SLIDE 1 ❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✶
❚♦❞❛②✬s ❡①❡r❝✐s❡s
- ✷✳✸✿ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ ✈❡rs✉s ❇❛❧❛♥❝❡❞
- ✷✳✽✿ ▼❛♥② ❱❛r✐❛❜❧❡s
- ✷✳✶✷✿ ❍♦✇ ♠✉❝❤ ■♥❞❡♣❡♥❞❡♥❝❡ ✐s ◆❡❝❡ss❛r②❄
- ✷✳✶✹✿ ❙✉♣r❡♠✉♠ ✈s ▼❛①✐♠✉♠
- ✸✳✷✿ ✭■♥ ❝❧❛ss✮ ■♠♣❧❡♠❡♥t✐♥❣ ❘❡s♦❧✉t✐♦♥ t♦ ❉❡❝✐❞❡ ✷✲❙❆❚
- ✭■♥ ❝❧❛ss✮ ❘❡s♦❧✉t✐♦♥ ❊①❡r❝✐s❡
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 2
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✷
✷✳✸✳ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ ❱❡rs✉s ❇❛❧❛♥❝❡❞
✭✶✮ ❋r♦♠ ✷✳✷ ✇❡ ❦♥♦✇ t❤❛t ❡✈❡r② t♦t❛❧ ❛ss✐❣♥♠❡♥t ❢♦r F s❛t✐s✜❡s ❛❧❧ ❜✉t ♦♥❡ ❝❧❛✉s❡ ✐♥ F✳ ❚❤❛t ✐s✱ ❢♦r ❛♥② ❛ss✐❣♥♠❡♥t α✱ ❡①❛❝t❧② ♦♥❡ ❝❧❛✉s❡ ✐s ✏r❡s♣♦♥s✐❜❧❡✑ ❢♦r t❤❡ ✉♥s❛t✐s✜❛❜✐❧✐t② ♦❢ F✳ ❉❡♥♦t❡ t❤✐s ❝❧❛✉s❡ ❜② C(α)✳ ▲❡t x ∈ vbl(F)✳ P❛rt✐t✐♦♥ t❤❡ ❝❧❛✉s❡s ✭✐✳❡✳ F✮ ❛♥❞ t❤❡ ❛ss✐❣♥♠❡♥ts ✐♥t♦ t❤r❡❡ ♣❛rts✿ ✶✳ F0✿ ❈❧❛✉s❡s ✇✐t❤ x ∈ vbl(C)✳ ▲❡t A0 = {α | C(α) ∈ F0}✳ ✷✳ Fx✿ ❈❧❛✉s❡s ✇✐t❤ x ∈ C✳ ▲❡t Ax = {α | C(α) ∈ Fx}✳ ✸✳ F¯
x✿ ❈❧❛✉s❡s ✇✐t❤ ¯
x ∈ C✳ ▲❡t A¯
x = {α | C(α) ∈ F¯ x}✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 3
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✸
❋✐rst ✇❡ s❡❡ t❤❛t |Ax| = |A¯
x|✿
Pr♦♦❢✿ ❋♦r α ∈ Ax✱ α(x) = 0✱ ❛♥❞ ❢♦r α ∈ A¯
x✱ α(x) = 1✳
A0 ✐s ✏s②♠♠❡tr✐❝✑ ✇✐t❤ r❡s♣❡❝t t♦ x✳ ❍❡♥❝❡ ❢♦r {0, 1}V \ A0✱ t❤❡r❡ ❛r❡ ❛s ♠❛♥② ❛ss✐❣♥♠❡♥ts s❡tt✐♥❣ x t♦ 0 ❛s s❡tt✐♥❣ x t♦ 1✳ ❖❜s❡r✈❛t✐♦♥✿ ❊❛❝❤ k✲❝❧❛✉s❡ ✐s ♥♦t s❛t✐s✜❡❞ ❜② ❡①❛❝t❧② 2|V |−k ❛ss✐❣♥✲ ♠❡♥ts✳ ❚❤❡r❡❢♦r❡ |Fx| = |F¯
x|✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 4
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✹
✷✳✸✳ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ ❱❡rs✉s ❇❛❧❛♥❝❡❞ ✭✷✮
✭✷✮ ❋♦r ❛♥ ❛r❜✐tr❛r② ✉♥s❛t✐s✜❛❜❧❡ ❈◆❋✱ t❤✐s ✐s ♥♦t tr✉❡✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ F = {{¯ x}, {x, ¯ y}, {x, y}}✳
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 5
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✺
✷✳✽✿ ▼❛♥② ✈❛r✐❛❜❧❡s
❈♦♥s✐❞❡r t❤❡ ❜✐♣❛rt✐t❡ ❣r❛♣❤ G = (V, E) ✇❤❡r❡ t❤❡ ✈❡rt❡① s❡t V ✐s ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ A∪B✱ ❝❧❛✉s❡ ✈❡rt✐❝❡s ❛♥❞ ✈❛r✐❛❜❧❡ ✈❡rt✐❝❡s r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡❛❝❤ ❝❧❛✉s❡ C ∈ F ❛♥❞ ❡❛❝❤ ✈❛r✐❛❜❧❡ i ∈ ✈❜❧(C)✱ E ❝♦♥t❛✐♥s ❛♥ ❡❞❣❡ ❝♦♥♥❡❝t✐♥❣ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛✉s❡ ❛♥❞ ✈❛r✐❛❜❧❡ ♣❛✐rs ✭❛♥❞ ♦♥❧② t❤♦s❡ ❡❞❣❡s✮✳ ❋♦r ❛♥② s✉❜s❡t S ⊆ A ♦❢ ❝❧❛✉s❡ ✈❡rt✐❝❡s✱ ✇❡ ❤❛✈❡ t❤❛t |N(S)| ≥ |S| s✐♥❝❡ N(S) ❝♦rr❡s♣♦♥❞s ♣r❡❝✐s❡❧② t♦ t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❝❧❛✉s❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ S✳ ❇② ❍❛❧❧✬s ❚❤❡♦r❡♠✱ t❤❡r❡ ✐s ❛ ♣❡r❢❡❝t ♠❛t❝❤✐♥❣ t❤❛t s❛t✉r❛t❡s A✱ ❛♥❞ s♦ F ✐s s❛t✐s✜❛❜❧❡ ✭❡❛❝❤ ❝❧❛✉s❡ ❝❛♥ ✉s❡ ❛ ✉♥✐q✉❡ ✈❛r✐❛❜❧❡ t♦ s❛t✐s❢② ✐t ✐♥ ❛♥ ❛ss✐❣♥♠❡♥t✮✳
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 6
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✻
✷✳✶✷✿ ❍♦✇ ▼✉❝❤ ■♥❞❡♣❡♥❞❡♥❝❡ ✐s ◆❡❝❡ss❛r②❄
▼✉t✉❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡✿ ❊❛❝❤ ✈❛r✐❛❜❧❡✬s tr✉t❤ ✈❛❧✉❡ ✐s s❡t t♦ 1 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② p ❛♥❞ t♦ 0 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② (1−p) ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ❛❧❧ ♦t❤❡r ✈❛r✐❛❜❧❡s✳ P❛✐r✇✐s❡ ✐♥❞❡♣❡♥❞❡♥❝❡✿ ❋♦r ❡❛❝❤ ♣❛✐r x, y ♦❢ ✈❛r✐❛❜❧❡s✱ ✇❡ ❤❛✈❡ Pr(y → 1) = Pr(y → 1|x → 0) = Pr(y → 1|x → 1) = p ✭t❤❡ ❛♥❛❧♦❣♦✉s st❛t❡♠❡♥t ❢♦r y → 0 ❢♦❧❧♦✇s ❜② t❛❦✐♥❣ ❝♦♠♣❧❡♠❡♥ts✮ ❊①❛♠♣❧❡✿ s❛♠♣❧❡ x, y ✉✳❛✳r✳ ✐♥❞❡♣❡♥❞❡♥t❧② ❛♥❞ t❤❡♥ z := x ⊕ y✳ ◗✿ ❈❛♥ ②♦✉ ♠❛❦❡ ❛ ❧❛r❣❡ s♣❛❝❡ ✇✐t❤ ♣❛✐r✇✐s❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s❄
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 7
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✼
✷✳✶✷✿ ❍♦✇ ▼✉❝❤ ■♥❞❡♣❡♥❞❡♥❝❡ ✐s ◆❡❝❡ss❛r②❄ ✭✷✮
■♥ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✹✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❡❛❝❤ ❝❧❛✉s❡ t♦ ❜❡ s❛t✐s✜❡❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳✳✳ ✏■❢ C ∈ F ❝♦♥t❛✐♥s ❛ ♣♦s✐t✐✈❡ ❧✐t❡r❛❧ t❤❡♥ C ✐s s❛t✐s✜❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ❧❡❛st p ❬✳✳✳❪✑ ✭t❤❡ r❡♠❛✐♥✐♥❣ ❧✐t❡r❛❧s ❝❛♥ ❜❡ ✐❣♥♦r❡❞✦✮✳ ✏❖t❤❡r✇✐s❡ C ❝♦♥t❛✐♥s t✇♦ ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧s ❛♥❞ C ✐s s❛t✐s✜❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ❧❡❛st 1 − p2 ❬✳✳✳❪✑ ✭t❤❡ r❡♠❛✐♥✐♥❣ ❧✐t❡r❛❧s ❛r❡ ✐❣♥♦r❡❞✦✮✳ ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r ❛t ♠♦st t✇♦ ✈❛r✐❛❜❧❡s ❛t t❤❡ s❛♠❡ t✐♠❡✱ ♣❛✐r✇✐s❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ✐s s✉✣❝✐❡♥t✳
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 8
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✽
✷✳✶✹✿ ❙✉♣r❡♠✉♠ ✈s ▼❛①✐♠✉♠
▲❡t (F, µ) ❜❡ ❛ ❣✐✈❡♥ ❈◆❋ ❢♦r♠✉❧❛ ❢♦r ✇❤✐❝❤ ✇❡ ❛✐♠ t♦ ✜♥❞ ❛♥ ❛s✲ s✐❣♥♠❡♥t α s✉❝❤ t❤❛t µ[α](F) ≥ rkµ(F)✳ ǫ := minS,T⊆F,µ(S)=µ(T)
|µ(S)−µ(T)| µ(F)
> 0✳ ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ rk t❤❡r❡ ✐s ❛♥ ❛ss✐❣♥♠❡♥t α s✉❝❤ t❤❛t µ[α](F) ≥ (rk − ǫ)µ(F)✳ ❊✐t❤❡r α ❛❧r❡❛❞② s❛t✐s✜❡s ❝❧❛✉s❡s ♦❢ ✇❡✐❣❤t ❛t ❧❡❛st rkµ(F)✱ ♦r µ[α](F) < rkµ(F)✳ ■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ ❛❣❛✐♥ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ rk✱ t❤❡r❡ ♠✉st ❜❡ ❛♥ ❛ss✐❣♥♠❡♥t β t❤❛t s❛t✐s✜❡s ♠♦r❡ t❤❛♥ t❤❡ ✇❡✐❣❤t ♦❢ ❝❧❛✉s❡s s❛t✐s✜❡❞ ❜② α ✐♥ F✳ ❇✉t ❛♥② s✉❝❤ ❛ss✐❣♥♠❡♥t s❛t✐s✜❡s ❝❧❛✉s❡s ♦❢ ✇❡✐❣❤t ❛t ❧❡❛st rkµ(F) ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ǫ✳
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 9
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✾
✸✳✷✿ ✭■♥ ❝❧❛ss✮ ■♠♣❧❡♠❡♥t✐♥❣ ❘❡s♦❧✉t✐♦♥ t♦ ❉❡❝✐❞❡ ✷✲❙❆❚
❋♦r ❛ ❣✐✈❡♥ ❢♦r♠✉❧❛ F ✇✐t❤ m ❝❧❛✉s❡s ❛♥❞ n ✈❛r✐❛❜❧❡s✱ ✜rst ♣❡r❢♦r♠ ✉♥✐t ❝❧❛✉s❡ r❡❞✉❝t✐♦♥ t♦ ❡♥s✉r❡ t❤❛t ❛❧❧ ❝❧❛✉s❡s ❛r❡ ♦❢ s✐③❡ 2✳ ❈♦♥str✉❝t t❤❡ ❞✐r❡❝t❡❞ ✐♠♣❧✐❝❛t✐♦♥ ❣r❛♣❤✱ G(F) = (V, E) ✇✐t❤ 2n ❧✐t❡r❛❧ ✈❡rt✐❝❡s ❛♥❞ 2m ❝❧❛✉s❡ ❡❞❣❡s ❛s ❢♦❧❧♦✇s✿ ❢♦r ❡❛❝❤ ❝❧❛✉s❡ {u, v} ∈ F ❛❞❞ t✇♦ ❞✐r❡❝t❡❞ ❡❞❣❡s ❢r♦♠ u t♦ v ❛♥❞ v t♦ u✳ ▲❡♠♠❛✿ F ✐s s❛t✐s✜❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ✐s ♥♦ ♣❛✐r ♦❢ ❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s s✉❝❤ t❤❛t ❜♦t❤ ❛r❡ r❡❛❝❤❛❜❧❡ ❢r♦♠ ❡❛❝❤ ♦t❤❡r ✐♥ G(F)✳ ❘✉♥♥✐♥❣ t✐♠❡ ♦❢ 2n ❜r❡❛❞t❤ ✜rst s❡❛r❝❤ ♦♣❡r❛t✐♦♥s ❂ O(n3)✳
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 10
❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✶✵
✭■♥ ❝❧❛ss✮ ❘❡s♦❧✉t✐♦♥ ❊①❡r❝✐s❡
▲❡t F ❜❡ ❛ ❈◆❋ ❢♦r♠✉❧❛✳ ▲❡t G ❜❡ ❛ ❈◆❋ ❢♦r♠✉❧❛ s✳t✳ ❢♦r ❛♥② ❝❧❛✉s❡ C ∈ F t❤❡r❡ ❡①✐sts ❛ ❝❧❛✉s❡ D ∈ G ✇✐t❤ D ⊆ C✳ ◆♦t❡✿ G ❤❛s ♥♦ ♠♦r❡ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥ts t❤❛♥ F ✭♦✈❡r vbl(F) ∪ vbl(G)✮ ❙❤♦✇✿ ■❢ t❤❡r❡ ✐s ❛ r❡s♦❧✉t✐♦♥ ❞❡❞✉❝t✐♦♥ ♦❢ C′ ❢r♦♠ F✱ t❤❡♥ t❤❡r❡ ✐s ❛ r❡s♦❧✉t✐♦♥ ❞❡❞✉❝t✐♦♥ ♦❢ D′ ❢r♦♠ G ✇✐t❤ D′ ⊆ C′
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
SLIDE 11 ❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✶✶
✭■♥ ❝❧❛ss✮ ❘❡s♦❧✉t✐♦♥ ❊①❡r❝✐s❡ ✭✷✮
▲❡t (C0, . . . , Cm) = C′ ❜❡ ❛ r❡s♦❧✉t✐♦♥ ❞❡❞✉❝t✐♦♥ ♦❢ C′ ❢r♦♠ F✳ ◆♦✇ ✇❡ ❜✉✐❧❞ ❛ r❡s♦❧✉t✐♦♥ ❞❡❞✉❝t✐♦♥ (D0, . . . , Dm = D′) ❢♦r D′ ⊆ C′✳ ❋♦r i = 0, . . . , m✱ ❝❤♦♦s❡ Di ⊆ Ci ❛s ❢♦❧❧♦✇s✿
- ■❢ Ci ∈ F✱ t❤❡♥ ❧❡t Di ∈ G ✇✐t❤ Di ⊆ Ci✳
- ❖t❤❡r✇✐s❡ Ci ✐s t❤❡ r❡s♦❧✈❡♥t ♦❢ Cj ❛♥❞ Ck ✇✐t❤ j < k < i✳ ❆ss✉♠❡
✇✳❧✳♦✳❣✳ Cj = {x} ˙ ∪ C′
j ❛♥❞ Ck = {¯
x} ˙ ∪ C′
k✱ s♦ Ci = C′ j ∪ C′ k✳
■❢ x ∈ Dj ❛♥❞ ¯ x ∈ Dk✱ t❤❡♥ ❧❡t Di ❜❡ t❤❡ r❡s♦❧✈❡♥t ♦❢ Dj ❛♥❞ Dk✳ ❖t❤❡r✇✐s❡ ❡✐t❤❡r Dj ♦r Dk ❞♦❡s ♥♦t ❝♦♥t❛✐♥ t❤❡ ✈❛r✐❛❜❧❡ x✳ ▲❡t Di ❜❡ t❤✐s ❝❧❛✉s❡✳
❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐
