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Pr t trr rs - - PowerPoint PPT Presentation

Nov 01, 2023 38 likes •769 views

Pr t trr rs Prst s t r Pr r

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SLIDE 1

P♦✇❡r ❉♦♠✐♥❛t✐♦♥ ✐♥ tr✐❛♥❣✉❧❛r ❣r✐❞s

Pr♦s❡♥❥✐t ❇♦s❡✶ ❱❛❧❡♥t✐♥ ●❧❡❞❡❧✷ ❈❧❛✐r❡ P❡♥♥❛r✉♥✸ ❙❛♥❞❡r ❱❡r❞♦♥s❝❤♦t✶

✶❙❝❤♦♦❧ ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❈❛r❧❡t♦♥ ❯♥✐✈❡rs✐t②✱ ❈❛♥❛❞❛ ✷▲■❘■❙✱ ❯♥✐✈❡rs✐té ▲②♦♥ ✶✱ ❋r❛♥❝❡ ✸▲■❘▼▼✱ ❈◆❘❙ ✫ ❯♥✐✈✳ ▼♦♥t♣❡❧❧✐❡r✱ ❋r❛♥❝❡ ✶✴✷✵

slide-2
SLIDE 2

❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S✳

❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥

✷

✷✴✷✵

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SLIDE 3

❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S✳

❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥

✷

✷✴✷✵

slide-4
SLIDE 4

❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S✳

❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥

✷

✷✴✷✵

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SLIDE 5

❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S✳

❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ γ(G)✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥ G

✷

✷✴✷✵

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SLIDE 6

❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ S ❞♦♠✐♥❛t❡s G ✐❢ ❛❧❧ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ✐♥ S ♦r ❛❞❥❛❝❡♥t t♦ ❛ ✈❡rt❡① ♦❢ S✳

❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ γ(G)✱ t❤❡ s✐③❡ ♦❢ ❛ ♠✐♥✐♠✉♠ ❞♦♠✐♥❛t✐♥❣ s❡t ✐♥ G

γ(G) = ✷

✷✴✷✵

slide-7
SLIDE 7

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N[S]✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ (uv) ∈ E ❛♥❞ N[u] \ {v} ⊆ M✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✶

✸✴✷✵

slide-8
SLIDE 8

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N[S]✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ (uv) ∈ E ❛♥❞ N[u] \ {v} ⊆ M✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✶

✸✴✷✵

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SLIDE 9

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N[S]✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ (uv) ∈ E ❛♥❞ N[u] \ {v} ⊆ M✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✶

✸✴✷✵

slide-10
SLIDE 10

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N[S]✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ (uv) ∈ E ❛♥❞ N[u] \ {v} ⊆ M✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✶

✸✴✷✵

slide-11
SLIDE 11

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N[S]✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ (uv) ∈ E ❛♥❞ N[u] \ {v} ⊆ M✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✶

✸✴✷✵

slide-12
SLIDE 12

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N[S]✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ (uv) ∈ E ❛♥❞ N[u] \ {v} ⊆ M✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ ✶

✸✴✷✵

slide-13
SLIDE 13

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

▲❡t G = (V , E) ❜❡ ❛ ❣r❛♣❤ ❛♥❞ S ⊆ V ✳ ❆t ✜rst M = N[S]✳ ❆ ✈❡rt❡① u ♣r♦♣❛❣❛t❡s t♦ ❛ ✈❡rt❡① v ✐❢ (uv) ∈ E ❛♥❞ N[u] \ {v} ⊆ M✳ S ✐s ❛ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ♦❢ G ✐❢ ❛t s♦♠❡ ♣♦✐♥t M = V ✳ γP(G) = ✶

✸✴✷✵

slide-14
SLIDE 14

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

  • ■♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠♦♥✐t♦r✐♥❣ ♣♦✇❡r ❣r✐❞s

◮ ▼✐❧✐✱ ❇❛❧❞✇✐♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✵✮ ◮ ❇❛❧❞✇✐♥✱ ▼✐❧✐✱ ❇♦✐s❡♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✸✮

❘❡❢♦r♠✉❧❛t❡❞ ✐♥ ❣r❛♣❤ t❡r♠s ❛♥❞ ♣r♦✈❡♥ t♦ ❜❡ ✲❝♦♠♣❧❡t❡

❍❛②♥❡s✱ ❍❡❞❡t♥✐❡♠✐✱ ❍❡❞❡t♥✐❡♠✐ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮

❙♦❧✈❡❞ ♦♥ sq✉❛r❡ ❣r✐❞s ❛♥❞ ♦t❤❡r ♣r♦❞✉❝ts ♦❢ ♣❛t❤s

❉♦r✐♥❣ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✻✮ ❉♦r❜❡❝✱ ▼♦❧❧❛r❞✱ ❑❧❛✈➸❛r ❛♥❞ ➆♣❛❝❛♣❛♥ ✭✷✵✵✽✮

❙♦❧✈❡❞ ♦♥ ❤❡①❛❣♦♥❛❧ ❣r✐❞s

❋❡rr❡r♦✱ ❱❛r❣❤❡s❡ ❛♥❞ ❱✐❥❛②❛❦✉♠❛ ✭✷✵✶✶✮

✹✴✷✵

slide-15
SLIDE 15

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

  • ■♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠♦♥✐t♦r✐♥❣ ♣♦✇❡r ❣r✐❞s

◮ ▼✐❧✐✱ ❇❛❧❞✇✐♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✵✮ ◮ ❇❛❧❞✇✐♥✱ ▼✐❧✐✱ ❇♦✐s❡♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✸✮

  • ❘❡❢♦r♠✉❧❛t❡❞ ✐♥ ❣r❛♣❤ t❡r♠s ❛♥❞ ♣r♦✈❡♥ t♦ ❜❡ NP✲❝♦♠♣❧❡t❡

◮ ❍❛②♥❡s✱ ❍❡❞❡t♥✐❡♠✐✱ ❍❡❞❡t♥✐❡♠✐ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮

❙♦❧✈❡❞ ♦♥ sq✉❛r❡ ❣r✐❞s ❛♥❞ ♦t❤❡r ♣r♦❞✉❝ts ♦❢ ♣❛t❤s

❉♦r✐♥❣ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✻✮ ❉♦r❜❡❝✱ ▼♦❧❧❛r❞✱ ❑❧❛✈➸❛r ❛♥❞ ➆♣❛❝❛♣❛♥ ✭✷✵✵✽✮

❙♦❧✈❡❞ ♦♥ ❤❡①❛❣♦♥❛❧ ❣r✐❞s

❋❡rr❡r♦✱ ❱❛r❣❤❡s❡ ❛♥❞ ❱✐❥❛②❛❦✉♠❛ ✭✷✵✶✶✮

✹✴✷✵

slide-16
SLIDE 16

P♦✇❡r ❉♦♠✐♥❛t✐♦♥

  • ■♥tr♦❞✉❝❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♠♦♥✐t♦r✐♥❣ ♣♦✇❡r ❣r✐❞s

◮ ▼✐❧✐✱ ❇❛❧❞✇✐♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✵✮ ◮ ❇❛❧❞✇✐♥✱ ▼✐❧✐✱ ❇♦✐s❡♥ ❛♥❞ ❆❞❛♣❛ ✭✶✾✾✸✮

  • ❘❡❢♦r♠✉❧❛t❡❞ ✐♥ ❣r❛♣❤ t❡r♠s ❛♥❞ ♣r♦✈❡♥ t♦ ❜❡ NP✲❝♦♠♣❧❡t❡

◮ ❍❛②♥❡s✱ ❍❡❞❡t♥✐❡♠✐✱ ❍❡❞❡t♥✐❡♠✐ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✷✮

  • ❙♦❧✈❡❞ ♦♥ sq✉❛r❡ ❣r✐❞s ❛♥❞ ♦t❤❡r ♣r♦❞✉❝ts ♦❢ ♣❛t❤s

◮ ❉♦r✐♥❣ ❛♥❞ ❍❡♥♥✐♥❣ ✭✷✵✵✻✮ ◮ ❉♦r❜❡❝✱ ▼♦❧❧❛r❞✱ ❑❧❛✈➸❛r ❛♥❞ ➆♣❛❝❛♣❛♥ ✭✷✵✵✽✮

  • ❙♦❧✈❡❞ ♦♥ ❤❡①❛❣♦♥❛❧ ❣r✐❞s

◮ ❋❡rr❡r♦✱ ❱❛r❣❤❡s❡ ❛♥❞ ❱✐❥❛②❛❦✉♠❛ ✭✷✵✶✶✮ ✹✴✷✵

slide-17
SLIDE 17

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ Hk

▲❡t Hk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Hk) = k ✸

  • ✳

❚❤❡♦r❡♠

k − ✶

✺✴✷✵

slide-18
SLIDE 18

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ Hk

▲❡t Hk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Hk) = k ✸

  • ✳

❚❤❡♦r❡♠

k − ✶

✺✴✷✵

slide-19
SLIDE 19

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ Hk

▲❡t Hk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Hk) = k ✸

  • ✳

❚❤❡♦r❡♠

k − ✶

✺✴✷✵

slide-20
SLIDE 20

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ Hk

▲❡t Hk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Hk) = k ✸

  • ✳

❚❤❡♦r❡♠

k − ✶

✺✴✷✵

slide-21
SLIDE 21

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ Hk

▲❡t Hk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Hk) = k ✸

  • ✳

❚❤❡♦r❡♠

k − ✶

✺✴✷✵

slide-22
SLIDE 22

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ Hk

▲❡t Hk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Hk) = k ✸

  • ✳

❚❤❡♦r❡♠

k − ✶

✺✴✷✵

slide-23
SLIDE 23

❘❡s✉❧t ♦♥ ❤❡①❛❣♦♥❛❧ s❤❛♣❡❞ ❣r✐❞ Hk

▲❡t Hk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛ r❡❣✉❧❛r ❤❡①❛❣♦♥❛❧✲s❤❛♣❡❞ ❜♦r✲ ❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Hk) = k ✸

  • ✳

❚❤❡♦r❡♠

k − ✶

✺✴✷✵

slide-24
SLIDE 24

❘❡s✉❧t ♦♥ tr✐❛♥❣✉❧❛r s❤❛♣❡❞ ❣r✐❞ Tk

▲❡t Tk ❜❡ ❛ tr✐❛♥❣✉❧❛r ❣r✐❞ ✇✐t❤ ❛♥ ❡q✉✐❧❛t❡r❛❧ tr✐❛♥❣✉❧❛r✲s❤❛♣❡❞ ❜♦r❞❡r ♦❢ ❧❡♥❣t❤ k − ✶✳ ❚❤❡♥✱ γP(Tk) = k ✹

  • ✳

❚❤❡♦r❡♠

k − ✶

✻✴✷✵

slide-25
SLIDE 25

▲♦✇❡r ❇♦✉♥❞

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦❧❧♦✇s t❤❡s❡ st❡♣s ✿ ❲❡ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ ♦♥ ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ✶✷ ❲❡ s❤♦✇ t❤❛t ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❡♥❞✱ ✐❢ t❤❡ ❣r✐❞ ✐s ❢✉❧❧② ♠♦♥✐t♦r❡❞✱ ✸ ❚❤✐s ♣r♦✈❡ t❤❛t ✇❡ ♠✉st ❤❛✈❡

✹

✼✴✷✵

slide-26
SLIDE 26

▲♦✇❡r ❇♦✉♥❞

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦❧❧♦✇s t❤❡s❡ st❡♣s ✿

  • ❲❡ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ Q ♦♥ M

❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ✶✷ ❲❡ s❤♦✇ t❤❛t ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❡♥❞✱ ✐❢ t❤❡ ❣r✐❞ ✐s ❢✉❧❧② ♠♦♥✐t♦r❡❞✱ ✸ ❚❤✐s ♣r♦✈❡ t❤❛t ✇❡ ♠✉st ❤❛✈❡

✹

✼✴✷✵

slide-27
SLIDE 27

▲♦✇❡r ❇♦✉♥❞

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦❧❧♦✇s t❤❡s❡ st❡♣s ✿

  • ❲❡ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ Q ♦♥ M
  • ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ Q(N[S]) ≤ ✶✷|S|

❲❡ s❤♦✇ t❤❛t ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❡♥❞✱ ✐❢ t❤❡ ❣r✐❞ ✐s ❢✉❧❧② ♠♦♥✐t♦r❡❞✱ ✸ ❚❤✐s ♣r♦✈❡ t❤❛t ✇❡ ♠✉st ❤❛✈❡

✹

✼✴✷✵

slide-28
SLIDE 28

▲♦✇❡r ❇♦✉♥❞

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦❧❧♦✇s t❤❡s❡ st❡♣s ✿

  • ❲❡ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ Q ♦♥ M
  • ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ Q(N[S]) ≤ ✶✷|S|
  • ❲❡ s❤♦✇ t❤❛t Q ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣

❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❡♥❞✱ ✐❢ t❤❡ ❣r✐❞ ✐s ❢✉❧❧② ♠♦♥✐t♦r❡❞✱ ✸ ❚❤✐s ♣r♦✈❡ t❤❛t ✇❡ ♠✉st ❤❛✈❡

✹

✼✴✷✵

slide-29
SLIDE 29

▲♦✇❡r ❇♦✉♥❞

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦❧❧♦✇s t❤❡s❡ st❡♣s ✿

  • ❲❡ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ Q ♦♥ M
  • ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ Q(N[S]) ≤ ✶✷|S|
  • ❲❡ s❤♦✇ t❤❛t Q ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣
  • ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❡♥❞✱ ✐❢ t❤❡ ❣r✐❞ ✐s ❢✉❧❧② ♠♦♥✐t♦r❡❞✱

Q(M) = ✸k ❚❤✐s ♣r♦✈❡ t❤❛t ✇❡ ♠✉st ❤❛✈❡

✹

✼✴✷✵

slide-30
SLIDE 30

▲♦✇❡r ❇♦✉♥❞

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❢♦❧❧♦✇s t❤❡s❡ st❡♣s ✿

  • ❲❡ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ Q ♦♥ M
  • ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ Q(N[S]) ≤ ✶✷|S|
  • ❲❡ s❤♦✇ t❤❛t Q ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣
  • ❲❡ s❤♦✇ t❤❛t ❛t t❤❡ ❡♥❞✱ ✐❢ t❤❡ ❣r✐❞ ✐s ❢✉❧❧② ♠♦♥✐t♦r❡❞✱

Q(M) = ✸k ❚❤✐s ♣r♦✈❡ t❤❛t ✇❡ ♠✉st ❤❛✈❡ |S| ≥ k

✹

✼✴✷✵

slide-31
SLIDE 31

❚✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s

  • ❆♥ ❡❞❣❡ (uv) ✐s ❛ t✐♣ ❡❞❣❡ ✐❢ u ❛♥❞ v ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r

♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t✐♣ ✐s ♥♦t✳ ❆♥ ❡❞❣❡ ✐s ❛ ❜❛s❡ ❡❞❣❡ ✐❢ ❛♥❞ ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r ♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❜❛s❡ ✐s ♥♦t✳ u v

✽✴✷✵

slide-32
SLIDE 32

❚✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s

  • ❆♥ ❡❞❣❡ (uv) ✐s ❛ t✐♣ ❡❞❣❡ ✐❢ u ❛♥❞ v ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r

♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t✐♣ ✐s ♥♦t✳

  • ❆♥ ❡❞❣❡ (uv) ✐s ❛ ❜❛s❡ ❡❞❣❡ ✐❢ u ❛♥❞ v ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r

♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❜❛s❡ ✐s ♥♦t✳ u v

✽✴✷✵

slide-33
SLIDE 33

❚✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s

  • ❆♥ ❡❞❣❡ (uv) ✐s ❛ t✐♣ ❡❞❣❡ ✐❢ u ❛♥❞ v ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r

♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t✐♣ ✐s ♥♦t✳

  • ❆♥ ❡❞❣❡ (uv) ✐s ❛ ❜❛s❡ ❡❞❣❡ ✐❢ u ❛♥❞ v ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r

♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❜❛s❡ ✐s ♥♦t✳ u v

✽✴✷✵

slide-34
SLIDE 34

❚✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s

  • ❆♥ ❡❞❣❡ (uv) ✐s ❛ t✐♣ ❡❞❣❡ ✐❢ u ❛♥❞ v ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r

♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ t✐♣ ✐s ♥♦t✳

  • ❆♥ ❡❞❣❡ (uv) ✐s ❛ ❜❛s❡ ❡❞❣❡ ✐❢ u ❛♥❞ v ❛r❡ ♠♦♥✐t♦r❡❞ ❜✉t t❤❡✐r

♥❡✐❣❤❜♦r ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ t❤❡ ❜❛s❡ ✐s ♥♦t✳ u v

✽✴✷✵

slide-35
SLIDE 35

❍♦❧❡s

  • ❆ ❤♦❧❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ V \ (M) t❤❛t ❞♦❡s ♥♦t

❝♦♥t❛✐♥ ♣♦✐♥ts ♦❢ t❤❡ ❜♦r❞❡r ♦❢ t❤❡ ❣r✐❞✳ ❤♦❧❡

✾✴✷✵

slide-36
SLIDE 36

❍♦❧❡s

  • ❆ ❤♦❧❡ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ V \ (M) t❤❛t ❞♦❡s ♥♦t

❝♦♥t❛✐♥ ♣♦✐♥ts ♦❢ t❤❡ ❜♦r❞❡r ♦❢ t❤❡ ❣r✐❞✳ ❤♦❧❡

✾✴✷✵

slide-37
SLIDE 37

❚❤❡ q✉❛♥t✐t② Q

❲❡ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ Q ❛s ❢♦❧❧♦✇s ✿ Q(M) = ✷T + B + ✸C − ✸H ❲❤❡r❡ ✿

  • T ✐s t❤❡ ♥✉♠❜❡r ♦❢ t✐♣ ❡❞❣❡s
  • B ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❜❛s❡ ❡❞❣❡s
  • C ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ♦❢ M
  • H ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❤♦❧❡s

✶✵✴✷✵

slide-38
SLIDE 38

❆t t❤❡ ❡♥❞

❲❡ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ Q ✇❤❡♥ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞ ✿ k − ✶ Q = ✸(k − ✶) × ✶ + ✸ = ✸k ❲❤❛t r❡♠❛✐♥s t♦ ❞♦ ✿ Pr♦✈✐♥❣ t❤❛t ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❋✐♥❞✐♥❣ t❤❡ st❛rt✐♥❣ ✈❛❧✉❡ ♦❢ ✇✐t❤ r❡s♣❡❝t t♦

✶✶✴✷✵

slide-39
SLIDE 39

❆t t❤❡ ❡♥❞

❲❡ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ Q ✇❤❡♥ ❛❧❧ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞ ✿ k − ✶ Q = ✸(k − ✶) × ✶ + ✸ = ✸k ❲❤❛t r❡♠❛✐♥s t♦ ❞♦ ✿

  • Pr♦✈✐♥❣ t❤❛t Q ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣
  • ❋✐♥❞✐♥❣ t❤❡ st❛rt✐♥❣ ✈❛❧✉❡ ♦❢ Q ✇✐t❤ r❡s♣❡❝t t♦ S

✶✶✴✷✵

slide-40
SLIDE 40

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

Q′ = Q − ✷ − ✶ + . . . ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-41
SLIDE 41

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ Q′ = Q − ✷ − ✶ + ✷ + ✶ = Q ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-42
SLIDE 42

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ Q′ = Q − ✷ × ✷ − ✶ + . . . ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-43
SLIDE 43

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ Q′ = Q − ✷ × ✷ − ✶ + ✷ × ✶ = Q − ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-44
SLIDE 44

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ Q′ = Q − ✷ − ✶ + . . . ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-45
SLIDE 45

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ Q′ = Q − ✷ − ✶ + ✷ × ✷ + ✷ × ✶ . . . = Q ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-46
SLIDE 46

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ Q′ = Q − ✷ − ✶ + ✷ × ✷ + ✷ × ✶ − ✸ = Q ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-47
SLIDE 47

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜

×

❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ Q′ = Q − ✷ − ✶ + . . . ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✷ ✶

✶✷✴✷✵

slide-48
SLIDE 48

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t

×✲❝

t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ Q′ = Q − ✷ − ✶ + ✷ × ✷ + ✷ × ✶ − ✸ = Q ✷ ✷ ✶

✶✷✴✷✵

slide-49
SLIDE 49

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ Q′ = Q − ✷ + . . . ✷ ✷ ✶

✶✷✴✷✵

slide-50
SLIDE 50

Q ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣

Q ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ ✇❤❡♥ ♥❡✇ ✈❡rt✐❝❡s ❛r❡ ♠♦♥✐t♦r❡❞✳ ▲❡♠♠❛ ❲❡ ♣r♦✈❡ t❤✐s st❛t❡♠❡♥t ❜② ❧♦♦❦✐♥❣ ❛t ❡✈❡r② ❝❛s❡✿

t ❜ ❜ t t ❜ t ❜ ❜ t ❜ ❜ t ❜ t ❤ t ❜ ❜ t ❜ t ✲❝ t ❜ ❜

✷ ✶ ✷ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✸ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ ✶ ✷ ✷ ✷ ✶ ✸ ✷ Q′ = Q − ✷ + ✷ × ✶ = Q

✶✷✴✷✵

slide-51
SLIDE 51

❙t❛rt✐♥❣ ✈❛❧✉❡ ♦❢ Q

❆t t❤❡ ❜❡❣✐♥♥✐♥❣✱ Q(N[S]) ≤ ✶✷|S| ▲❡♠♠❛ ❲❡ s✉♣♣♦s❡ ❤❡r❡ t❤❛t N[S] ✐s ❝♦♥♥❡❝t❡❞✳ ❲❡ ❞❡✜♥❡ GS = (VS, ES) ❛s ❢♦❧❧♦✇s ✿

  • VS = S
  • (xy) ∈ ES ✐❢ x ❛♥❞ y ❢♦r♠ ❛ ❜r✐❞❣❡ ♦r ❛ ❞♦✉❜❧❡✲❜r✐❞❣❡ ✿

x y x y

❜r✐❞❣❡ ❞♦✉❜❧❡✲❜r✐❞❣❡

✶✸✴✷✵

slide-52
SLIDE 52

❙t❛rt✐♥❣ ✈❛❧✉❡ ♦❢ Q

GS ✐s ♣❧❛♥❛r ▲❡♠♠❛ ❲❡ ❝❛♥ ❛♣♣❧② ❊✉❧❡r✬s ❢♦r♠✉❧❛✿ |ES| − f (GS) + ✶ = |VS| − c(GS) ❲❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t✿ ✶ ✶ ✶

✶✹✴✷✵

slide-53
SLIDE 53

❙t❛rt✐♥❣ ✈❛❧✉❡ ♦❢ Q

GS ✐s ♣❧❛♥❛r ▲❡♠♠❛ ❲❡ ❝❛♥ ❛♣♣❧② ❊✉❧❡r✬s ❢♦r♠✉❧❛✿ |ES| − f (GS) + ✶ = |VS| − c(GS) ❲❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t✿

  • |VS| = |S|
  • c(GS) ≥ ✶
  • f (GS) − ✶ ≤ H

✶

✶✹✴✷✵

slide-54
SLIDE 54

❙t❛rt✐♥❣ ✈❛❧✉❡ ♦❢ Q

GS ✐s ♣❧❛♥❛r ▲❡♠♠❛ ❲❡ ❝❛♥ ❛♣♣❧② ❊✉❧❡r✬s ❢♦r♠✉❧❛✿ |ES| − f (GS) + ✶ = |VS| − c(GS) ❲❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t✿

  • |VS| = |S|
  • c(GS) ≥ ✶
  • f (GS) − ✶ ≤ H

|ES| − H + ✶ ≤ |S|

✶✹✴✷✵

slide-55
SLIDE 55

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆t t❤❡ ❜❡❣✐♥♥✐♥❣✱ ✷T + B ≤ ✾|S| + ✸|ES| ▲❡♠♠❛ ❲❡ ❣✐✈❡✿

  • ❛ ✇❡✐❣❤t ♦❢ ✾ t♦ ❡❛❝❤ ✈❡rt❡① ♦❢ S
  • ❛ ✇❡✐❣❤t ♦❢ ✸ t♦ ❡❛❝❤ ❜r✐❞❣❡ ❛♥❞ ❞♦✉❜❧❡✲❜r✐❞❣❡

❆t t❤❡ ❡♥❞ ✇❡ ✇❛♥t✿ ❆ ✇❡✐❣❤t ♦❢ ✷ ♦♥ ❡❛❝❤ t✐♣ ❡❞❣❡ ❆ ✇❡✐❣❤t ♦❢ ✶ ♦♥ ❡❛❝❤ ❜❛s❡ ❡❞❣❡ ❆ ♥♦♥✲♥❡❣❛t✐✈❡ ✇❡✐❣❤t ♦♥ ❡❛❝❤ ✈❡rt❡①

✶✺✴✷✵

slide-56
SLIDE 56

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆t t❤❡ ❜❡❣✐♥♥✐♥❣✱ ✷T + B ≤ ✾|S| + ✸|ES| ▲❡♠♠❛ ❲❡ ❣✐✈❡✿

  • ❛ ✇❡✐❣❤t ♦❢ ✾ t♦ ❡❛❝❤ ✈❡rt❡① ♦❢ S
  • ❛ ✇❡✐❣❤t ♦❢ ✸ t♦ ❡❛❝❤ ❜r✐❞❣❡ ❛♥❞ ❞♦✉❜❧❡✲❜r✐❞❣❡

❆t t❤❡ ❡♥❞ ✇❡ ✇❛♥t✿

  • ❆ ✇❡✐❣❤t ♦❢ ✷ ♦♥ ❡❛❝❤ t✐♣ ❡❞❣❡
  • ❆ ✇❡✐❣❤t ♦❢ ✶ ♦♥ ❡❛❝❤ ❜❛s❡ ❡❞❣❡
  • ❆ ♥♦♥✲♥❡❣❛t✐✈❡ ✇❡✐❣❤t ♦♥ ❡❛❝❤ ✈❡rt❡①

✶✺✴✷✵

slide-57
SLIDE 57

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

  • ■❢ u ✐s ✐♥ S✱ t❤❡♥ ✐t ❣✐✈❡s ❛ ✇❡✐❣❤t ♦❢ ✶✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts

♥❡✐❣❤❜♦rs ■❢ ✐s ✐♥❝✐❞❡♥t t♦ ❛ t✐♣ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✶ ■❢ ✐s ✐♥❝✐❞❡♥t t♦ ❛ ❜❛s❡ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✵✳✺ ❖t❤❡r✇✐s❡✱ ✐t ❣✐✈❡s ✵✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts ♥❡✐❣❤❜♦rs t❤❛t ✐t s❤❛r❡s ✇✐t❤ ❛ ✈❡rt❡① ♦❢ ✳ ❇r✐❞❣❡s ❛♥❞ ❞♦✉❜❧❡✲❜r✐❞❣❡s ❣✐✈❡ ✷ t♦ t❤❡✐r t✐♣ ❡❞❣❡ ❛♥❞ ✶ t♦ t❤❡✐r ❜❛s❡ ❡❞❣❡

✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶ ✶ ✵✳✺ ✵✳✺ ✵✳✺ ✵✳✺ ✷ ✶ ✷ ✶ ✶✻✴✷✵

slide-58
SLIDE 58

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

  • ■❢ u ✐s ✐♥ S✱ t❤❡♥ ✐t ❣✐✈❡s ❛ ✇❡✐❣❤t ♦❢ ✶✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts

♥❡✐❣❤❜♦rs

  • ■❢ u ✐s ✐♥❝✐❞❡♥t t♦ ❛ t✐♣ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✶

■❢ ✐s ✐♥❝✐❞❡♥t t♦ ❛ ❜❛s❡ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✵✳✺ ❖t❤❡r✇✐s❡✱ ✐t ❣✐✈❡s ✵✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts ♥❡✐❣❤❜♦rs t❤❛t ✐t s❤❛r❡s ✇✐t❤ ❛ ✈❡rt❡① ♦❢ ✳ ❇r✐❞❣❡s ❛♥❞ ❞♦✉❜❧❡✲❜r✐❞❣❡s ❣✐✈❡ ✷ t♦ t❤❡✐r t✐♣ ❡❞❣❡ ❛♥❞ ✶ t♦ t❤❡✐r ❜❛s❡ ❡❞❣❡

✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶ ✶ ✵✳✺ ✵✳✺ ✵✳✺ ✵✳✺ ✷ ✶ ✷ ✶ ✶✻✴✷✵

slide-59
SLIDE 59

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

  • ■❢ u ✐s ✐♥ S✱ t❤❡♥ ✐t ❣✐✈❡s ❛ ✇❡✐❣❤t ♦❢ ✶✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts

♥❡✐❣❤❜♦rs

  • ■❢ u ✐s ✐♥❝✐❞❡♥t t♦ ❛ t✐♣ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✶
  • ■❢ u ✐s ✐♥❝✐❞❡♥t t♦ ❛ ❜❛s❡ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✵✳✺

❖t❤❡r✇✐s❡✱ ✐t ❣✐✈❡s ✵✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts ♥❡✐❣❤❜♦rs t❤❛t ✐t s❤❛r❡s ✇✐t❤ ❛ ✈❡rt❡① ♦❢ ✳ ❇r✐❞❣❡s ❛♥❞ ❞♦✉❜❧❡✲❜r✐❞❣❡s ❣✐✈❡ ✷ t♦ t❤❡✐r t✐♣ ❡❞❣❡ ❛♥❞ ✶ t♦ t❤❡✐r ❜❛s❡ ❡❞❣❡

✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶ ✶ ✵✳✺ ✵✳✺ ✵✳✺ ✵✳✺ ✷ ✶ ✷ ✶ ✶✻✴✷✵

slide-60
SLIDE 60

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

  • ■❢ u ✐s ✐♥ S✱ t❤❡♥ ✐t ❣✐✈❡s ❛ ✇❡✐❣❤t ♦❢ ✶✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts

♥❡✐❣❤❜♦rs

  • ■❢ u ✐s ✐♥❝✐❞❡♥t t♦ ❛ t✐♣ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✶
  • ■❢ u ✐s ✐♥❝✐❞❡♥t t♦ ❛ ❜❛s❡ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✵✳✺
  • ❖t❤❡r✇✐s❡✱ ✐t ❣✐✈❡s ✵✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts ♥❡✐❣❤❜♦rs t❤❛t ✐t s❤❛r❡s

✇✐t❤ ❛ ✈❡rt❡① ♦❢ S✳ ❇r✐❞❣❡s ❛♥❞ ❞♦✉❜❧❡✲❜r✐❞❣❡s ❣✐✈❡ ✷ t♦ t❤❡✐r t✐♣ ❡❞❣❡ ❛♥❞ ✶ t♦ t❤❡✐r ❜❛s❡ ❡❞❣❡

✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶ ✶ ✵✳✺ ✵✳✺ ✵✳✺ ✵✳✺ ✷ ✶ ✷ ✶ ✶✻✴✷✵

slide-61
SLIDE 61

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

  • ■❢ u ✐s ✐♥ S✱ t❤❡♥ ✐t ❣✐✈❡s ❛ ✇❡✐❣❤t ♦❢ ✶✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts

♥❡✐❣❤❜♦rs

  • ■❢ u ✐s ✐♥❝✐❞❡♥t t♦ ❛ t✐♣ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✶
  • ■❢ u ✐s ✐♥❝✐❞❡♥t t♦ ❛ ❜❛s❡ ❡❞❣❡✱ t❤❡♥ ✐t ❣✐✈❡s ✐t ❛ ✇❡✐❣❤t ♦❢ ✵✳✺
  • ❖t❤❡r✇✐s❡✱ ✐t ❣✐✈❡s ✵✳✺ t♦ ❡❛❝❤ ♦❢ ✐ts ♥❡✐❣❤❜♦rs t❤❛t ✐t s❤❛r❡s

✇✐t❤ ❛ ✈❡rt❡① ♦❢ S✳

  • ❇r✐❞❣❡s ❛♥❞ ❞♦✉❜❧❡✲❜r✐❞❣❡s ❣✐✈❡ ✷ t♦ t❤❡✐r t✐♣ ❡❞❣❡ ❛♥❞ ✶ t♦

t❤❡✐r ❜❛s❡ ❡❞❣❡

✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶✳✺ ✶ ✶ ✵✳✺ ✵✳✺ ✵✳✺ ✵✳✺ ✷ ✶ x y ✷ ✶ x y ✶✻✴✷✵

slide-62
SLIDE 62

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

✶✼✴✷✵

slide-63
SLIDE 63

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ v ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

u v

✶✼✴✷✵

slide-64
SLIDE 64

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ × ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

u v

✶✼✴✷✵

slide-65
SLIDE 65

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ × ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

u v

✶✼✴✷✵

slide-66
SLIDE 66

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ × ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

u v

✶✼✴✷✵

slide-67
SLIDE 67

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

u v

✶✼✴✷✵

slide-68
SLIDE 68

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

u v

✶✼✴✷✵

slide-69
SLIDE 69

❯s❡ ♦❢ ❛ ❞✐s❝❤❛r❣✐♥❣ ♠❡t❤♦❞

❆❧❧ t✐♣ ❡❞❣❡s ❛♥❞ ❜❛s❡ ❡❞❣❡s ❤❛✈❡ t❤❡ ❣♦♦❞ ✇❡✐❣❤t✳ ❲❡ ❤❛✈❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t ♥♦ ✈❡rt❡① ❤❛s ❛ ♥❡❣❛t✐✈❡ ✇❡✐❣❤t✳ ❚❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐ss✉❡ ✐s ✇❤❡♥ ❛ ✈❡rt❡① ✐s ❛❞❥❛❝❡♥t t♦ t✇♦ t✐♣ ❡❞❣❡s✳

❄ ❄

✶✳✺ ✵✳✺

❄

✵✳✺

❖♥❡ ♦❢ t❤❡ ♥❡✐❣❤❜♦r ♦❢ ✐s ✐♥ ❙ ❞♦✉❜❧❡ ❜r✐❞❣❡

❄ ❄ ❄

u v

✶✼✴✷✵

slide-70
SLIDE 70

❙t❛rt✐♥❣ ✈❛❧✉❡ ♦❢ Q

❲❡ ❤❛✈❡ s❡❡♥ t❤❛t✿

  • ✷T + B ≤ ✾|S| + ✸|ES|
  • |ES| − H + ✶ ≤ |S|

s♦ ✷T + B ≤ ✾|S| + ✸|S| + ✸H − ✸ t❤✐s ✐s tr✉❡ ❢♦r ❡❛❝❤ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t s♦ Q = ✷T + B + ✸C − ✸H ≤ ✶✷|S|

✶✽✴✷✵

slide-71
SLIDE 71

❈♦♥❝❧✉s✐♦♥

  • ❆t t❤❡ ❜❡❣✐♥♥✐♥❣✱ Q ≤ ✶✷|S|
  • ❆t t❤❡ ❡♥❞✱ Q = ✸k
  • Q ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣

s♦ ✿ |S| ≥ k ✹ ❚❤✐s ❣✐✈❡s ✉s t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❛♥❞ ✇❡ ❝❛♥ r❡❛❝❤ ✐t s♦✿ ✹ ❚❤❡♦r❡♠

✶✾✴✷✵

slide-72
SLIDE 72

❈♦♥❝❧✉s✐♦♥

  • ❆t t❤❡ ❜❡❣✐♥♥✐♥❣✱ Q ≤ ✶✷|S|
  • ❆t t❤❡ ❡♥❞✱ Q = ✸k
  • Q ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣

s♦ ✿ |S| ≥ k ✹ ❚❤✐s ❣✐✈❡s ✉s t❤❡ ❧♦✇❡r ❜♦✉♥❞ ❛♥❞ ✇❡ ❝❛♥ r❡❛❝❤ ✐t s♦✿ γP(Tk) = k ✹

  • ❚❤❡♦r❡♠

✶✾✴✷✵

slide-73
SLIDE 73

✷✵✴✷✵