desired Need to show: (∃f ′)(|f ′−1(0)| < 9n/10) f ′ leaves < 9n/10 vertices uncolored

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

slide-93
SLIDE 93

The “standard deviation bound”

Lemma If m ≥ n then ∃ partial coloring f s.t. disc(H, f) = O(

  • n ln(2m/n)) and

|f −1(0)| ≤ 0.9n (at most 90% of vertices is uncolored) We found a set of > 24n/5 partial colorings f ′ satisfying the discrepancy condition. Need to show at least one of them colors at least 10%

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

slide-94
SLIDE 94

The “standard deviation bound”

Lemma If m ≥ n then ∃ partial coloring f s.t. disc(H, f) = O(

  • n ln(2m/n)) and

|f −1(0)| ≤ 0.9n (at most 90% of vertices is uncolored) We found a set of > 24n/5 partial colorings f ′ satisfying the discrepancy condition. Need to show at least one of them colors at least 10% # partial functions that color ≤ 10% is < 24n/5 In fact, much fewer. So pick f ′ in the difference. QED[Standard deviation bound]

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

slide-95
SLIDE 95

The “standard deviation bound”

Lemma If m ≥ n then ∃ partial coloring f s.t. disc(H, f) = O(

  • n ln(2m/n)) and

|f −1(0)| ≤ 0.9n (at most 90% of vertices is uncolored) We found a set of > 24n/5 partial colorings f ′ satisfying the discrepancy condition. Need to show at least one of them colors at least 10% # partial functions that color ≤ 10% is < 24n/5 In fact, much fewer. So pick f ′ in the difference. QED[Standard deviation bound] Theorem (Spencer, Beck) If m = O(n) then disc(H) = O( √ n)

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

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SLIDE 96

The “standard deviation bound”

T(k) = {partial functions that color ≤ k vertices} Claim: |T(n/10)| < 23n/5

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

slide-97
SLIDE 97

The “standard deviation bound”

T(k) = {partial functions that color ≤ k vertices} Claim: |T(n/10)| < 23n/5 Exercise: |T(k)| =

k

  • j=1
  • n

j

  • · 2j <

2en k k

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

slide-98
SLIDE 98

The “standard deviation bound”

T(k) = {partial functions that color ≤ k vertices} Claim: |T(n/10)| < 23n/5 Exercise: |T(k)| =

k

  • j=1
  • n

j

  • · 2j <

2en k k Set k = n/10 so |T(n/10)| < 2en n/10 n/10 = (20e)n/10 < 1.5n < 20.585n

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