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r r rt - - PowerPoint PPT Presentation

Jul 28, 2023 212 likes •632 views

r r rt r t s rt rt Pr

slide-1
SLIDE 1

❚r✐♣❧❡ ❛♥❞ ◗✉❛❞r✉♣❧❡ ❊♥❝r②♣t✐♦♥✿ ❇r✐❞❣✐♥❣ t❤❡ ●❛♣s

❇❛rt ▼❡♥♥✐♥❦ ❛♥❞ ❇❛rt Pr❡♥❡❡❧ ❑❯ ▲❡✉✈❡♥ ✭❇❡❧❣✐✉♠✮

❉❛❣st✉❤❧ ✖ ❏❛♥✉❛r② ✻✱ ✷✵✶✹

✶ ✴ ✶✷

slide-2
SLIDE 2

■♥tr♦❞✉❝t✐♦♥

m c

E

k

✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ ✶✾✽✹ ❉❊❙❳ ✶✾✾✶ ■❉❊❆ ✷✵✵✶ ❆❊❙

✷ ✴ ✶✷

❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ✲ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ✲ ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳

slide-3
SLIDE 3

■♥tr♦❞✉❝t✐♦♥

m c

E

k

✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 ✶✾✽✹ ❉❊❙❳ ✶✾✾✶ ■❉❊❆ ✷✵✵✶ ❆❊❙

✷ ✴ ✶✷

❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ✲ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ✲ ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳

slide-4
SLIDE 4

■♥tr♦❞✉❝t✐♦♥

m c

E

k

✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 ✶✾✽✹ ❉❊❙❳ κ = 184 n = 64 ✶✾✾✶ ■❉❊❆ ✷✵✵✶ ❆❊❙

✷ ✴ ✶✷

❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ✲ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ✲ ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳

slide-5
SLIDE 5

■♥tr♦❞✉❝t✐♦♥

m c

E

k

✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 ✶✾✽✹ ❉❊❙❳ κ = 184 n = 64 ✶✾✾✶ ■❉❊❆ κ = 128 n = 64 ✷✵✵✶ ❆❊❙ κ ≥ 128 n = 128

✷ ✴ ✶✷

❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ✲ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ✲ ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳

slide-6
SLIDE 6

■♥tr♦❞✉❝t✐♦♥

m c

E

k

✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 ✶✾✽✹ ❉❊❙❳ κ = 184 n = 64 ✶✾✾✶ ■❉❊❆ κ = 128 n = 64 ✷✵✵✶ ❆❊❙ κ ≥ 128 n = 128

✷ ✴ ✶✷

  • ❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞

✲

  • ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

✲

  • ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳
slide-7
SLIDE 7

■♥tr♦❞✉❝t✐♦♥✿ ❚r✐♣❧❡✲❉❊❙

m c

E E E

k1 k2 k3

  • ❉♦✉❜❧❡✲❉❊❙✿ ♦♥❧② ♠❛r❣✐♥❛❧ s❡❝✉r✐t② ✐♥❝r❡❛s❡
  • ❚r✐♣❧❡✲❉❊❙
  • E ◦ D ◦ E ✈❡rs✉s E ◦ E ◦ E
  • k1 = k3 ✈❡rs✉s k1 = k3

✸ ✴ ✶✷

slide-8
SLIDE 8

■♥tr♦❞✉❝t✐♦♥✿ ❈❛s❝❛❞❡ ❊♥❝r②♣t✐♦♥

m c

· · · · · ·

E E E

k1 k2 kr

  • κ, n ❛r❜✐tr❛r②
  • r′ := ⌈r/2⌉

✲ ■❞❡❛❧ ❝✐♣❤❡r ♠♦❞❡❧ ■♥❢♦r♠❛t✐♦♥✲t❤❡♦r❡t✐❝ ❞✐st✐♥❣✉✐s❤❡r ❤❛s ❛❝❝❡ss t♦

✹ ✴ ✶✷

slide-9
SLIDE 9

■♥tr♦❞✉❝t✐♦♥✿ ❈❛s❝❛❞❡ ❊♥❝r②♣t✐♦♥

m c

· · · · · ·

E E E

k1 k2 kr

  • κ, n ❛r❜✐tr❛r②
  • r′ := ⌈r/2⌉

✲

  • ■❞❡❛❧ ❝✐♣❤❡r ♠♦❞❡❧
  • ■♥❢♦r♠❛t✐♦♥✲t❤❡♦r❡t✐❝ ❞✐st✐♥❣✉✐s❤❡r ❤❛s ❛❝❝❡ss t♦ E

✹ ✴ ✶✷

slide-10
SLIDE 10

❙t❛t❡ ♦❢ t❤❡ ❆rt

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗

  • ❬▲❡❡✶✸❪✿ κ + min{κ, n} − 16

r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16

✲ ❋♦r ✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r ❚r✐♣❧❡✲❉❊❙✿ ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥

✺ ✴ ✶✷

slide-11
SLIDE 11

❙t❛t❡ ♦❢ t❤❡ ❆rt

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗

  • ❬▲❡❡✶✸❪✿ κ + min{κ, n} − 16

r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16

✲

  • ❋♦r r = 3, 4✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r κ ≤ n

❚r✐♣❧❡✲❉❊❙✿ ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥

✺ ✴ ✶✷

slide-12
SLIDE 12

❙t❛t❡ ♦❢ t❤❡ ❆rt

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗

  • ❬▲❡❡✶✸❪✿ κ + min{κ, n} − 16

r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16

✲

  • ❋♦r r = 3, 4✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r κ ≤ n

❚r✐♣❧❡✲❉❊❙✿ 284 ≤ 288 ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥

✺ ✴ ✶✷

slide-13
SLIDE 13

❙t❛t❡ ♦❢ t❤❡ ❆rt

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗

  • ❬▲❡❡✶✸❪✿ κ + min{κ, n} − 16

r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16

✲

  • ❋♦r r = 3, 4✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r κ ≤ n

❚r✐♣❧❡✲❉❊❙✿ 284 ≤ 288 ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥

✺ ✴ ✶✷

slide-14
SLIDE 14

■♠♣r♦✈✐♥❣ ❆tt❛❝❦s

m c

· · · · · ·

E E E

k1 k2 kr

  • ❬●❛➸✶✸❪✿ ❛tt❛❝❦ ✐♥ 2κ+ r′−1

r′

n q✉❡r✐❡s

✲ ✿ ❉✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ r❛♥❞♦♠ ✐♥ ❝♦♥st❛♥t ★q✉❡r✐❡s

✻ ✴ ✶✷

slide-15
SLIDE 15

■♠♣r♦✈✐♥❣ ❆tt❛❝❦s

m c

· · · · · ·

E E E

k1 k2 kr

  • ❬●❛➸✶✸❪✿ ❛tt❛❝❦ ✐♥ 2κ+ r′−1

r′

n q✉❡r✐❡s

✲

  • κ = 0✿

m c

· · · · · ·

π π π

  • ❉✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ r❛♥❞♦♠ ✐♥ ❝♦♥st❛♥t ★q✉❡r✐❡s

✻ ✴ ✶✷

slide-16
SLIDE 16

■♠♣r♦✈✐♥❣ ❆tt❛❝❦s

m c

· · · · · ·

E E E

k1 k2 kr

  • ❬●❛➸✶✸❪✿ ❛tt❛❝❦ ✐♥ 2κ+ r′−1

r′

n q✉❡r✐❡s

❘❡s✉❧t ✶✿ ❛tt❛❝❦ ✐♥ 2r′κ q✉❡r✐❡s ❆tt❛❝❦ ✐❞❡❛✿

❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♠❡❡t✲✐♥✲t❤❡✲♠✐❞❞❧❡ ❛tt❛❝❦

❬❉❉❑❙✶✷❪✿ ❛tt❛❝❦ ✐♥

✐♥ ✐♥❝♦♠♣❛r❛❜❧❡ ♠♦❞❡❧

❈♦r♦❧❧❛r②✿ ❛tt❛❝❦ ✐♥ q✉❡r✐❡s

✼ ✴ ✶✷

slide-17
SLIDE 17

■♠♣r♦✈✐♥❣ ❆tt❛❝❦s

m c

· · · · · ·

E E E

k1 k2 kr

  • ❬●❛➸✶✸❪✿ ❛tt❛❝❦ ✐♥ 2κ+ r′−1

r′

n q✉❡r✐❡s

❘❡s✉❧t ✶✿ ❛tt❛❝❦ ✐♥ 2r′κ q✉❡r✐❡s

  • ❆tt❛❝❦ ✐❞❡❛✿
  • ❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♠❡❡t✲✐♥✲t❤❡✲♠✐❞❞❧❡ ❛tt❛❝❦
  • ❬❉❉❑❙✶✷❪✿ ❛tt❛❝❦ ✐♥ 2(r−

√ 2r)κ ✐♥ ✐♥❝♦♠♣❛r❛❜❧❡ ♠♦❞❡❧

❈♦r♦❧❧❛r②✿ ❛tt❛❝❦ ✐♥ q✉❡r✐❡s

✼ ✴ ✶✷

slide-18
SLIDE 18

■♠♣r♦✈✐♥❣ ❆tt❛❝❦s

m c

· · · · · ·

E E E

k1 k2 kr

  • ❬●❛➸✶✸❪✿ ❛tt❛❝❦ ✐♥ 2κ+ r′−1

r′

n q✉❡r✐❡s

❘❡s✉❧t ✶✿ ❛tt❛❝❦ ✐♥ 2r′κ q✉❡r✐❡s

  • ❆tt❛❝❦ ✐❞❡❛✿
  • ❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♠❡❡t✲✐♥✲t❤❡✲♠✐❞❞❧❡ ❛tt❛❝❦
  • ❬❉❉❑❙✶✷❪✿ ❛tt❛❝❦ ✐♥ 2(r−

√ 2r)κ ✐♥ ✐♥❝♦♠♣❛r❛❜❧❡ ♠♦❞❡❧

❈♦r♦❧❧❛r②✿ ❛tt❛❝❦ ✐♥ 2κ+ r′−1

r′

min{r′κ,n} q✉❡r✐❡s

✼ ✴ ✶✷

slide-19
SLIDE 19

◆❡✇ ❙t❛t❡ ♦❢ t❤❡ ❆rt

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ κ + min{κ, n/2} ✗ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗ κ + r′−1

r′

min{r′κ, n} ✗

❚r✐♣❧❡✲❉❊❙✿ ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥

✽ ✴ ✶✷

slide-20
SLIDE 20

◆❡✇ ❙t❛t❡ ♦❢ t❤❡ ❆rt

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ κ + min{κ, n/2} ✗ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗ κ + r′−1

r′

min{r′κ, n} ✗

❚r✐♣❧❡✲❉❊❙✿ 284 ≤ 288 ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥

✽ ✴ ✶✷

slide-21
SLIDE 21

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s

m c

E E E

k1 k2 k3

✳

  • ❬❇❘✵✻✱●▼✵✾❪✿ s❡❝✉r✐t② ✉♣ t♦ 2κ+min{κ/2,n/2} q✉❡r✐❡s
  • ❆tt❛❝❦ ✐♥ 2κ+min{κ,n/2} q✉❡r✐❡s ✭♣r❡✈✐♦✉s s❧✐❞❡✮

❘❡s✉❧t ✷✿ t✐❣❤t s❡❝✉r✐t② ✉♣ t♦ q✉❡r✐❡s Pr♦♦❢ ✐❞❡❛✿

  • ❛♣ ❞✉❡ t♦ r❛t❤❡r ✐s♦❧❛t❡❞ ❧❡♠♠❛ ♦❢ ❬❇❘✵✻✱●▼✵✾❪

■♠♣r♦✈❡♠❡♥t ♦❢ ❧❡♠♠❛ ❧❡❛❞s t♦ t✐❣❤t s❡❝✉r✐t②

✾ ✴ ✶✷

slide-22
SLIDE 22

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s

m c

E E E

k1 k2 k3

✳

  • ❬❇❘✵✻✱●▼✵✾❪✿ s❡❝✉r✐t② ✉♣ t♦ 2κ+min{κ/2,n/2} q✉❡r✐❡s
  • ❆tt❛❝❦ ✐♥ 2κ+min{κ,n/2} q✉❡r✐❡s ✭♣r❡✈✐♦✉s s❧✐❞❡✮

❘❡s✉❧t ✷✿ t✐❣❤t s❡❝✉r✐t② ✉♣ t♦ 2κ+min{κ,n/2} q✉❡r✐❡s Pr♦♦❢ ✐❞❡❛✿

  • ❛♣ ❞✉❡ t♦ r❛t❤❡r ✐s♦❧❛t❡❞ ❧❡♠♠❛ ♦❢ ❬❇❘✵✻✱●▼✵✾❪

■♠♣r♦✈❡♠❡♥t ♦❢ ❧❡♠♠❛ ❧❡❛❞s t♦ t✐❣❤t s❡❝✉r✐t②

✾ ✴ ✶✷

slide-23
SLIDE 23

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s

m c

E E E

k1 k2 k3

✳

  • ❬❇❘✵✻✱●▼✵✾❪✿ s❡❝✉r✐t② ✉♣ t♦ 2κ+min{κ/2,n/2} q✉❡r✐❡s
  • ❆tt❛❝❦ ✐♥ 2κ+min{κ,n/2} q✉❡r✐❡s ✭♣r❡✈✐♦✉s s❧✐❞❡✮

❘❡s✉❧t ✷✿ t✐❣❤t s❡❝✉r✐t② ✉♣ t♦ 2κ+min{κ,n/2} q✉❡r✐❡s

  • Pr♦♦❢ ✐❞❡❛✿
  • ●❛♣ ❞✉❡ t♦ r❛t❤❡r ✐s♦❧❛t❡❞ ❧❡♠♠❛ ♦❢ ❬❇❘✵✻✱●▼✵✾❪
  • ■♠♣r♦✈❡♠❡♥t ♦❢ ❧❡♠♠❛ ❧❡❛❞s t♦ t✐❣❤t s❡❝✉r✐t②

✾ ✴ ✶✷

slide-24
SLIDE 24

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

♥♦✇

slide-25
SLIDE 25

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇

slide-26
SLIDE 26

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← →

q ♦♣t✐♦♥s ♦♣t✐♦♥s q ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇

slide-27
SLIDE 27

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s

(α1 = max{2e2κ−n, 2n + κ})

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇

slide-28
SLIDE 28

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇

slide-29
SLIDE 29

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇                              

slide-30
SLIDE 30

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

− →

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

← −

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇                              

slide-31
SLIDE 31

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

− → ← →

♦♣t✐♦♥s ♦♣t✐♦♥s q ♦♣t✐♦♥s

← −

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇                              

slide-32
SLIDE 32

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

− → ← →

♦♣t✐♦♥s 2α2 ♦♣t✐♦♥s q ♦♣t✐♦♥s

← −

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

(α2 = max{2eq/2n, n + κ})

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇                              

slide-33
SLIDE 33

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

← → − → ← →

2κ ♦♣t✐♦♥s 2α2 ♦♣t✐♦♥s q ♦♣t✐♦♥s

← −

♦♣t✐♦♥s ♦♣t✐♦♥s ♦♣t✐♦♥s

(α2 = max{2eq/2n, n + κ})

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇                              

slide-34
SLIDE 34

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

← → − → ← →

2κ ♦♣t✐♦♥s 2α2 ♦♣t✐♦♥s q ♦♣t✐♦♥s

← → ← − ← →

q ♦♣t✐♦♥s 2α2 ♦♣t✐♦♥s 2κ ♦♣t✐♦♥s

(α2 = max{2eq/2n, n + κ})

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇                              

slide-35
SLIDE 35

❚✐❣❤t❡♥✐♥❣ ❙❡❝✉r✐t② ❇♦✉♥❞s✿ Pr♦♦❢ ■❞❡❛

m c

E E E

k1 k2 k3

✳

← → ← → ← →

q ♦♣t✐♦♥s 2α1 ♦♣t✐♦♥s q ♦♣t✐♦♥s E = 2α1q2

(α1 = max{2e2κ−n, 2n + κ})

← → − → ← →

2κ ♦♣t✐♦♥s 2α2 ♦♣t✐♦♥s q ♦♣t✐♦♥s

← → ← − ← →

q ♦♣t✐♦♥s 2α2 ♦♣t✐♦♥s 2κ ♦♣t✐♦♥s E = 4α22κq

(α2 = max{2eq/2n, n + κ})

✶✵ ✴ ✶✷

❬❇❘✵✻✱

  • ▼✵✾❪

          ♥♦✇                              

slide-36
SLIDE 36

❈♦♥❝❧✉s✐♦♥s

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ κ + min{κ, n/2} κ + min{κ, n/2} ✓ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗ κ + min{κ, n/2} κ + r′−1

r′

min{r′κ, n} ✗

❚✐❣❤t s❡❝✉r✐t② ❢♦r ✭♥♦♥✲tr✐✈✐❛❧✮❄

❬▲❡❡✶✸❪✿ ❛s②♠♣t♦t✐❝

s❡❝✉r✐t②

✶✶ ✴ ✶✷

slide-37
SLIDE 37

❈♦♥❝❧✉s✐♦♥s

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1, 2 κ κ ❬❉❍✼✼❪ ✓ r = 3, 4 κ + min{κ/2, n/2} ❬❇❘✵✻✱●▼✵✾❪ κ + n/2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ κ + min{κ, n/2} κ + min{κ, n/2} ✓ r ≥ 5 κ + min

  • (r′−1)

r′

κ, n/2

  • ❬●▼✵✾❪

κ + r′−1

r′ n ❬●❛➸✶✸❪

✗ κ + min{κ, n/2} κ + r′−1

r′

min{r′κ, n} ✗

  • ❚✐❣❤t s❡❝✉r✐t② ❢♦r r ≥ 5 ✭♥♦♥✲tr✐✈✐❛❧✮❄
  • ❬▲❡❡✶✸❪✿ ❛s②♠♣t♦t✐❝ κ + min{κ, n} s❡❝✉r✐t②

✶✶ ✴ ✶✷

slide-38
SLIDE 38

❈♦♥❝❧✉s✐♦♥s

  • ❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❞✐✛❡r❡♥t ♠♦❞❡❧
  • ❈♦♥s✐❞❡r ❝❛s❝❛❞❡❞ ❉❊❙

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ ❛tt❛❝❦ t✐♠❡ ♠❡♠♦r②

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✶✷ ✴ ✶✷

slide-39
SLIDE 39

❈♦♥❝❧✉s✐♦♥s

  • ❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❞✐✛❡r❡♥t ♠♦❞❡❧
  • ❈♦♥s✐❞❡r ❝❛s❝❛❞❡❞ ❉❊❙

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ ❛tt❛❝❦ t✐♠❡ ♠❡♠♦r② r = 2 256 256 r = 3 288 288 r = 4 288 288

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✶✷ ✴ ✶✷

slide-40
SLIDE 40

❈♦♥❝❧✉s✐♦♥s

  • ❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❞✐✛❡r❡♥t ♠♦❞❡❧
  • ❈♦♥s✐❞❡r ❝❛s❝❛❞❡❞ ❉❊❙

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ ❛tt❛❝❦ t✐♠❡ ♠❡♠♦r② r = 2 256 256 257 256 r = 3 288 288 2112 256 r = 4 288 288 2121 256

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✶✷ ✴ ✶✷

slide-41
SLIDE 41

❈♦♥❝❧✉s✐♦♥s

  • ❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❞✐✛❡r❡♥t ♠♦❞❡❧
  • ❈♦♥s✐❞❡r ❝❛s❝❛❞❡❞ ❉❊❙

r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ ❛tt❛❝❦ t✐♠❡ ♠❡♠♦r② r = 2 256 256 257 256 r = 3 288 288 2112 256 r = 4 288 288 2121 256

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✶✷ ✴ ✶✷