rs r Pr s - - PowerPoint PPT Presentation

SLIDE 1

✺✵ ②❡❛rs ♦❢ ▲✐♥❡❛r Pr♦❜✐♥❣ ❍❛s❤✐♥❣

❆❧❢r❡❞♦ ❱✐♦❧❛

❯♥✐✈❡rs✐❞❛❞ ❞❡ ❧❛ ❘❡♣ú❜❧✐❝❛✱ ❯r✉❣✉❛②

❆♦❢❆ ✷✵✶✸

❉❡❞✐❝❛t❡❞ t♦ P❤✐❧✐♣♣❡ ❋❧❛❥♦❧❡t

SLIDE 2

❚❤❡ ♣r♦❜❧❡♠✳

❚❤❡ t❛❜❧❡ ❤❛s ♠ ♣❧❛❝❡s t♦ ❤❛s❤ ✭♣❛r❦✮ ❢r♦♠ ✵ t♦ ♠ ✶✱ ❛♥❞ ♥ ❡❧❡♠❡♥ts ✭❝❛rs✮✳ ❊❛❝❤ ❡❧❡♠❡♥t ✐s ❣✐✈❡♥ ❛ ❤❛s❤ ✈❛❧✉❡ ✭♣r❡❢❡rr❡❞ ♣❛r❦✐♥❣ ❧♦t✮✳ ■❢ ♣❧❛❝❡ ✐s ❡♠♣t②✱ t❤❡♥ t❤❡ ❡❧❡♠❡♥t ✐s st♦r❡❞ t❤❡r❡✳ ❖t❤❡r✇✐s❡✱ ❧♦♦❦s s❡q✉❡♥t✐❛❧❧② ❢♦r ❛♥ ❡♠♣t② ♣❧❛❝❡✳ ■❢ ♥♦ ❡♠♣t② ♣❧❛❝❡ ✉♣ t♦ t❤❡ ❡♥❞ ♦❢ t❤❡ t❛❜❧❡✱ t❤❡ s❡❛r❝❤ ❢♦❧❧♦✇s ❛t ❧♦❝❛t✐♦♥ ✵✳ ❙❡✈❡r❛❧ ❘✳❱✳ t♦ st✉❞②✱ ♠❛✐♥❧② r❡❧❛t❡❞ ✇✐t❤ ❝♦st ♦❢ ✐♥❞✐✈✐❞✉❛❧ s❡❛r❝❤❡s ❛♥❞ t♦t❛❧ ❝♦♥str✉❝t✐♦♥ ❝♦st✳ ❱❡r② ✐♠♣♦rt❛♥t s♣❡❝✐❛❧ ❝❛s❡✿ P❛r❦✐♥❣ Pr♦❜❧❡♠✳ ■♥ ♣❛r❦✐♥❣ t❤❡ ❝❛r ✐s ❧♦st ✐❢ ♥♦ ❡♠♣t② ♣❧❛❝❡ ✉♣ t♦ t❤❡ ❡♥❞✳ ▼❛✐♥ ❘✳❱✳ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❧♦st ❝❛rs✳ ■♠♣♦rt❛♥t ✈❛r✐❛♥t✿ ❡❛❝❤ ❧♦❝❛t✐♦♥ ❝❛♥ ❤♦❧❞ ✉♣ t♦ ❜ ✭♦r ❦✮ ❝❛rs✳

SLIDE 3

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 4

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 5

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 6

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 7

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 8

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 9

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 10

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 11

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 12

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 13

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 14

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 15

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

5 / 37

SLIDE 16

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 17

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 18

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

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SLIDE 19

A discrete parking problem Limiting distribution results Analysis Further research

A discrete parking problem: Example

Example: 8 parking lots, 8 cars Parking sequence: 3, 6, 3, 8, 6, 7, 4, 5

1 2 3 4 5 6 7 8

⇒ 2 cars are unsuccessful

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SLIDE 20

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

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SLIDE 21

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

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SLIDE 22

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

SLIDE 23

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

SLIDE 24

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

SLIDE 25

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

SLIDE 26

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

SLIDE 27

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

SLIDE 28

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

32 / 37

SLIDE 29

A discrete parking problem Limiting distribution results Analysis Further research

Further research: Bucket parking scheme

Bucket parking scheme Blake and Konheim [1976]: Each parking lots can hold up to r cars Related to analysis of bucket hashing algorithms r

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SLIDE 30

▲✐♥❡❛r Pr♦❜✐♥❣ ❍❛s❤✐♥❣✳

SLIDE 31

❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❜❡❛✉t② ♦❢ ▲✐♥❡❛r Pr♦❜✐♥❣✦

SLIDE 32

❙♦♠❡ ♣❡rs♦♥❛❧ ♠♦t✐✈❛t✐♦♥s ✳✳✳

❙✐♠♣❧❡st ❝♦❧❧✐s✐♦♥ r❡s♦❧✉t✐♦♥ str❛t❡❣② ❢♦r ♦♣❡♥ ❛❞❞r❡ss✐♥❣ ❬P❡t❡rs♦♥ ✶✾✺✼❪✳ ❲♦r❦s ✇❡❧❧ ❢♦r t❛❜❧❡s t❤❛t ❛r❡ ♥♦t t♦♦ ❢✉❧❧✳ ❇❡❝❛✉s❡ ♦❢ ♣r✐♠❛r② ❝❧✉st❡r✐♥❣✱ ✐ts ♣❡r❢♦r♠❛♥❝❡ ❞❡t❡r✐♦r❛t❡s ✇❤❡♥ t❤❡ ❧♦❛❞ ❢❛❝t♦r ✐s ❤✐❣❤✳ ■ts ❛♥❛❧②s✐s ❧❡❛❞s t♦ ♥♦♥tr✐✈✐❛❧ ❛♥❞ ✐♥t❡r❡st✐♥❣ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s✳ ❚❤❡r❡ ❛r❡ ❝♦♥♥❡❝t✐♦♥s ✇✐t❤ tr❡❡ ✐♥✈❡rs✐♦♥s✱ tr❡❡ ♣❛t❤ ❧❡♥❣❤ts✱ ❣r❛♣❤ ❝♦♥♥❡❝t✐✈✐t②✱ ❛r❡❛ ✉♥❞❡r ❡①❝✉rs✐♦♥✱ ❡t❝✳ ❊q✉✐✈❛❧❡♥t ❢♦r♠✉❧❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ♣❛r❦✐♥❣ ♣r♦❜❧❡♠✳ ❋✐rst ♣r♦❜❧❡♠ t❤❛t ❉✳ ❑♥✉t❤ ❛♥❛❧②③❡❞ ❬❑♥✉t❤ ✶✾✻✷❪ ✇✐t❤ ❜✉❝❦❡t s✐③❡ ✶✱ ❛♥❞ ♠♦t✐✈❛t❡❞ t❤❡ ❝♦❧❧❡❝t✐♦♥ ✧❚❤❡ ❆rt ♦❢ ❈♦♠♣✉t❡r Pr♦❣r❛♠♠✐♥❣✧✳ ❚❤❡ ❛♥❛❧②s✐s ❢♦r ❣❡♥❡r❛❧ ❜✉❝❦❡t s✐③❡ ❜ ♣r❡s❡♥ts ✈❡r② ✐♥t❡r❡st✐♥❣ ❝❤❛❧❧❡♥❣❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ❝❛♥ s②♠❜♦❧✐❝ ♠❡t❤♦❞s ❜❡ ✉s❡❞❄

SLIDE 33

✳✳✳ ❛♥❞ ❛ ❜♦① ❢✉❧❧ ♦❢ s✉r♣r✐s❡s ✳✳✳

❚❤❡ st✉❞② ♦❢ ♣❛r❦✐♥❣ s❡q✉❡♥❝❡s ❛♥❞ t❤❡✐r ❞❡❡♣ r❡❧❛t✐♦♥s ✇✐t❤ ♦t❤❡r ♣r♦❜❧❡♠s ✐♥ ❜♦t❤ ❞✐s❝r❡t❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s ♠❛t❤❡♠❛t✐❝s ❤❛s ❜❡❡♥ ❝❛rr✐❡❞ ♦✉t ❜② ❞✐✛❡r❡♥t r❡s❡❛r❝❤ ❝♦♠♠✉♥✐t✐❡s ✐♥ ♣❛r❛❧❧❡❧ ❛♥❞ ✇✐t❤ ❧✐tt❧❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❛♠♦♥❣ t❤❡♠✳ ❙❡✈❡r❛❧ r❡❧❛t❡❞ ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ st✉❞✐❡❞ ❜② ❡①♣❡rts ✐♥ ♣r♦❜❛❜✐❧✐t②✱ ❝♦♠❜✐♥❛t♦r✐❝s ❛♥❞ ❝♦♠♣✉t❡r s❝✐❡♥❝❡s✳ ❚❤❡ ♠❡t❤♦❞♦❧♦❣✐❝❛❧ t❡❝❤♥✐q✉❡s t♦ st✉❞② t❤❡s❡ ♣r♦❜❧❡♠s ❛r❡ ✈❡r② ❞✐✈❡rs❡✱ ❛♥❞ ❝♦✈❡r ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ r❡s❡❛r❝❤ ❛r❡❛s✳ ❆s ✐t ✐s s❛✐❞ ✐♥ ❬❈❤❛ss❛✐♥❣ ❛♥❞ ❋❧❛❥♦❧❡t ✷✵✵✸❪✱ ❛ s②st❡♠❛t✐❝ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ t❤❡ ♣r♦❜❧❡♠ ✐s ✈❡r② ❞✐✣❝✉❧t✳ ■ ✇✐❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❝♦♥tr✐❜✉t✐♦♥s ♠❛❞❡ ❜② ♣❡♦♣❧❡ ❢r♦♠ ♦✉r ❝♦♠♠✉♥✐t② ❛s ✇❡❧❧ ❛s s♦♠❡ r❡❧❛t❡❞ ✇♦r❦ t❤❛t ❤❛✈❡ ✐♥s♣✐r❡❞ t❤❡✐r r❡s❡❛r❝❤✳

SLIDE 34

✶✾✻✷✿ ❙✉♠♠❡r ✇♦r❦ ❜② ❉♦♥ ❑♥✉t❤ ✳✳✳

SLIDE 35

❋✐rst ♣✉❜❧✐s❤❡❞ ♣❛♣❡r✳

SLIDE 36

❍✐st♦r✐❝❛❧ ♥♦t❡ ♦♥ ✜rst ♣✉❜❧✐s❤❡❞ ♣❛♣❡r✳

SLIDE 37

❖r✐❣✐♥❛❧ r❡s✉❧ts✳

▲❡t ❛ ❤❛s❤ t❛❜❧❡ ✇✐t❤ ♠ ♣♦s✐t✐♦♥s ❛♥❞ ♥ ✐♥s❡rt❡❞ ❡❧❡♠❡♥ts✳ ▲❡t P♠❀♥ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❧❛st ♣♦s✐t✐♦♥ ❜❡✐♥❣ ❡♠♣t②✳

P♠❀♥ ❂

• ✶ ♥

♠

✁ ✳

▲❡t ❈♠❀♥ t❤❡ ❘✳❱✳ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❝❡ss❢✉❧ s❡❛r❝❤❡s ♦❢ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t✳

❊ ❬❈♠❀♥❪ ❂ ✶

✷ ✭✶ ✰ ◗✵✭✭♠❀ ♥ ✶✮✮✳

❊ ❬❈♠❀☛♠❪ ❂ ✶

✷

✏ ✶ ✰

✶ ✶☛

✑ ✇✐t❤ ✵ ✔ ☛ ❁ ✶✳ ❊ ❬❈♥❀♥❪ ❂ ♣ ✙♥

✽ ✰ ❖✭✶✮ ✭♣r♦✈❡❞ ♦♥ ♠❛② ✷✵✱ ✶✾✻✺✮✳

▲❡t ❯♠❀♥ t❤❡ ❘✳❱✳ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ✉♥s✉❝❝❡ss❢✉❧ s❡❛r❝❤❡s ♦❢ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t✳

❊ ❬❯♠❀♥❪ ❂ ✶

✷ ✭✶ ✰ ◗✶✭✭♠❀ ♥ ✶✮✮✳

❊ ❬❯♠❀☛♠❪ ❂ ✶

✷

✏ ✶ ✰

✶ ✭✶☛✮✷

✑ ✇✐t❤ ✵ ✔ ☛ ❁ ✶✳

❚❤❡ ❘❛♠❛♥✉❥❛♥ ◗ ❢✉♥❝t✐♦♥ ✐s t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ◗✵✭♥❀ ♥✮ ♦❢ ◗r✭♠❀ ♥✮ ❂

♥

❳

❦❂✵

✥

❦ ✰ r ❦

✦

♥❦ ♠♥ ✿

SLIDE 38

✳✳✳ q✉❡st✐♦♥ ❜② ❘❛♠❛♥✉❥❛♥ t♦ ❍❛r❞② ✐♥ ✶✾✶✸✳

SLIDE 39

❚❤❡ ♣r♦❜❧❡♠ ✳✳✳✳

SLIDE 40

✳✳✳ ❛♥❞ t❤❡ ❘❛♠❛♥✉❥❛♥✬s ◗ ❢✉♥❝t✐♦♥ ✐♥t♦ ♣❧❛②✦

SLIDE 41

▼❡t❤♦❞♦❧♦❣② ✳ ✳ ✳

SLIDE 42

✳ ✳ ✳ ❛♥❞ ♠♦r❡ ♠❡t❤♦❞♦❧♦❣②✦

SLIDE 43

❈♦❧❧✐s✐♦♥ ❘❡s♦❧✉t✐♦♥ ❙tr❛t❡❣✐❡s✳

■♥ ♦♣❡♥ ❛❞❞r❡ss✐♥❣✱ ✇❤❡♥ t✇♦ ❦❡②s ❝♦❧❧✐❞❡✱ ❡✐t❤❡r ♦♥❡ ♦❢ t❤❡♠ ♠❛② st❛② ✐♥ t❤❛t ❧♦❝❛t✐♦♥✱ ✇❤✐❧❡ t❤❡ ♦t❤❡r ♦♥❡ ❦❡❡♣s ♣r♦❜✐♥❣✳ ❋✐rst✲❈♦♠❡✲❋✐rst✲❙❡r✈❡❞ ✭st❛♥❞❛r❞✮✳

❊❛❝❤ ❝♦❧❧✐s✐♦♥ ✐s r❡s♦❧✈❡❞ ✐♥ ❢❛✈♦r ♦❢ t❤❡ ✜rst r❡❝♦r❞ t❤❛t ♣r♦❜❡❞ t❤❡ ❧♦❝❛t✐♦♥✳

▲❛st✲❈♦♠❡✲❋✐rst✲❙❡r✈❡❞ ❬P♦❜❧❡t❡ ❛♥❞ ▼✉♥r♦ ✶✾✽✾❪✳

❊❛❝❤ ❝♦❧❧✐s✐♦♥ ✐s r❡s♦❧✈❡❞ ✐♥ ❢❛✈♦r ♦❢ t❤❡ ✐♥❝♦♠✐♥❣ r❡❝♦r❞✳

❘♦❜✐♥ ❍♦♦❞ ❬❈❡❧✐s✱ ▲❛rs♦♥ ❛♥❞ ▼✉♥r♦ ✶✾✽✺❪✳

❊❛❝❤ ❝♦❧❧✐s✐♦♥ ✐s r❡s♦❧✈❡❞ ✐♥ ❢❛✈♦r ♦❢ t❤❡ r❡❝♦r❞ t❤❛t ✐s ❢✉rt❤❡r ❛✇❛② ❢r♦♠ ✐ts ❤♦♠❡ ❧♦❝❛t✐♦♥✳

SLIDE 44

❖r✐❣✐♥❛❧ ♣r♦♣♦s❛❧ ♦❢ ❘♦❜✐♥ ❍♦♦❞✳

✧❆♥❛❧②s✐s ♦❢ ❛ ✜❧❡ ❛❞❞r❡ss✐♥❣ ♠❡t❤♦❞✧ ❬❙❝❤❛② ❛♥❞ ❙♣r✉t❤ ✶✾✻✷❪✳ ▼♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ♣r♦♣♦s❡❞ ❜② P❡t❡rs♦♥ ✐♥ ✶✾✺✼✳ ❚❤✐s ✐s t❤❡ ❘♦❜✐♥ ❍♦♦❞ str❛t❡❣② ✦ ❊①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ s❡❛r❝❤ ❝♦st ❞♦❡s ♥♦t ❝❤❛♥❣❡✱ ❛s ❛❧r❡❛❞② ♥♦t❡❞ ❜② ❉✳ ❑♥✉t❤ ✐♥ ❤✐s ♦r✐❣✐♥❛❧ ♥♦t❡✳ ❚❤❡② ❞♦ ❛♥ ❛♥❛❧②s✐s ❜❛s❡❞ ♦♥ ❛ P♦✐ss♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✱ ❛♥❞ ✜♥❞ ❊ ❬❈♠❀☛♠❪ ❂ ✶

✷

✏

✶ ✰

✶ ✶☛

✑

✇✐t❤ ✵ ✔ ☛ ❁ ✶✳

SLIDE 45

❚✇♦ ♠♦❞❡❧s t♦ ❛♥❛❧②③❡ t❤❡ ♣r♦❜❧❡♠✳

❊①❛❝t ✜❧❧✐♥❣ ♠♦❞❡❧✳

❆ ✜①❡❞ ♥✉♠❜❡r ♦❢ ❦❡②s ♥✱ ❛r❡ ❞✐str✐❜✉t❡❞ ❛♠♦♥❣ ♠ ❧♦❝❛t✐♦♥s✱ ❛♥❞ ❛❧❧ ♠♥ ♣♦ss✐❜❧❡ ❛rr❛♥❣❡♠❡♥ts ❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧② t♦ ♦❝❝✉r✳

P♦✐ss♦♥ ♠♦❞❡❧✳

❊❛❝❤ ❧♦❝❛t✐♦♥ r❡❝❡✐✈❡s ❛ ♥✉♠❜❡r ♦❢ ❦❡②s t❤❛t ✐s P♦✐ss♦♥ ❞✐str✐❜✉t❡❞ ✇✐t❤ ♣❛r❛♠❡t❡r ❜☛✱ ❛♥❞ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❦❡②s ❣♦✐♥❣ ❡❧s❡✇❤❡r❡✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❦❡②s✱ ◆✱ ✐s ✐ts❡❧❢ ❛ P♦✐ss♦♥ ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡r ❜☛♠✿ Pr❬◆ ❂ ♥❪ ❂ ❡❜☛♠✭❜☛♠✮♥ ♥✦ ✿

SLIDE 46

P♦✐ss♦♥ ❚r❛♥s❢♦r♠✳

❘❡s✉❧ts ✐♥ ♦♥❡ ♠♦❞❡❧ ❝❛♥ ❜❡ tr❛♥s❢❡r❡❞ ✐♥t♦ t❤❡ ♦t❤❡r ♠♦❞❡❧ ❜② t❤❡ P♦✐ss♦♥ ❚r❛♥s❢♦r♠✿ P♠❬❢♠❀♥❀ ❜☛❪ ❂

❳

♥✕✵

Pr❬◆ ❂ ♥❪❢♠❀♥ ❂ ❡❜☛♠ ❳

♥✕✵

✭❜☛♠✮♥ ♥✦ ❢♠❀♥✿ ■♥✈❡rs✐♦♥ ❚❤❡♦r❡♠✿ ❬●♦♥♥❡t ❛♥❞ ▼✉♥r♦ ✶✾✽✹❪ ■❢ P♠❬❢♠❀♥❀ ❜☛❪ ❂

❳

❦✕✵

❛♠❀❦✭❜♠☛✮❦ t❤❡♥ ❢♠❀♥ ❂

❳

❦✕✵

❛♠❀❦ ♥❦ ✭❜♠✮❦ ✿ ❚❤❡ P♦✐ss♦♥ ♠♦❞❡❧ ✐s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❡①❛❝t ✜❧❧✐♥❣ ♠♦❞❡❧ ✇❤❡♥ ♥❀ ♠ ✦ ✶ ✇✐t❤ ♥❂♠ ❂ ❜☛ ✇✐t❤ ✵ ✔ ☛ ❁ ✶✳

SLIDE 47

❉✐❛❣♦♥❛❧ P♦✐ss♦♥ ❚r❛♥s❢♦r♠✳

✱ ❬▼✉♥r♦✱ P♦❜❧❡t❡✱ ❱✐♦❧❛ ✶✾✾✼❪ ▲❡t ❛ ❤❛s❤ t❛❜❧❡ ♦❢ s✐③❡ ♠✱ ✇✐t❤ ♥ ✰ ✶ ❦❡②s✱ ❛♥❞ ❧❡t P ❜❡ ❛ ♣r♦♣❡rt② ❢♦r ✎ ✭❝❤♦s❡♥ ✉♥✐❢♦r♠✐❧② ❛t r❛♥❞♦♠✮✳ ▲❡t ❢♠❀♥ ❜❡ t❤❡ r❡s✉❧t ♦❢ ❛♣♣❧②✐♥❣ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r ✭❡✳❣✳ ❛♥ ❡①♣❡❝t❡❞ ✈❛❧✉❡✮ t♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ P✳

❢♠❀♥ ❂ ❳

✐✕✵

Pr❬✎ ✷ ❝❧✉st❡r ♦❢ s✐③❡ ✐ ✰ ✶❪ ❢✐✰✷❀✐ ❂ ❳

✐✕✵

✒♥ ✐ ✓✭♠ ✐ ✷✮♥✐✶✭♠ ♥ ✷✮✭✐ ✰ ✷✮✐ ♠♥ ❢✐✰✷❀✐✿

❚❤❡♥ P♠❬❢♠❀♥❀ ☛❪ ❂ ❉✷❬❢♥✰✷❀♥❀ ☛❪ ✇✐t❤ ❉❝❬❢♥❀ ☛❪ ❂ ✭✶ ☛✮

❳

♥✕✵

❡✭♥✰❝✮☛ ✭✭♥ ✰ ❝✮☛✮♥ ♥✦ ❢♥✿

SLIDE 48

❈♦♠❜✐♥❛t♦r✐❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✳

❆♥② ▲✐♥❡❛r Pr♦❜✐♥❣ ❍❛s❤ t❛❜❧❡ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ❛❧♠♦st ❢✉❧❧ t❛❜❧❡s ✭❛ s✉❜t❛❜❧❡ ✇✐t❤ ❛❧❧ ❜✉t t❤❡ ❧❛st ❜✉❝❦❡t ❢✉❧❧✮✳ ❊①❛♠♣❧❡✿ ❬✸✲✸❪✱❬✹✲✹❪✱❬✺✲✺❪✱❬✻✲✷❪✳ ❚❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ ❝❛♥ ❜❡ ♥✐❝❡❧② ❤❛♥❞❧❡❞ ❜② ❆♥❛❧②t✐❝ ❈♦♠❜✐♥❛t♦r✐❝s✱ s✐♥❝❡ ❢♦r ❡①❛♠♣❧❡✱ ✐t ✐♠♣❧✐❡s t❤❛t ✐t ✐s ❡♥♦✉❣❤ t♦ st✉❞② ❛❧♠♦st ❢✉❧❧ t❛❜❧❡s✱ ❛♥❞ t❤❡♥ ✉s❡ t❤❡ s❡q✉❡♥❝❡ ❝♦♥str✉❝t✐♦♥✳

SLIDE 49

❉✐str✐❜✉t✐♦♥ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❞✐s♣❧❛❝❡♠❡♥ts✳

✎ ❈♦♠♣❧❡♠❡♥t❛r② ♣r♦❜❛❜✐❧✐st✐❝ ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛♣♣r♦❛❝❤❡s✳ ✎ ❬❏❛♥s♦♥ ✷✵✵✺✱ ❱✐♦❧❛ ✷✵✵✺❪✳ ▲❡t P ☎

☛ ✭③✮ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r t❤❡

❞✐s♣❧❛❝❡♠❡♥t ♦❢ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t ✇t❤ ✵ ✔ ☛ ❁ ✶ ❛♥❞ ☎ ✷ ❢❋❈❋❙❀ ❘❍❣✳ ❚❤❡♥✱ ✇✐t❤ ❚✭①✮ ❂ ③❡❚✭①✮ t❤❡ tr❡❡ ❢✉♥❝t✐♦♥✱ P ❋❈❋❙

☛

✭③✮ ❂ ✭✶ ❚✭③☛❡☛✮✮✷ ✭✶ ☛✮✷ ✷☛✭✶ ③✮ ✿ P ❘❍

☛

✭③✮ ❂ ✶ ☛ ☛ ❡③☛ ❡☛ ③❡☛ ❡③☛ ✿ ▲❈❋❙ ✐s ♠✉❝❤ ♠♦r❡ ❝❤❛❧❧❡♥❣✐♥❣ ❛♥❞ ❬❏❛♥s♦♥ ✷✵✵✺❪ ♣r❡s❡♥ts ❛♥ ❡①❛❝t ❡①♣r❡ss✐♦♥✳ ❊①❛❝t r❡s✉❧ts ❢♦❧❧♦✇ ❜② ❞❡♣♦✐ss♦♥✐③❛t✐♦♥✳

SLIDE 50

P❛r❦✐♥❣ s❡q✉❡♥❝❡s✱ ❛❝②❝❧✐❝ ♠❛♣s✱ ♣r✐♦r✐t② q✉❡✉❡s✳

✎ ❬❙❡✐t③ ✷✵✵✾❪ s✉r✈❡②s s♦♠❡ ♦❢ t❤❡s❡ r❡❧❛t✐♦♥s✳ ✎ ❬●✐❧❜❡② ❛♥❞ ❑❛❧✐❦♦✇ ✶✾✾✾❪✿ ♣r✐♦r✐t② q✉❡✉❡s✳ ✎ Pr❡s❡♥ts ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ♣❛r❦✐♥❣ ♣r♦❜❧❡♠✳ ✎ ❉✐str✐❜✉t✐♦♥❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡ ♦✈❡r✢♦✇✳ ✎ ❆♥❛❧②s✐s ♦❢ t❤❡ ♦✈❡r✢♦✇ ✐♥ ♣❛r❦✐♥❣ ✇✐t❤ ❜✉❝❦❡ts✳ ❣✭♠❀ ♥❀ ❦✮✿ t❤❡ ♥✉♠❜❡r ♦❢ ❞❡❢❡❝t✐✈❡ ♣❛r❦✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ ❞❡❢❡❝t ❦✳ ▲❡t ●✭③❀ ✉❀ ✈✮ ❂

❳

♠✕✵

❳

♥✕✵

❳

❞✕✵

❣✭♠❀ ♥❀ ❦✮ ③♥

♥✦ ✉♠✈❞✳

❚❤❡♥

• ✭③❀ ✉❀ ✈✮ ❂

✶ ❚✭③✉✮

③✈

✏

✶ ❚✭③✉✮

③

✑ ✶ ✉

✈ ❡③✈✁✿

❬❈❛♠❡r♦♥✱ ❏♦❤❛♥♥s❡♥✱ Pr❡❧❧❜❡r❣ ❛♥❞ ❙❝❤✇❡✐t③❡r ✷✵✵✽❪✳ ■♥ ❬❱✐♦❧❛ ✷✵✵✺❪ ♣❛r❦✐♥❣ ✐s ✉s❡❞ ❛s ❛ s✉❜♣r♦❜❧❡♠ ❢♦r ❘❍✳

SLIDE 51

P❛r❦✐♥❣ ♣r♦❜❧❡♠✿ ❧✐♠✐t ❞✐str✐❜✉t✐♦♥s

❬P❛♥❤♦❧③❡r ✷✵✵✽❪ ✎ ❙t✉❞✐❡s Pr❢❳♠❀♥ ❂ ❦❣ ❂ ❣✭♠❀♥❀❦✮

♠♥

✳ ✎ ▲✐♠✐t ❞✐str✐❜✉t✐♦♥s ❢♦r ❳♠❀♥✳ ◆✐♥❡ r❡❣✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ ❣r♦✇t❤ ♦❢ ♠❀ ♥✳ ♥ ✓ ♠✿ ❳♠❀♥ ▲

• ✦❳✚ ✇✐t❤ Pr❢❳ ❂ ✵❣ ❂ ✶ ✭❞❡❣❡♥❡r❛t❡❞ ❧❛✇✮✳

♥ ✘ ✚♠❀ ✵ ❁ ✚ ❁ ✶✿ ❳♠❀♥ ▲

• ✦❳✚✱ ❞✐s❝r❡t❡ ❧✐♠✐t ❧❛✇✳

♣♠ ✓ ✁ ✿❂ ♠ ♥ ✓ ♠✿ ✁

♠❳♠❀♥ ▲

• ✦❳✭❞✮

❂ ❊❳P✭✷✮✳

✁ ✿❂ ♠ ♥ ✘ ✚♣♠❀ ✚ ❃ ✵✿

✶ ♣♠❳♠❀♥ ▲

• ✦❳✭❞✮

❂ ▲■◆❊❳P✭✷✱✚✮✳

✵ ✔ ✁ ✿❂ ♠ ♥ ✓ ♣♠✿

✶ ♣♠❳♠❀♥ ▲

• ✦❳✭❞✮

❂ ❘❆❨▲❊■●❍✭✷✮✳

✵ ✔ ✁ ✿❂ ♥ ♠ ✓ ♣♥✿ ❳♠❀♥✰♠♥

♣♥

❳♠❀♥ ▲

• ✦❳✭❞✮

❂ ❘❆❨✭✷✮✳

✁ ✿❂ ♥♠ ✘ ✚♣♥❀ ✚ ❃ ✵✿ ❳♠❀♥✰♠♥

♣♥

❳♠❀♥ ▲

• ✦❳✚✭❞✮

❂ ▲❊✭✷✱✚✮✳

♣♥ ✓ ✁ ✿❂ ♥ ♠ ✓ ♥✿ ✁

♠✭❳♠❀♥ ✰ ♠ ♥✮ ▲

• ✦❳✭❞✮

❂ ❊❳P✭✷✮✳

♥ ✘ ✚♠❀ ✚ ❃ ✶✿ ✭❳♠❀♥ ✰ ♠ ♥✮ ▲

• ✦❳✚✱ ❞✐s❝r❡t❡ ❧✐♠✐t ❧❛✇✳

♠ ✓ ♥✿ ❳♠❀♥ ▲

• ✦❳✚ ✇✐t❤ Pr❢❳ ❂ ✵❣ ❂ ✵ ✭❞❡❣❡♥❡r❛t❡❞ ❧❛✇✮✳

SLIDE 52

❚♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✭❜♦① ❢✉❧❧ ♦❢ s✉r♣r✐s❡s✦✮✳

SLIDE 53

SLIDE 54 4 Analytic Com binatorics Tw

• basic

p rincipl es 7! \dictiona ries" SYMBOLIC METHODS Generating functions 7! z 11 z + z 2 + z 3 + 2 z 4 + 2 z 5 + 4 z 6 + 5 z 7 + 9 z 8 +

• Analytic

functions and singula riti es 10

SLIDE 55 CONSTR UCTIONS Dictionary (I) F 7! ff n g 7! f (z ) = X n f n z n n! : 1 1

• f

= 1 + f + f 2 + f 3 +

• exp

(f ) = 1 + f + 1 2! f 2 + 1 3! f 3 +

• A[

B 7! A(z )+B (z ) AB 7! A(z )B (z ) Seq A 7! 1 1

• A(z

) Set A 7! exp (A(z )) Cycle A 7! log 1 1

• A(z

) 11

SLIDE 56 COMPLEX ASYMPTOTICS Dictionary (I I) P

• in

t

• f

regularit y. f (z )

• f

(z ) + f (z )(z

• z

) exp (z ) 2.5 2 1.5 1 0.5 2 1

• 1
• 2

P

• in

t

• f

singularit y. :9 d dz f (z )

• z

(1

• z

)

• 7!

n

• 1

( )

• p

1

• z

7! n 3=2

• (1
• z

) 3=2 7! n 5=2

• 0.2
• 0.4
• 0.6
• 0.8
• 1
• 1.2
• 1.4

0.6 0.4 0.2

• 0.2
• 0.4
• 0.6

0.5

• 0.5
• 1
• 1.5
• 2
• 2.5

2 1

• 1
• 2

12

SLIDE 57 P erm utations: (1

• z

) 1

0.8 0.9 1 1.1 1.2 x

• 0.2
• 0.1

0.1 0.2 y

• 30
• 20
• 10

10 20 30

T rees: 1

• p

1

• z

0.6 0.8 1 1.2 1.4 x

• 0.4
• 0.2

0.2 0.4 y

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

13

SLIDE 58 Distributional analysis (almost full tables)

• Reco

rd construction cost = total displaceme nt F n (q ) = n1 X k =0

• n
• 1

k

• F

k (q )(1 + q +

• +

q k )F n1k (q ):

• q

{Calculus X n q n 2 z 2 The construction cost is w

• rst-case

quadrati c 17

SLIDE 59 Lemma . The generating functions asso ciated with momen ts f r (z ) = U @ r q F (z ; q ) satisfy a linear ODE that in v

• lv

es T (z ). z f 1 = 1 2 T 3 (1

• T

) 2 ; z f 2 = 1 12 T 4 (24

• 11T

+ 2T 2 ) (1

• T

) 5 Theo rem. [F ull tables, exact fo rm

• f

moments] E[d n;n ] = n 2 (Q(n)

• 1)

E[d 2 n;n ] = n 12 (5n 2 + 4n

• 1
• 8n

Q(n)): Q(n) := 1 + n

• 1

n + (n

• 1)(n
• 2)

n 2 + (n

• 1)(n
• 2)(n
• 3)

n 3 +

• Theo

rem. [F ull tables, asymptotic fo rm

• f

moments] E[d n;n ] = p 2 4 n 3=2

• 2

3 n + p 2 48 n 1=2

• 2

135 + O (n 1 ); V ar[d n;n ] = 10

• 3

24 n 3 + 16

• 3

144 n 2 + p 2 135 n 3=2

• Knuth

(1962-3); Flajolet-P

• blete-Viol

a (1997); Knuth (1997) 20

SLIDE 60 6 Limit distribution A metho d

• f

\pumping" moments | Sta rt from nonlinear combinato ri al decomp

• siti
• n

(BGF) [F (z ; q )] = | Apply derivatives U @ r q to get r th moment. | Exp ect linear

• p

erato r L with Lf r =

• r

[f ; f 1 ; : : : ; f r 1 ] | Solve exactly and/o r

• r

asymptotically (singula riti es) Metho d used

• n
• Quickso

rt, Hennequin 1989: 100 moments; nonGaussian la w

• P

ath length in trees, T ak acs 1990 +

• Area

b elo w w alks, Loucha rd 1984

• In

situ p ermutation , Knuth 1972, Pro dinger et alii P ath length in Ca yley trees: F (z ; q )

• z

e F (q z ;q ) = 21

SLIDE 61 Moment pr

• blem

A classical theo rem : if the \Moment generating function" M (z ) = X r

• r

z r r ! has nonzero radius

• f

convergence, then the la w is uniquely determined b y its moments. A co rolla ry : Convergence

• f

moments implies convergence

• f

distributi

• ns

Example. w (x) = e x ,

• r
• r

! = ) M (z ) = 1 1z Theo rem. F

• r

almost full tables, convergence to the Airy distributi

• n,

Pr f d n;n1 (n=2) 3=2

• xg

! PrfX

• xg

(n ! 1); where X is Airy distributed . E[X r ] =

• (

1 2 ) ( 3r 1 2 )

• r

. X r

• r

w r r ! =

• 2=3

(w )

• 1=3

(w )

• (w

) = 1

• (4

2

• 1)
• w

24

• +

(4 2

• 1)(4

2

• 9)

2!

• w

24

• 2
• (4

2

• 1)(4

2

• 9)(4

2

• 25)

3!

• w

24

• 3

+

• :

24

SLIDE 62

❚❤❡ ❆✐r② ❞❡♥s✐t② ✭❛r❡❛ ✉♥❞❡r ❡①❝✉rs✐♦♥s✮✳

❬▲♦✉❝❤❛r❞ ✶✾✽✹❪✱❬❚❛❦á❝s ✶✾✾✶❪ ❚❤❡ ❝♦♥st❛♥ts ✡❦ s❛t✐s❢② ❚❤❡ ❞❡♥s✐t② ✦✭①✮ ❤❛s ❜❡❡♥ ❝❛❧❝✉❧❛t❡❞ ✐♥ ❬❚❛❦á❝s ✶✾✾✶❪✳ ❚❤❡ ❝♦♥st❛♥ts ☛❦ ❛r❡ t❤❡ ③❡r♦s ♦❢ t❤❡ ❆✐r② ❢✉♥❝t✐♦♥ ❆✐✭③✮✳

SLIDE 63 7 Sparse tables A table with m cells and n elements has g = m

• n

\gaps". SparseTable := <Full> * ... * <Full> (---- g times

• ---)

Biva riate GF is: (F (z ; q )) g The analysis can b e \recycled" Theo rem. F

• r
• spa

rse tables,

• =

n m , mean and va riance: E[d m;n ] = n 2 (Q (m; n

• 1)
• 1);

E[d 2 m;n ] = n 12

• (m
• n)

3 + (n + 3)(m

• n)

2 + (8n + 1)(m

• n)

+ 5n 2 + 4n

• 1

((m

• n)

3 + 4(m

• n)

2 + (6n + 3)(m

• n)

+ 8n)Q (m; n

• 1)
• :

Q (m; n) := 1 + n

• 1

m + (n

• 1)(n
• 2)

m 2 + (n

• 1)(n
• 2)(n
• 3)

m 3 +

• E[d

m;n ] =

• 2(1
• )

n

• 2(1
• )

3 + O (n 1 ); V ar [d m;n ] = 6

• 6

2 + 4 3

• 4

12(1

• )

4 n

• Flajolet-P
• blete-Vi
• la

(1997); Knuth (1997) 26

SLIDE 64 The limit distribution Theo rem. A Gaussian la w. Pro

• f.

Integral

• f

la rge p

• w

ers b y saddle p

• int

[z n ](F (z ; q )) mn = 1 2i I (F (z ; q )) mn dz z n+1

1e-05 8e-06 6e-06 4e-06 2e-06 y 10 5

• 5
• 10

x 20 15 10 5

• 5
• 10

Plus continuit y theo rem fo r cha racterist ic functions q = e i Co rolla ry . W

• rks

fo r la rge assemblies! E.g. [Mahmoud] Distribution so rts with O (n) buck ets. 27

SLIDE 65

▲✐♥❡❛r Pr♦❜✐♥❣ ❛♥❞ r❛♥❞♦♠ ❣r❛♣❤s ✭■✮✳

SLIDE 66

▼❛✐❧s ❡①❝❤❛♥❣❡❞ ✇✐t❤ ❉♦♥ ❑♥✉t❤✳

❉❛t❡✿ ▼♦♥✱ ✷✾ ❙❡♣ ✶✾✾✼ ✶✸✿✶✺✿✷✶ ✲✵✼✵✵ ✭P❉❚✮ ✳ ✳ ✳ ❚♦✿ P❤✐❧✐♣♣❡✳❋❧❛❥♦❧❡t❅✐♥r✐❛✳❢r ❙✉❜❥❡❝t✿ ♥♦t❡ ❢r♦♠ ❉♦♥ ❑♥✉t❤ ❉❡❛r P❤✱ ❖r❞✐♥❛r✐❧② ■ ❛♠ ♥♦t ❤❛♣♣② t♦ r❡❝❡✐✈❡ ❡♠❛✐❧✱ ❜✉t ✐♥ t❤✐s ❝❛s❡ ✐t ✇❛s ✈❡r② t♦✉❝❤✐♥❣ t♦ ❧❡❛r♥ t❤❛t ②♦✉ ❤❛❞ ❞❡❝✐❞❡❞ t♦ ❞❡❞✐❝❛t❡ s✉❝❤ ❛ ♥✐❝❡ ♣❛♣❡r t♦ ♠❡✱ ❥✉st ❛❢t❡r ■ ❤❛❞ ✭s❡❝r❡t❧②✮ ❞❡❝✐❞❡❞ t♦ ❞❡❞✐❝❛t❡ r❡❢❡r❡♥❝❡ ❬✷✷❪ t♦ ②♦✉✦ ❇✉t ■ ❤❛✈❡♥✬t t✐♠❡ t♦ st✉❞② ✐t ✐♥ ❞❡t❛✐❧ ♥♦✇✱ ❛s ■✬♠ ✇♦r❦✐♥❣ ✶✺✵✪ t✐♠❡ ♦♥ t❤❡ ♥❡✇ ❡❞✐t✐♦♥ ♦❢ ❱♦❧✉♠❡ ✸✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇❡st r❡❣❛r❞s✱ ❉♦♥

SLIDE 67

❈♦♠❜✐♥❛t♦r✐❛❧ ❛♣♣r♦❛❝❤ t♦ ▲✐♥❡❛r Pr♦❜✐♥❣✳

SLIDE 68

❈♦♠❜✐♥❛t♦r✐❛❧ ❆♥❛❧②s✐s ✭❋P❱✮✳

SLIDE 69

❙♦❧✉t✐♦♥ t♦ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡❝✉rr❡♥❝❡ ✭❑♥✉t❤✮✳

SLIDE 70

❈♦♥❝❧✉s✐♦♥s ✭❑♥✉t❤✮✳

SLIDE 71

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ♣❛r❦✐♥❣ ❢✉♥❝t✐♦♥ ❋♥✭q✮✳

✎ ❙❡✈❡r❛❧ ❝♦♠❜✐♥❛t♦r✐❛❧ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ✈❡r② ✐♠♣♦rt❛♥t ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦❜❧❡♠s✳ ❬❑r❡✇❡r❛s ✶✾✽✵❪

SLIDE 72

P❛r❦✐♥❣✱ r❛♥❞♦♠ ❣r❛♣❤s ❛♥❞ r❛♥❞♦♠ tr❡❡s✳

❬❙♣❡♥❝❡r ✶✾✾✼❪✳ ❇❋❙ tr❛✈❡rs❛❧ ♦❢ ❛ r❛♥❞♦♠ ❣r❛♣❤ ✌ ✇✐t❤ ✈❡rt✐❝❡s ④✵✱ ✶✱ ✳ ✳ ✳ ✱ ♥⑥✳ ■♥❞✉❝❡s ❛ q✉❡✉❡ ✭◗❦✭✜✮✮✶✔❦✔♥ ❛♥❞ ❛ s♣❛♥♥✐♥❣ tr❡❡ ✜✳ ❇❋❙ ✐♥❞✉❝❡s ❛ ♣❛r❦✐♥❣ s❡q✉❡♥❝❡ ✭❆❦✭✜✮✮✶✔❦✔♥✳ ❊①✿ ✭❆❦✭✜✮✮ ❂ ❢❢✻❀ ✽❣❀ ❢✷❀ ✸❣❀ ✣❀ ❢✼❣❀ ❢✶❀ ✹❣❀ ❢✺❣❀ ❢✾❣❀ ✣❀ ✣❣✳ ①❦✭✜✮ ❂ ❥❆❦✭✜✮❥✳ ❊①✿ ✭①❦✭✜✮✮ ❂ ❢✷❀ ✷❀ ✵❀ ✶❀ ✷❀ ✶❀ ✶❀ ✵❀ ✵❣✳ ②❦✭✜✮ ❂ ①✶✭✜✮ ✰ ①✷✭✜✮ ✰ ✿ ✿ ✿ ✰ ①❦✭✜✮ ❦ ✰ ✶✱ s✐③❡ ♦❢ q✉❡✉❡ ✭◗✭✜✮✮ ❜❡❢♦r❡ st❡♣ ❦✳ ❊①✿ ✭②❦✭✜✮✮ ❂ ❢✷❀ ✸❀ ✷❀ ✷❀ ✸❀ ✸❀ ✸❀ ✷❀ ✶❣✳ ②✶✭✜✮ ✰ ✿ ✿ ✿ ②♥✭✜✮ ♥ ✐s t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t✳ ❊①✿ ✶✷✳ ❇❋❙ ✐♥❞✉❝❡s ❛ r❛♥❞♦♠ ✇❛❧❦ ❡①❝✉rs✐♦♥✳ ❊①✿ ❜✳

SLIDE 73

P❛r❦✐♥❣ s❡q✉❡♥❝❡s ❛♥❞ r❛♥❞♦♠ ❣r❛♣❤s✳

✎ ❈♥❀❦✿ ★ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✇✐t❤ ♥ ✈❡rt✐❝❡s ❛♥❞ ♥ ✰ ❦ ✶ ❡❞❣❡s✳ ✎ ❉♥✰✶❀♥✿ ❘❱ ❢♦r t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✐♥ ♣❛r❦✐♥❣ ✇✐t❤ ♥ ❝❛rs✳ ❚❤❡♦r❡♠✿

❈♥❀❦ ❈♥❀✵ ❂ ❊

❤❉♥❀♥✶

❦

✁✐

✳ ❙❦❡t❝❤✿ ❍♦✇ ♠❛② ❣r❛♣❤s ❣✐✈❡ t❤❡ s❛♠❡ ✜ ✇✐t❤ t❤✐s ❇❋❙❄

• r❛♣❤ ❤❛s t❤❡ ❡❞❣❡s ♦❢ ✜ ♣❧✉s s♦♠❡ ♦❢ t❤❡ ✭②✶✭✜✮ ✶✮✰

✿ ✿ ✿ ✰ ✭②✶✭✜✮ ✶✮ ❡❞❣❡s ❥♦✐♥✐♥❣ ♣❛r❦❡❞ ❝❛r ✇✐t❤ ❝♦❧❧✐s✐♦♥s✳ ❈♥✰✶❀❦ ❂

❳

✜

②✶✭✜✮✰✿✿✿✰②✶✭✜✮♥

❦

✁ ❂ ❊ ❤❉♥✰✶❀♥

❦

✁✐

✭♥ ✰ ✶✮♥✶✳ ✭♥ ✰ ✶✮♥✶ ✐s ❜♦t❤ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛r❦✐♥❣ ❢✉♥❝t✐♦♥s ✇✐t❤ ♥ ❝❛rs ❛♥❞ ❧❛❜❡❧❧❡❞ tr❡❡s ✇✐t❤ ♥ ✰ ✶ ♥♦❞❡s✳

SLIDE 74

❇❛❝❦ t♦ ❑♥✉t❤ ✭✶✾✾✼✮✳

P✳❣✳❢✳ ❢♦r t❤❡ t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✐♥ ♣❛r❦✐♥❣ ✇✐t❤ ♥ ❝❛rs✿ ❋♥✭q✮ ❋♥✭✶✮✿ ▼♦r❡♦✈❡r ❊

✧✥

❉♥✰✶❀♥ ❦

✦★

❂ ✶ ❦✦ ❋ ✭❦✮

♥ ✭✶✮

❋♥✭✶✮ ❀ ❛♥❞ ❋✭✶ ✰ q✮ ❂

❳

❦✕✵

✶ ❦✦❋ ✭❦✮

♥ ✭✶✮q❦✿

✎ ❙♦✱ ✐❢ ❋♥✭q✮ ❡♥✉♠❡r❛t❡s t♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥ts ✐♥ ♣❛r❦✐♥❣ ✇✐t❤ ♥ ❝❛rs✱ t❤❡♥ ❋♥✭✶ ✰ q✮ ❡♥✉♠❡r❛t❡s ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✇✐t❤ ♥ ✰ ✶ ♥♦❞❡s ❞✐s❝r✐♠✐♥❛t❡❞ ❜② t❤❡✐r ❡①❝❡ss✦ ✳

SLIDE 75

■♥✈❡rs✐♦♥s ✐♥ ❈❛②❧❡② tr❡❡s✳

❬●❡ss❡❧ ❛♥❞ ❲❛♥❣ ✶✾✼✾❪ ❚❤❡ ✐♥✈❡rs✐♦♥s ❛r❡ ❢✹❀ ✸❣❀ ❢✹❀ ✷❣❀ ❢✻❀ ✷❣✱ ❛♥❞ ❢✻❀ ✺❣✳ ❇✐❥❡❝t✐♦♥ ✇✐t❤ ♣❛r❦✐♥❣ ♣r♦❜❧❡♠✳ ❙❛♠❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✳ ❇❋❙ st❛rt✐♥❣ ❛t ✶✱ ✈✐s✐t✐♥❣ t❤❡ ❣r❡❛t❡st ✉♥✈✐s✐t❡❞ ♥♦❞❡ ✜rst✳ ❍♦✇ ♠❛♥② ❣r❛♣❤s s❤❛r❡ t❤❡ s❛♠❡ s♣❛♥♥✐♥❣ tr❡❡❄✿ ■♥✈❡rs✐♦♥s✦ ✎ ■♥✭t✮✿ ★ ♦❢ ✐♥✈❡rs✐♦♥s ♦❢ ❛ tr❡❡ r♦♦t❡❞ ❛t ✶✳ ✎ ❈♥✭t✮ ❂ t♥✶■♥✭✶ ✰ t✮✳

SLIDE 76

P❛r❦✐♥❣ s❡q✉❡♥❝❡s✱ r❛♥❞♦♠ ❡①❝✉rs✐♦♥s✱ ✳ ✳ ✳

❬❙♣❡♥❝❡r ✶✾✾✼❪✳ ✭❆❦✭✜✮✮ ❂ ❢❢✻❀ ✽❣❀ ❢✷❀ ✸❣❀ ✣❀ ❢✼❣❀ ❢✶❀ ✹❣❀ ❢✺❣❀ ❢✾❣❀ ✣❀ ✣❣✳ ①❦✭✜✮ ❂ ❥❆❦✭✜✮❥✳ ❊①✿ ✭①❦✭✜✮✮ ❂ ❢✷❀ ✷❀ ✵❀ ✶❀ ✷❀ ✶❀ ✶❀ ✵❀ ✵❣✳ ②❦✭✜✮ ❂ ①✶✭✜✮ ✰ ①✷✭✜✮ ✰ ✿ ✿ ✿ ✰ ①❦✭✜✮ ❦ ✰ ✶✱ s✐③❡ ♦❢ q✉❡✉❡ ✭❆✭✜✮✮ ❜❡❢♦r❡ st❡♣ ❦✳ ❊①✿ ✭②❦✭✜✮✮ ❂ ❢✷❀ ✸❀ ✷❀ ✷❀ ✸❀ ✸❀ ✸❀ ✷❀ ✶❣✳ ❚❤❡r❡ ❛r❡

♥✦ ①✶✦✿✿✿①♥✦ ♣❛r❦✐♥❣s s❡qs✳ ❛ss♦❝✐❛t❡❞ t♦ ✭②✶❀ ✿ ✿ ✿ ❀ ②❦✮✱

✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣r♦♣♦rt✐♦♥❛❧ t♦ ◗♥

✐❂✶ ❡✶ ①✐✦ ✱ ✭♥ P♦✭✶✮ ✐✳✐✳❞✳✮✳

P♦✐ss♦♥ ❘✳❲✳ ✇✐t❤ ②✵ ❂ ✵✱ st❡♣s ①✐ ✶✱ ❝♦♥❞✐t✐♦♥❡❞ t♦ ❜❡ ♣♦s✐t✐✈❡ ♦♥ t✐♠❡ ✶❀ ✷❀ ✿ ✿ ✿ ❀ ♥ ❛♥❞ ③❡r♦ ♦♥ t✐♠❡ ♥ ✰ ✶✳

SLIDE 77

✳ ✳ ✳ ❛♥❞ t❤❡ ✐♥❡✛❛❜❧❡ ❇r♦✇♥✐❛♥ ❡①❝✉rs✐♦♥✦

✭①❦✭✜✮✮✵✔❦✔♥ ❛r❡ P♦✭✶✮ ✐✳✐✳❞✳ ❝♦♥❞✐t✐♦♥❡❞ ❛s ❛❜♦✈❡✳ ❈♦rr❡s♣♦♥❞✐♥❣ ✉♥❧❛❜❡❧❧❡❞ tr❡❡ ✜ ✐s ❛ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡ ✇✐t❤ P♦✭✶✮ ♣r♦❣❡♥②✱ ❝♦♥str❛✐♥❡❞ t♦ ❤❛✈❡ ♥ ✰ ✶ ♥♦❞❡s✱ ❛♥❞ ①❦ ✐s t❤❡ ♣r♦❣❡♥② ♦❢ ❦t❤ ♥♦❞❡ ✈✐s✐t❡❞ ❜② t❤❡ ❇❋❙✳ ■t ✐s ❦♥♦✇♥ t❤❛t

✏ ②❜♥t❝

♣♥

✑

✵✔t✔✶ ▲

• ✦✭❡✭t✮✮✵✔t✔✶✱ ✭ ▲
• ✦ ❞❡♥♦t❡s

❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❧❛✇✮ ✇✐t❤ ✭❡✭t✮✮ t❤❡ ❇r♦✇♥✐❛♥ ❡①❝✉rs✐♦♥✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ❉♥✰✶❀♥

♥♣♥ ▲

• ✦

❘ ✶

✵ ❡✭t✮❞t✿ ♠❛①❦ ②❦ ♣♥ ▲

• ✦♠ ❂ ♠❛①

✵✔t✔✶❡✭t✮✿

❬❈❤❛ss❛✐♥❣ ❛♥❞ ▼❛r❝❦❡rt ✷✵✵✶❪✳ ✎ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ♠♦♠❡♥ts✳ ❈♦✉♣❧✐♥❣ ❧❛❜❡❧❡❞ tr❡❡s✲❡♠♣✐r✐❝❛❧ ♣r♦❝❡ss❡s ✉s✐♥❣ ♣❛r❦✐♥❣ ❢✉♥❝t✐♦♥s✳ ❆❧t❡r♥❛t✐✈❡ ♣r♦❜❛❜✐❧✐st✐❝ ♣r♦♦❢ t♦ ❬❋❧❛❥♦❧❡t✱ P♦❜❧❡t❡✱ ❱✐♦❧❛ ✶✾✾✽❪ ❢♦r t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♠♦♠❡♥ts t♦✇❛r❞s ♠♦♠❡♥ts ♦❢ t❤❡ ❆✐r② ❧❛✇✳

SLIDE 78

▼♦r❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥s✿ ♥❡✇ ❝❛s❡s✳

❬❏❛♥s♦♥ ✷✵✵✶❪ ❋♦✉r ❞✐✛❡r❡♥t ♠❡t❤♦❞s ✉s❡❞ ✐♥ t❤❡ ♣r♦♦❢✦ ❚❤❡r❡ ✐s ❛ ♣❤❛s❡ tr❛♥s✐t✐♦♥ ❛r♦✉♥❞ ♠ ♥ ✏ ♣♠✳ ▲❡t ❯♠❀♥ t❤❡ ❛✈❡r❛❣❡ ❝♦st ♦❢ ❛♥ ✉♥s✉❝❝❡ss❢✉❧ s❡❛r❝❤ ❜❡❣✐♥♥✐♥❣ ❛t ❛ r❛♥❞♦♠ ❝❡❧❧ ❤ ✭❛✈❡r❛❣❡❞ ♦✈❡r ❤✮✳ ❚❤❡♥

SLIDE 79

✧▼♦♥❦❡② ❙❛❞❞❧❡✧ ❛s t❤❡ ❆▲●❖ ♣r♦❥❡❝t✬s ❧♦❣♦✳

SLIDE 80

❉✐str✐❜✉t✐♦♥ ♦❢ ❧❡♥❣t❤s ♦❢ ♣❛r❦✐♥❣ ❜❧♦❝❦s✳

❬P✐tt❡❧ ✶✾✽✺❪✳ ✎ ❚❛❜❧❡ ✇✐t❤ ❵ ❂ ♠ ♥ ❡♠♣t② ♣❧❛❝❡s✳ ✎ ❇♠❀❵

❦

❧❡♥❣t❤ ♦❢ ❦t❤ ❧❛r❣❡st ❝❧✉st❡r✳ ✎ ❲❤❡♥ ❵❂♠ ✦ ✕ ✇✐t❤ ✕ ❃ ✵ t❤❡♥

❇♠❀❵

✶

❂ ✷ ❧♦❣ ♠✸ ❧♦❣ ❧♦❣ ♠✰☎♠

✷✭✕✶❧♦❣ ✕✮

❀ ✇❤❡r❡ ☎♠

❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ❛♥ ❡①tr❡♠❡✲✈❛❧✉❡ ❞✐str✐❜✉t✐♦♥✳ ▼♦r❡♦✈❡r✱ ❢r♦♠ ❬❈❤❛ss❛✐♥❣ ❛♥❞ ▲♦✉❝❤❛r❞ ✷✵✵✷❪ ✇❡ ❤❛✈❡ ❚❤❡♦r❡♠✿ ❋♦r ♥❀ ♠ ❥♦✐♥t❧② t♦ ✰✶✱

✶

✐❢ ♣♠ ❂ ♦✭❵✮❀ ❇♠❀❵

✶

❂♠

▲

• ✦ ✵❀

✷

✐❢ ❵ ❂ ♦✭♣♠✮❀ ❇♠❀❵

✶

❂♠

▲

• ✦ ✶✿

P❤❛s❡ tr❛♥s✐t✐♦♥ ♦❝❝✉rs ✇❤❡♥ ❵ ❂ ✂✭♣♠✮✳ ▲❛r❣❡st ❜❧♦❝❦ r❡❛❝❤❡s ❖✭♠✮ ❛❢t❡r ♣♠ ❝❛rs ❛rr✐✈❡ ❛t t❤❡ ❝r✐t✐❝❛❧ r❡❣✐♦♥✳ ❙♦✱ ✇❡ ❤❛✈❡ ❛ ❝♦❛❧❡s❝❡♥❝❡ ♣❤❡♥♦♠❡♥❛✳ ❙t✉❞✐❡❞ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ t❤❡♦r②✳

SLIDE 81

❙✐③❡ ♦❢ t❤❡ ✜rst ❝r❡❛t❡❞ ❝❧✉st❡r✳

❬❈❤❛ss❛✐♥❣ ❛♥❞ ▲♦✉❝❤❛r❞ ✷✵✵✷❪✳ ❘♠❀♥✱ s✐③❡ ♦❢ t❤❡ ❝❧✉st❡r ♦❢ t❤❡ ✜rst ❛rr✐✈❡❞ ❝❛r✳ Pr❬❘♠❀♥ ❂ ❦❪ ❂

♥✶

❦✶

✁✭❦✰✶✮❦✶✭♠❦✶✮♥❦✶✭♠♥✶✮

♠♥✶

✳ Pr❬❘♠❀♥ ❂ ❦❪ ✘ ✭❦ ✰ ✶✮❦✶ ❡☛✭❦✰✶✮☛❦✶

✭❦✶✮✦

✭✶ ☛✮✱ ✭✵ ❁ ☛ ❁ ✶✮✳ Pr❬❘♠❀♥ ❂ ❦❪ ❂ ✶

♠❢

✏

✕❀ ❦

♠

✑

✰ ❖✭♠✸❂✷✮❀ ♠ ♥ ❂ ✕♣♠✱ ✇❤❡r❡ ❢ ✐s t❤❡ ❞❡♥s✐t② ♦❢

◆✷ ✕✷✰◆✷ ✭◆ st❛♥❞❛r❞ ●❛✉ss✐❛♥✮✱ ✇✐t❤

❢ ✭✕❀ ①✮ ❂

✕ ✷✙①✶❂✷✭✶ ①✮✸❂✷❡①♣

✏

• ✕✷①

✷✭✶①✮

✑

✶❪✵❀✶❬✭①✮✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❡♥ ♠ ♥ ❂ ✕♣♠ t❤❡♥ ❘♠❀♥

♠ ▲

• ✦✿

◆✷ ✕✷✰◆✷ ✳

▲✐♠✐t ❝❛s❡s✿ ❘♠❀♥

♠ P

• ✦✵ ✭✕ ✦ ✶✮ ❛♥❞ ❘♠❀♥

♠ P

• ✦✶ ✭✕ ✦ ✵✮✳

SLIDE 82

❉✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝❧✉st❡rs✳

❇♠❀❵

❦

❧❡♥❣t❤ ♦❢ t❤❡ ❦t❤ ❧❛r❣❡st ❝❧✉st❡r ❛♥❞ ❇♠❀❵ ❂

✏

❇♠❀❵

❦

✑

❦✕✶✳

▲❡t ❡✭t✮ t❤❡ ♥♦r♠❛❧✐③❡❞ ❇r♦✇♥✐❛♥ ❡①❝✉rs✐♦♥✳ ▲❡t ✠✕❡✭①✮ ❂ ❡✭①✮ ✕①

s✉♣ ✵✔②✔①✭❡✭②✮ ✕②✮✳

❇✭✕✮ ❂ ✭❇❦✭✕✮✮❦✕✶ ❞❡❝r❡❛s✐♥❣ ✇✐❞t❤s ♦❢ ✠✕❡✭①✮ ❡①❝✉sr✐♦♥s✳ ❚❤❡♦r❡♠✿ ■❢ ❧✐♠

❵ ♣♠ ❂ ✕ ❃ ✵✱ t❤❡♥ ❇♠❀❧ ♠ ▲

• ✦❇✭✕✮✳

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ❢r♦♠ ❬P❛✈❧♦✈ ✶✾✼✼❪ ✐♥ r❛♥❞♦♠ ❢♦r❡st ✭❛♥❞ ✐ts r❡❧❛t✐♦♥ ✇✐t❤ t❤❡ ♣❛r❦✐♥❣ ♣r♦❜❧❡♠✮✱ ❏♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ ❇✭✕✮ ♦✉t ♦❢ r❡❛❝❤✳

SLIDE 83

❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥s ♦❢ ❜❧♦❝❦ ❧❡♥❣❤ts✳

SLIDE 84

❙✐③❡✲❜✐❛s❡❞ ♣❡r♠✉t❛t✐♦♥s ❘✭✕✮✳

❘✭✕✮ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❇✭✕✮✳

✶

❈❤♦♦s❡ ❘✶✭✕✮ ✇✐t❤ Pr✭❘✶✭✕✮ ❂ ❇❦✭✕✮❥❇✭✕✮✮ ❂ ❇❦✭✕✮✳

✷

❙❛♠❡ ❢♦r ❘❦✭✕✮ ❜✉t ✇✐t❤ t❤❡ t❡r♠s t❤❛t ❞✐❞ ♥♦t ❛♣♣❡❛r ❜❡❢♦r❡✳

✎ ❘✭✕✮ ❛♣♣❡❛rs ✐♥ t❤❡ ✁✲✈❛❧✉❡❞ ❢r❛❣♠❡♥t❛t✐♦♥ ♣r♦❝❡ss ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❝♦♥t✐♥✉✉♠ r❛♥❞♦♠ tr❡❡ ✭❈❘❚✮ ✐♥ t❤❡ st❛♥❞❛r❞ ❛❞❞✐t✐✈❡ ❝♦❛❧❡s❝❡♥❝❡✳ ✎ ❬❆❧❞♦✉s ❛♥❞ P✐t♠❛♥ ✶✾✾✽❪✳ ▲❡t ❘♠❀❵ ❂

✏

❘♠❀❵

❦

✑

❦✕✶✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❜❧♦❝❦ s✐③❡s ♦r❞❡r❡❞ ❜②

❞❛t❡ ♦❢ ❜✐rt❤✱ ❛♥❞ ❘❦✭✕✮ ❂ ❘♠❀❵

❦

♠

✇✐t❤ ❵ ❂ ♥ ♠ ❂ ❞✕♣♠❡✳ ❚❤❡♦r❡♠✿ ■❢ ❧✐♠

❵ ♣♠ ❂ ✕ ❃ ✵✱ t❤❡♥ ❘♠❀❧ ♠ ▲

• ✦❘✭✕✮✳

SLIDE 85

❈♦❛❧❡s❝❡♥❝❡✿ ❡♠❡r❣❡♥❝❡ ♦❢ ❛ ❣✐❛♥t ❜❧♦❝❦✳

❯♣ t♦ ♥♦✇ ♣❛r❦✐♥❣ ❢r♦③❡♥ ❛t ✜①❡❞ ✕✿ ♥✭✕✮ ❂ ♠ ❜✕♣♠❝✳ ❚♦ ✉♥❞❡rst❛♥❞ ❝♦❛❧❡s❝❡♥❝❡✿ ✉♥❞❡rst❛♥❞ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ ♣❛r❦✐♥❣ s❝❤❡♠❡s ❛t t✐♠❡s ♥✭✕✶✮ ❁ ✿ ✿ ✿ ❁ ♥✭✕❦✮✳ ❏♦✐♥t ❞✐str✐❜✉t✐♦♥✿ ✭s✐③❡✭✕✮✱ ✐♥✐t✐❛❧ ♣♦s✐t✐♦♥✭✕✮✮✕✕✵ ♦❢ ❡❛❝❤ ❜❧♦❝❦✳ ❈♦❛❧❡s❝❡♥❝❡ ✐♥ t❤❡ ❞✐s❝r❡t❡ ♠♦❞❡❧✱ tr❛♥s❧❛t❡s ✐♥t♦ ❝♦❛❧❡s❝❡♥❝❡ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s ♠♦❞❡❧ ❛t t❤❡ ❧✐♠✐t✳ ❚✇♦ ❝♦♥str✉❝t✐♦♥s ♦❢ t❤❡ ❛❞❞✐t✐✈❡ ❝♦❛❧❡s❝❡♥t✳ ❬❆❧❞♦✉s ❛♥❞ P✐t♠❛♥ ✶✾✾✽❪ ❈❘❚ ❛s ❛ ❧✐♠✐t ♦❢ ❛ ❞✐s❝r❡t❡ ♠♦❞❡❧ ♦❢ ❝♦❛❧❡s❝❡♥❝❡✲❢r❛❣♠❡♥t❛t✐♦♥ ♣r♦❝❡ss t❤❛t st❛rts ✇✐t❤ ❛ r❛♥❞♦♠ ✉♥r♦♦t❡❞ ❧❛❜❡❧❧❡❞ tr❡❡ ✭r❡✈❡rs❡ ♦❢ ♣r♦❝❡ss ♦❢ ❞❡❧❡t✐♥❣ ❡❞❣❡s ❛t r❛♥❞♦♠✮✳ ✎ ❬ ❇❡rt♦✐♥ ✷✵✵✵❪ ❜❛s❡❞ ♦♥ ①❝✉rs✐♦♥s ♦❢ t❤❡ ❢❛♠✐❧② ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✭✠✕❡✮✕✕✵✳ ✎ ❬❈❤❛ss❛✐♥❣ ❛♥❞ ▲♦✉❝❤❛r❞ ✷✵✵✷❪ ♣r♦✈❡ t❤❛t ❛s②♠♣t♦t✐❝❛❧❧② ♣❛r❦✐♥❣ s❝❤❡♠❡s ❧❡❛❞ t♦ t❤✐s ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❛❞❞✐t✐✈❡ ❝♦❛❧❡s❝❡♥t✳

SLIDE 86

P❛r❦✐♥❣ s❝❤❡♠❡s ❛♥❞ P❛✈❧♦✈✬s ❢♦r❡sts✳

SLIDE 87

P❛r❦✐♥❣ ✇✐t❤ ❜✉❝❦❡ts✿ ❞✐str✐❜✉t✐♦♥ ♦❢ ♦✈❡r✢♦✇✳

▲❡t ❣❜✭♠❀ ♥❀ ❞✮ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛r❦✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ ❞❡❢❡❝t ❞✱ ✇✐t❤ ❜✉❝❦❡ts ♦❢ s✐③❡ ❜✳ ▲❡t ●❜✭③❀ ✉❀ ✈✮ ❂

❳

♠✕✵

❳

♥✕✵

❳

❞✕✵

❣❜✭♠❀ ♥❀ ❞✮ ③♥

♥✦ ✉♠✈❞✳

❚❤❡♥ ❬❙❡✐t③ ✷✵✵✾❪

• ❜✭③❀ ✉❀ ✈✮ ❂

✥

✶ ✶ ✉

✈❦ ❡③✈

✦

❜✶

❨

❥❂✶

✒

✶ ❜

③✈❚✭ ✦❥✉

✶ ❜ ③

❜

✮

✓

❜✶

❨

❥❂✶

✒

✶ ❜

③❚✭ ✦❥✉

✶ ❜ ③

❜

✮

✓ ✿

▲❡t ✇♠❀❜☛❀❦ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❤❛✈✐♥❣ ❦ ❝❛rs ❣♦✐♥❣ t♦ ♦✈❡r✢♦✇ ✐♥ ❛ ❜☛✲❢✉❧❧ t❛❜❧❡ ✇✐t❤ ♠ ❜✉❝❦❡ts ♦❢ s✐③❡ ❜ ❛♥❞ ☛ ❁ ✶✱ ❛♥❞ ✡♠✭❜☛❀ ③✮ ❂ P

❦✕✵ ✇♠❀❜☛❀❦③❦✳

❚❤❡♥ ❬❱✐♦❧❛ ✷✵✶✵❪ ✡♠✭❜☛❀ ③✮ ❂

✒❜✭✶ ☛✮✭③ ✶✮

③❜ ❡❜☛✭③✶✮

✓ ◗❜✶

❥❂✶

✏

③ ❚✭✦❥☛❡☛✮

☛

✑ ◗❜✶

❥❂✶

✏

✶ ❚✭✦❥☛❡☛✮

☛

✑ ✰ ❖ ✏

☛❜♠✑ ✿

SLIDE 88

▲✐♥❡❛r Pr♦❜✐♥❣ ✇✐t❤ ❇✉❝❦❡ts✿ ❜✉❝❦❡t ♦❝❝✉♣❛♥❝②✳

▲❡t ❚❞✭❜☛✮ ❜❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ❣✐✈❡♥ ❜✉❝❦❡t ❤❛s ♠♦r❡ t❤❛♥ ❞ ❡♠♣t② ♣❧❛❝❡s✱ ✇❤❡♥ ♥❀ ♠ ✦ ✶✱ ✵ ✔ ♥❂❜♠ ❂ ☛ ❁ ✶✳ ❋r♦♠ ❬❱✐♦❧❛ ✷✵✶✵❪✱ ✐♥s♣✐r❡❞ ✐♥ ❬❇❧❛❦❡ ❛♥❞ ❑♦♥❤❡✐♠ ✶✾✼✼❪✿

❚❤❡♦r❡♠

❚❞✭❜☛✮ ❂ ❜✭✶ ☛✮ ❬✉❞❪ ◗❜✶

✐❂✶

✏

✶ ✉❚✭✦✐☛❡☛✮

☛

✑ ◗❜✶

❥❂✶

✏

✶ ❚✭✦❥☛❡☛✮

☛

✑

❀ ✵ ✔ ❞ ✔ ❜ ✶❀ ✇❤❡r❡ ❚ ✐s t❤❡ ❚r❡❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✦ ✐s ❛ ❜✲t❤ r♦♦t ♦❢ ✉♥✐t②✳ ❚❤❡ s❡q✉❡♥❝❡ ❚❦❀❞❀❜ ❂ ❦✦❬☛❦❪❚❞✭☛✮❀ ✵ ✔ ❞ ❁ ❜ ✐s s❡q✉❡♥❝❡ ❊■❙ ❆✶✷✹✹✺✸ ◆❡✐❧ ❙❧♦❛♥❡✬s ❊♥❝②❝❧♦♣❡❞✐❛ ♦❢ ■♥t❡❣❡r ❙❡q✉❡♥❝❡s✳

SLIDE 89

▲✐♥❡❛r Pr♦❜✐♥❣ ❛♥❞ P❛❦✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤ ❇✉❝❦❡ts✳

❋r♦♠ ❬❱✐♦❧❛ ✷✵✶✵❪✱ ❢♦❧❧♦✇✐♥❣ ❬❱✐♦❧❛ ❛♥❞ P♦❜❧❡t❡ ✶✾✾✽❪ ❛♥❞ t❤❡ ♣✐♦♥❡❡r✐♥❣ ✇♦r❦ ❜② ❬❇❧❛❦❡ ❛♥❞ ❑♦♥❤❡✐♠ ✶✾✼✼❪✿

❚❤❡♦r❡♠

▲❡t ✠♠❀❜☛ ❜❡ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❢♦r t❤❡ ❝♦st ♦❢ s❡❛r❝❤✐♥❣ ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t ✐♥ ❛ ❜☛✲❢✉❧❧ t❛❜❧❡ ✇✐t❤ ♠ ❜✉❝❦❡ts ♦❢ s✐③❡ ❜ ❛♥❞ ☛ ❁ ✶✱ ✉s✐♥❣ t❤❡ ❘♦❜✐♥ ❍♦♦❞ ❧✐♥❡❛r ♣r♦❜✐♥❣ ❤❛s❤✐♥❣ ❛❧❣♦r✐t❤♠✱ ❛♥❞ ❧❡t ✠♠✭❜☛❀ ③✮ ❜❡ ✐ts ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✳ ❚❤❡♥ ✠♠✭❜☛❀ ③✮ ❂ ③ ❜

❜✶

❳

❞❂✵

❈♠

✏

❜☛❀ ❡

✷✙i❞ ❜ ③✶❂❜✑ ❜✶

❳

♣❂✵

✏

❡

✷✙i❞ ❜ ③✶❂❜✑♣

❀ ✇✐t❤ ❈♠✭❜☛❀ ③✮ ❂ ❜✭✶ ☛✮✭✶ ❡❜☛✭③✶✮✮ ❜☛

③❜ ❡❜☛✭③✶✮✁ ◗❜✶

❥❂✶

✏

③ ❚✭r❥☛❡☛✮

☛

✑ ◗❜✶

❥❂✶

✏

✶ ❚✭r❥☛❡☛✮

☛

✑✿

SLIDE 90

❙♦♠❡ ✜♥❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s✳

Pr♦❜❧❡♠ ✇✐t❤ ❛ ✈❡r② r✐❝❤ ❤✐st♦r②✳ P❛r❛❞✐❣♠ ♦❢ ❛ ♣r♦❜❧❡♠ t❤❛t ♥✐❝❡❧② ✐♥t❡❣r❛t❡s ❛♥❛❧②t✐❝❛❧✱ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛♥❞ ♣r♦❜❛❜✐❧✐st✐❝ ❛♣♣r♦❛❝❤❡s✳ ❚❤✐s ✐♥t❡❣r❛t✐♦♥ ✭t♦❣❡t❤❡r ✇✐t❤ t❤❡ ✉s❡ ♦❢ s②♠❜♦❧✐❝ ♠❡t❤♦❞s✦✮ ❤❛s ❛❧❧♦✇❡❞ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❞❡❡♣ r❡❧❛t✐♦♥s ✇✐t❤ ♦t❤❡r ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠✳ Pr♦❜❧❡♠ t❤❛t ✐t ✐s ❝♦♥t✐♥✉♦✉s❧② ❡✈♦❧✈✐♥❣✱ ❛♥❞ ♥❡✇ ✐♠♣♦rt❛♥t r❡s✉❧ts ❛r❡ st✐❧❧ t♦ ❝♦♠❡✳ ❙♦♠❡ ♦♥❣♦✐♥❣ ✇♦r❦✿ ❚♦t❛❧ ❞✐s♣❧❛❝❡♠❡♥t ✇✐t❤ ❜✉❝❦❡ts✳ ❘❡❧❛t✐♦♥ ✇✐t❤ ♦t❤❡r ♣r♦❜❧❡♠s ❛s ❢♦r ❜❂✶❄ ❋❈❋❙ ✇✐t❤ ❜✉❝❦❡ts✳ ❙t✉❞② ♠♦r❡ ♣r♦♣❡rt✐❡s ♦❢ s❡q✉❡♥❝❡ ❚❞✭❜☛✮ ❛♥❞ ✜♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✳ ◆✉♠❜❡r ♦❢ ♠♦✈❡♠❡♥ts ✐♥ ❞❡❧❡t✐♦♥ ❛❧❣♦r✐t❤♠✳ ❲♦rst ❝❛s❡ ❘♦❜✐♥ ❍♦♦❞✳