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trt rs s sqt - - PowerPoint PPT Presentation

Jun 19, 2023 137 likes •775 views

trt rs s sqt t t r r

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SLIDE 1

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

◆✐❝♦❧❛s ❇♦✉sq✉❡t✱ ❆♥t♦✐♥❡ ❉❛✐❧❧②✱ ❱❛❧❡♥t✐♥ ●❧❡❞❡❧ ❛♥❞ ▼❛r❝ ❍❡✐♥r✐❝❤

❯♥✐✈❡rs✐té ❞❡ ▲②♦♥

❈●❚❈ ✲ ❏❛♥✉❛r② t❤❡ ✷✼t❤✱ ✷✵✶✼

Games And Graphs

✶✴✷✼

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SLIDE 2

❆r❝✲❑❛②❧❡s

✷✴✷✼

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SLIDE 3

❆r❝✲❑❛②❧❡s

❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛

❣r❛♣❤ G = (V , E)✳

❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t

♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳

❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s

♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽

✸✴✷✼

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SLIDE 4

❆r❝✲❑❛②❧❡s

❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛

❣r❛♣❤ G = (V , E)✳

❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t

♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳

❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s

♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽

✸✴✷✼

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SLIDE 5

❆r❝✲❑❛②❧❡s

❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛

❣r❛♣❤ G = (V , E)✳

❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t

♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳

❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s

♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽

✸✴✷✼

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SLIDE 6

❆r❝✲❑❛②❧❡s

❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛

❣r❛♣❤ G = (V , E)✳

❼ ❆t ❡❛❝❤ t✉r♥ t❤❡ ❝✉rr❡♥t

♣❧❛②❡r ❝❤♦♦s❡s ❛♥ ❡❞❣❡✳

❼ ■ts ❡♥❞♣♦✐♥ts ❛r❡ ❞❡❧❡t❡❞✳ ❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s

♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ❆r❝✲❑❛②❧❡s ✿ ❙❝❤❛❡❢❡r✱ ✶✾✼✽

✸✴✷✼

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SLIDE 7

❆r❝✲❑❛②❧❡s

❼ ❙♦❧✈❡❞ ♦♥ ♣❛t❤s ✭●✉② ❛♥❞ ❙♠✐t❤✱ ✶✾✺✻✮ ❼ ❙♦❧✈❡❞ ♦♥ ❡q✉✐♠❛t❝❤❛❜❧❡ ❣r❛♣❤s✱ ❝②❝❧❡s✱ ✇❤❡❡❧s ❛♥❞

❣❡♥❡r❛❧✐③❡❞ st❛r ❣r❛♣❤s ✇✐t❤ t❤r❡❡ ❜r❛♥❝❤❡s ✭❍✉❣❣❛♥ ❛♥❞ ❙t❡✈❡♥s✱ ✷✵✶✻✮

❼ ❋P❚ ✭▲❛♠♣✐s ❛♥❞ ▼✐ts♦✉✱ ✷✵✶✹✮

✹✴✷✼

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SLIDE 8

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

✺✴✷✼

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SLIDE 9

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛

❣r❛♣❤ G = (V , E) ✇✐t❤ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ω : V → N✳

❼ ❆t ❡❛❝❤ st❡♣ t❤❡ ❝✉rr❡♥t

♣❧❛②❡r ❝❤♦♦s❡ ❛♥ ❡❞❣❡✳

❼ ❚❤❡ ✇❡✐❣❤t ♦❢ ❜♦t❤

❡♥❞♣♦✐♥ts ✐s ❞❡❝r❡❛s❡❞ ❜② ✶✳

❼ ❱❡rt✐❝❡s ✇✐t❤ ✇❡✐❣❤t ③❡r♦ ❛r❡

r❡♠♦✈❡❞✳

❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s

♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s m n ✶ ✶

✻✴✷✼

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SLIDE 10

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛

❣r❛♣❤ G = (V , E) ✇✐t❤ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ω : V → N✳

❼ ❆t ❡❛❝❤ st❡♣ t❤❡ ❝✉rr❡♥t

♣❧❛②❡r ❝❤♦♦s❡ ❛♥ ❡❞❣❡✳

❼ ❚❤❡ ✇❡✐❣❤t ♦❢ ❜♦t❤

❡♥❞♣♦✐♥ts ✐s ❞❡❝r❡❛s❡❞ ❜② ✶✳

❼ ❱❡rt✐❝❡s ✇✐t❤ ✇❡✐❣❤t ③❡r♦ ❛r❡

r❡♠♦✈❡❞✳

❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s

♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s m n ✶ ✶

✻✴✷✼

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SLIDE 11

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

❼ ❚❤✐s ❣❛♠❡ ✐s ♣❧❛②❡❞ ♦♥ ❛

❣r❛♣❤ G = (V , E) ✇✐t❤ ❛ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ω : V → N✳

❼ ❆t ❡❛❝❤ st❡♣ t❤❡ ❝✉rr❡♥t

♣❧❛②❡r ❝❤♦♦s❡ ❛♥ ❡❞❣❡✳

❼ ❚❤❡ ✇❡✐❣❤t ♦❢ ❜♦t❤

❡♥❞♣♦✐♥ts ✐s ❞❡❝r❡❛s❡❞ ❜② ✶✳

❼ ❱❡rt✐❝❡s ✇✐t❤ ✇❡✐❣❤t ③❡r♦ ❛r❡

r❡♠♦✈❡❞✳

❼ ❚❤❡ ❣❛♠❡ ❡♥❞s ✇❤❡♥ t❤❡r❡ ✐s

♥♦ ♠♦r❡ ❡❞❣❡s t♦ ♣❧❛②✳ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s m − ✶ n − ✶

✻✴✷✼

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SLIDE 12

❊①❛♠♣❧❡

✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✵ ✸ ✸

✼✴✷✼

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SLIDE 13

❊①❛♠♣❧❡

✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✵ ✸ ✸

✼✴✷✼

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SLIDE 14

❊①❛♠♣❧❡

✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✵ ✸ ✸

✼✴✷✼

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SLIDE 15

❊①❛♠♣❧❡

✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✵ ✸ ✸

✼✴✷✼

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SLIDE 16

❊①❛♠♣❧❡

✸ ✹ ✶ ✷ ✸ ✶ ✸ ✸ ✶ ✵ ✸ ✸

✼✴✷✼

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SLIDE 17

◆♦♥✲❛tt❛❝❦✐♥❣ r♦♦❦s ✐♥ ❛ ❤♦❧❡❞ ❝❤❡ss❜♦❛r❞

✽✴✷✼

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SLIDE 18

❚❤❡ ❣❛♠❡

✻ 0Z0Z0Z0Z ✺ Z0Z0S0Z0 ✹ 0Z0Z0Z0Z ✸ S0Z0Z0Z0 ✷ 0Z0Z0Z0Z ✶ Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤

■♥s♣✐r❡❞ ❜② t❤❡ ♥♦♥✲❛tt❛❝❦✐♥❣ q✉❡❡♥s ❣❛♠❡✭◆♦♦♥ ❛♥❞ ❱❛♥ ❇r✉♠♠❡❧❡♥✱ ✷✵✵✻✮

✾✴✷✼

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SLIDE 19

❚❤❡ ❣❛♠❡

✶✵0Z0Z0Z0Z0Z ✾ Z0Z0Z0Z0Z0 ✽ 0Z0Z0Z0Z0Z ✼ Z0Z0Z0Z0Z0 ✻ 0S0Z0Z0S0Z ✺ Z0Z0Z0Z0Z0 ✹ 0Z0Z0Z0Z0Z ✸ Z0Z0ZRZ0Z0 ✷ 0Z0Z0Z0Z0Z ✶ Z0Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥

✶✵✴✷✼

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SLIDE 20

❚❤❡ ❣❛♠❡

✶✵0Z0Z0Z0Z0Z ✾ Z0Z0Z0Z0Z0 ✽ 0Z0Z0Z0Z0Z ✼ Z0Z0Z0Z0Z0 ✻ 0S0Z0Z0S0Z ✺ Z0Z0Z0Z0Z0 ✹ 0Z0Z0Z0Z0Z ✸ Z0Z0ZRZ0Z0 ✷ 0Z0Z0Z0Z0Z ✶ Z0Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥

✸ ✷ ✷ ✷ ✷ ✹ ✸ ✸ ✷ ✷

✶✵✴✷✼

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SLIDE 21

❚❤❡ ❣❛♠❡

✶✵0Z0Z0Z0Z0Z ✾ Z0Z0Z0Z0Z0 ✽ 0Z0Z0Z0Z0Z ✼ Z0Z0Z0Z0Z0 ✻ 0S0Z0Z0S0Z ✺ Z0Z0Z0Z0Z0 ✹ 0Z0Z0Z0Z0Z ✸ Z0Z0ZRZ0Z0 ✷ 0Z0ZRZ0Z0Z ✶ Z0Z0Z0Z0Z0 ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥

✸ ✷ ✷ ✷ ✷ ✹ ✸ ✸ ✷ ✷

✶✵✴✷✼

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SLIDE 22

❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

m✷ w✷ h✶ h✷ m✶ n✶ n✷ w✶

✶ ✷ ✶

✷ ✷ ✶ ✶ ✷ ✷ ✶ ✶ ✶✶✴✷✼

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SLIDE 23

❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

m✷ w✷ h✶ h✷ m✶ n✶ n✷ w✶

✶ ✷ ✶

✷ ✷ ✶ ✶ ✷ ✷ ✶ ✶ ✶✶✴✷✼

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SLIDE 24

❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

m✷ w✷ h✶ h✷ m✶ n✶ n✷ w✶

✶ ✷ ✶

m✷ w✷ h✶ n✶ n✷ h✷ w✶ m✶

✶✶✴✷✼

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SLIDE 25

❘❡❞✉❝t✐♦♥ t♦ ❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s

✷ ✷ ✶ ✷ ✶

n✶ n✷ w✶ h✶ h✷ m✶ ✷ ✷

h✶ n✶ n✷ h✷ w✶ m✶

✶✶✴✷✼

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SLIDE 26

❋✐rst r❡s✉❧ts

✶✷✴✷✼

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SLIDE 27

❆❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥

■❢ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥✱ t❤❡♥ t❤❡ ♦✉t❝♦♠❡ ✐s P ▲❡♠♠❛ ✷m ✷n ✷ ✶ ✷ ✶ ✷ ✷ ✷ ✷

✶✸✴✷✼

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SLIDE 28

❆❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥

■❢ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥✱ t❤❡♥ t❤❡ ♦✉t❝♦♠❡ ✐s P ▲❡♠♠❛ ✷ ✷ ✷m − ✶ ✷n − ✶ ✷ ✷ ✷ ✷

✶✸✴✷✼

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SLIDE 29

❆❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥

■❢ ❛❧❧ ✇❡✐❣❤ts ❛r❡ ❡✈❡♥✱ t❤❡♥ t❤❡ ♦✉t❝♦♠❡ ✐s P ▲❡♠♠❛ ✷ ✷ ✷ ✶ ✷ ✶ ✷m − ✷ ✷n − ✷

✶✸✴✷✼

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SLIDE 30

❘❡❞✉❝t✐♦♥ ❧❡♠♠❛s

■❢ t✇♦ ✈❡rt✐❝❡s ❤❛✈❡ t❤❡ s❛♠❡ ♥❡✐❣❤❜♦rs ❛♥❞ ❡✐t❤❡r ❤❛✈❡ ❜♦t❤ ❧♦♦♣s ♦r ♥♦♥❡ ♦❢ t❤❡♠ ❤❛✈❡ ❧♦♦♣s✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❞✉❝t✐♦♥ ❤♦❧❞s ✿ ❚✇✐♥ ✈❡rt✐❝❡s ❧❡♠♠❛ m ✶ n m + n ✶

✶✹✴✷✼

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SLIDE 31

❘❡❞✉❝t✐♦♥ ❧❡♠♠❛s

■❢ t✇♦ ✈❡rt✐❝❡s ❤❛✈❡ t❤❡ s❛♠❡ ♥❡✐❣❤❜♦rs ❛♥❞ ❡✐t❤❡r ❤❛✈❡ ❜♦t❤ ❧♦♦♣s ♦r ♥♦♥❡ ♦❢ t❤❡♠ ❤❛✈❡ ❧♦♦♣s✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❞✉❝t✐♦♥ ❤♦❧❞s ✿ ❚✇✐♥ ✈❡rt✐❝❡s ❧❡♠♠❛ m − ✶ n m + n − ✶

✶✹✴✷✼

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SLIDE 32

❘❡❞✉❝t✐♦♥ ❧❡♠♠❛s

■❢ t❤❡r❡ ✐s ❛ ✈❡rt❡① ✇✐t❤♦✉t ❧♦♦♣ ❛♥❞ s✉❝❤ t❤❛t ✐ts ✇❡✐❣❤t ✐s ❣r❡❛t❡r t❤❛♥ t❤❡ s✉♠ ♦❢ ❛❧❧ ♦❢ ✐ts ♥❡✐❣❤❜♦rs✬ ✇❡✐❣❤t✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❞✉❝t✐♦♥ ❤♦❧❞s ✿ ❍❡❛✈② ✈❡rt❡① ❧❡♠♠❛ m c b a e d c b a e d

m ≥ a + b + c + d + e ✶✺✴✷✼

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SLIDE 33

❘❡❞✉❝t✐♦♥ ❧❡♠♠❛s

■❢ t❤❡r❡ ✐s ❛ ✈❡rt❡① ✇✐t❤♦✉t ❧♦♦♣ ❛♥❞ s✉❝❤ t❤❛t ✐ts ✇❡✐❣❤t ✐s ❣r❡❛t❡r t❤❛♥ t❤❡ s✉♠ ♦❢ ❛❧❧ ♦❢ ✐ts ♥❡✐❣❤❜♦rs✬ ✇❡✐❣❤t✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❞✉❝t✐♦♥ ❤♦❧❞s ✿ ❍❡❛✈② ✈❡rt❡① ❧❡♠♠❛ ∞ c b a e d c b a e d

✶✺✴✷✼

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SLIDE 34

❘❡❞✉❝t✐♦♥ ❧❡♠♠❛s

■❢ t❤❡r❡ ✐s ❛ ✈❡rt❡① ✇✐t❤♦✉t ❧♦♦♣ ❛♥❞ s✉❝❤ t❤❛t ❛❧❧ ✐ts ♥❡✐❣❤❜♦rs ❤❛✈❡ ❧♦♦♣s t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❞✉❝t✐♦♥ ❤♦❧❞s ✿ ❯s❡❧❡ss ✈❡rt❡① ❧❡♠♠❛ m c b a e d c b a e d

✶✻✴✷✼

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SLIDE 35

❇❛❝❦ t♦ ❆r❝✲❑❛②❧❡s

❙✉❜tr❛❝t✐♦♥ ❆r❝✲❑❛②❧❡s ✷ ✸ ✶ ✶ ✷ ✶ ❆r❝✲❑❛②❧❡s

✶✼✴✷✼

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SLIDE 36

❙♦❧✈✐♥❣ s✐♠♣❧❡ ❝❛s❡s

✶✽✴✷✼

slide-37
SLIDE 37

❙✐♠♣❧❡ ❝❛s❡s

m ❚❤❡ ❣❛♠❡ ✐s ❛❧r❡❛❞② ❡♥❞❡❞✳ m ❚❤❡ ❣r✉♥❞② ✈❛❧✉❡ ✐s m ♠♦❞✉❧♦ ✷✳ m n ♠✐♥✭m✱n✮ m n m

✶✾✴✷✼

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SLIDE 38

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷ ❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 39

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷ ❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 40

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷ ❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 41

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷ ❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

slide-42
SLIDE 42

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷ ❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 43

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷ ❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 44

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷ ❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 45

❙✐♠♣❧❡ ❝❛s❡s

m n

m n

✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✷ ✷ ✷ ✸ ✸ ✶ ✶ ✵ ✷

  • • •

❚❤❡ ❣❛♠❡ ✐s P ✇❤❡♥ ❜♦t❤ m ❛♥❞ n ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 46

❙✐♠♣❧❡ ❝❛s❡s

m n

n m ∞ ∞

❚❤❡ ❣❛♠❡ ✐s ✇❤❡♥ ❜♦t❤ ❛♥❞ ❛r❡ ❡✈❡♥✳

✷✵✴✷✼

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SLIDE 47

❙✐♠♣❧❡ ❝❛s❡s

m n p ♠✐♥✭n✱m + p✮ m n p n

✷✶✴✷✼

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SLIDE 48

❙✐♠♣❧❡ ❝❛s❡s

m n p ♠✐♥✭n✱m + p✮ m n p n

  • ✷✶✴✷✼
slide-49
SLIDE 49

❋✐rst ❞✐✣❝✉❧t②

m n p

p ≥ n

m n

✷✷✴✷✼

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SLIDE 50

❋✐rst ❞✐✣❝✉❧t②

m n p

p ≥ n

m n

p < n

m + p

✷✷✴✷✼

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SLIDE 51
  • ❡♥❡r❛❧ ♠❡t❤♦❞

m n p ✶

p < n ✶

m + p ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✷

✷✸✴✷✼

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SLIDE 52
  • ❡♥❡r❛❧ ♠❡t❤♦❞

m n n − ✶

n − ✶ < n

m + n − ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✷

✷✸✴✷✼

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SLIDE 53
  • ❡♥❡r❛❧ ♠❡t❤♦❞

m n n − ✶

n − ✶ < n

m + n − ✶ m − ✶ n n − ✶ m − ✶ n − ✶ n − ✶ ✶ ✶ m n − ✶ n − ✷

✷✸✴✷✼

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SLIDE 54
  • ❡♥❡r❛❧ ♠❡t❤♦❞

m n n − ✶

n − ✶ < n

m + n − ✶ m − ✶ n n − ✶ ✶ ✶ ✶ m − ✶ n − ✶ m n − ✶ n − ✷

✷✸✴✷✼

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SLIDE 55

▼❛✐♥ r❡s✉❧ts

  • x✶

y✶ x✷ y✷ xn yn c a

✰ ✰

✷✹✴✷✼

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SLIDE 56

▼❛✐♥ r❡s✉❧ts

  • x✶

y✶ x✷ y✷ xn yn c a

  • i yi
  • i xi

c a

✰ ✰

✷✹✴✷✼

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SLIDE 57

▼❛✐♥ r❡s✉❧ts

  • x✶

y✶ x✷ y✷ xn yn c a

  • i yi
  • i xi

c a

  • i xi ≥ c +

i yi

  • i xi < c +

i yi

✰

  • i yi

c a a +

i xi

✰

✷✹✴✷✼

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SLIDE 58

▼❛✐♥ r❡s✉❧ts

  • x✶

y✶ x✷ y✷ xn yn c a

✰ ✰

✷✹✴✷✼

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SLIDE 59

▼❛✐♥ r❡s✉❧ts

  • x✶

y✶ x✷ y✷ xn yn c a

  • i yi
  • i xi

c a

✰ ✰

✷✹✴✷✼

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SLIDE 60

▼❛✐♥ r❡s✉❧ts

  • x✶

y✶ x✷ y✷ xn yn c a

  • i yi
  • i xi

c a

✰ ✰

  • i yi

c a

✷✹✴✷✼

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SLIDE 61

P❡r✐♦❞✐❝✐t②

❚❤❡ ❢✉♥❝t✐♦♥ x → outcome(G(x, ω✷, . . . ωn)) ✐s ✉❧t✐♠❛t❡❧② ✷✲♣❡r✐♦❞✐❝ ✇✐t❤ ♣r❡♣❡r✐♦❞ ❛t ♠♦st ✷

i≥✷ ωi✳

P❡r✐♦❞✐❝✐t② t❤❡♦r❡♠

✷✺✴✷✼

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SLIDE 62

❈♦♥❝❧✉s✐♦♥

❼ ■♥tr♦❞✉❝t✐♦♥ ♦❢ ❛ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ❆r❝✲❑❛②❧❡s ❼ ❆♣♣❧✐❝❛t✐♦♥ t♦ ♥♦♥✲❛tt❛❝❦✐♥❣ r♦♦❦s ♦♥ ❛ ❤♦❧❡❞ ❝❤❡ss❜♦❛r❞ ❼ ❈♦♠♣❧❡t❡ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ tr❡❡s ♦❢ ❞❡♣t❤ ✷ ❼ P❡r✐♦❞✐❝✐t② ❢♦r ♦♥❡ ✈❡rt❡①

❘❡s✉❧ts

❼ ❙♦❧✈✐♥❣ ♠♦r❡ ❝♦♠♣❧❡① ❣r❛♣❤s ❼ ❙♦❧✈✐♥❣ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ♦❢ r♦♦❦s ♦♥ ❛ ❤♦❧❡❞ ❝❤❡ss❜♦❛r❞ ❼ ❙t✉❞②✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❣❛♠❡

P❡rs♣❡❝t✐✈❡s

✷✻✴✷✼

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SLIDE 63

✷✼✴✷✼