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t t rrs - - PowerPoint PPT Presentation

May 17, 2023 45 likes •966 views

t t rrs rss r ts tr s 1 r 1

slide-1
SLIDE 1

❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ●r❛♠♠❛r✲❇❛s❡❞ ❈♦♠♣r❡ss✐♦♥ ♦✈❡r ❋✐①❡❞ ❆❧♣❤❛❜❡ts

❑❛tr✐♥ ❈❛s❡❧1✱ ❍❡♥♥✐♥❣ ❋❡r♥❛✉1✱ ❙❡r❣❡ ●❛s♣❡rs2✱ ❇❡♥❥❛♠✐♥ ●r❛s3✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞1

1 ❚r✐❡r ❯♥✐✈❡rs✐t②✱ ●❡r♠❛♥② 2 ❉❛t❛✻✶✱ ❈❙■❘❖✱ ❆✉str❛❧✐❛ 3 ➱❝♦❧❡ ◆♦r♠❛❧❡ ❙✉♣❡r✐❡✉r❡ ❞❡ ▲②♦♥✱ ❋r❛♥❝❡

■❈❆▲P ✷✵✶✻

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SLIDE 2

❈♦♥t❡①t✲❋r❡❡ ●r❛♠♠❛rs

G = (N, Σ, R, S)✱ N s❡t ♦❢ ♥♦♥t❡r♠✐♥❛❧s✱ Σ t❡r♠✐♥❛❧ ❛❧♣❤❛❜❡t✱ S ∈ N st❛rt s②♠❜♦❧✱ R ⊆ N × (N ∪ Σ)+ s❡t ♦❢ r✉❧❡s r✉❧❡s✳ ✏str❛✐❣❤t✲❧✐♥❡ ♣r♦❣r❛♠✑ ✭♦r s✐♠♣❧② ❣r❛♠♠❛r✮

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SLIDE 3

❈♦♥t❡①t✲❋r❡❡ ●r❛♠♠❛rs

G = (N, Σ, R, S)✱ N s❡t ♦❢ ♥♦♥t❡r♠✐♥❛❧s✱ Σ t❡r♠✐♥❛❧ ❛❧♣❤❛❜❡t✱ S ∈ N st❛rt s②♠❜♦❧✱ R ⊆ N × (N ∪ Σ)+ s❡t ♦❢ r✉❧❡s r✉❧❡s✳ G ✏str❛✐❣❤t✲❧✐♥❡ ♣r♦❣r❛♠✑ ✭♦r s✐♠♣❧② ❣r❛♠♠❛r✮ ⇔ |L(G)| = 1

slide-4
SLIDE 4

❈♦♥t❡①t✲❋r❡❡ ●r❛♠♠❛rs

G = (N, Σ, R, S)✱ N s❡t ♦❢ ♥♦♥t❡r♠✐♥❛❧s✱ Σ t❡r♠✐♥❛❧ ❛❧♣❤❛❜❡t✱ S ∈ N st❛rt s②♠❜♦❧✱ R ⊆ N × (N ∪ Σ)+ s❡t ♦❢ r✉❧❡s r✉❧❡s✳ G ✏str❛✐❣❤t✲❧✐♥❡ ♣r♦❣r❛♠✑ ✭♦r s✐♠♣❧② ❣r❛♠♠❛r✮ ⇔ |L(G)| = 1 |G| =

A→α∈R |α|

slide-5
SLIDE 5
  • r❛♠♠❛r✲❇❛s❡❞ ❈♦♠♣r❡ss✐♦♥
  • ❡♥❡r❛❧ ✐❞❡❛

■♥♣✉t✿ ✇♦r❞ w✱ ❖✉t♣✉t✿ ❣r❛♠♠❛r ❢♦r w✳ ❲❡ ♠❛② ❛s❦ ❢♦r ❛ s❤♦rt❡st ❣r❛♠♠❛r✳ ❛ s❤♦rt ❣r❛♠♠❛r ✭❜✉t ❝♦♠♣✉t❡❞ ❢❛st✮✳ ❛ ❣r❛♠♠❛r t❤❛t ✐s s❤♦rt❡st ✭s❤♦rt✮ ❛♠♦♥❣ ❛❧❧ ❣r❛♠♠❛rs ✇✐t❤

♦♥❧② ♥♦♥✲t❡r♠✐♥❛❧s ✭✐✳ ❡✳✱ ♦♥❧② r✉❧❡s✮✱ ❛ ❞❡r✐✈❛t✐♦♥s tr❡❡ ✇✐t❤ ❛t ♠♦st ❧❡✈❡❧s✱ r✉❧❡s t❤❛t ❤❛✈❡ r✐❣❤t s✐❞❡s ♦❢ s✐③❡ ❛t ♠♦st ✱

slide-6
SLIDE 6
  • r❛♠♠❛r✲❇❛s❡❞ ❈♦♠♣r❡ss✐♦♥
  • ❡♥❡r❛❧ ✐❞❡❛

■♥♣✉t✿ ✇♦r❞ w✱ ❖✉t♣✉t✿ ❣r❛♠♠❛r ❢♦r w✳ ❲❡ ♠❛② ❛s❦ ❢♦r ❛ s❤♦rt❡st ❣r❛♠♠❛r✳ ❛ s❤♦rt ❣r❛♠♠❛r ✭❜✉t ❝♦♠♣✉t❡❞ ❢❛st✮✳ ❛ ❣r❛♠♠❛r t❤❛t ✐s s❤♦rt❡st ✭s❤♦rt✮ ❛♠♦♥❣ ❛❧❧ ❣r❛♠♠❛rs ✇✐t❤

◮ ♦♥❧② k ♥♦♥✲t❡r♠✐♥❛❧s ✭✐✳ ❡✳✱ ♦♥❧② k r✉❧❡s✮✱ ◮ ❛ ❞❡r✐✈❛t✐♦♥s tr❡❡ ✇✐t❤ ❛t ♠♦st k ❧❡✈❡❧s✱ ◮ r✉❧❡s t❤❛t ❤❛✈❡ r✐❣❤t s✐❞❡s ♦❢ s✐③❡ ❛t ♠♦st k✱ ◮ . . .

slide-7
SLIDE 7

❆❧❣♦r✐t❤♠✐❝s ♦♥ ❈♦♠♣r❡ss❡❞ ❙tr✐♥❣s

❘❡q✉✐r❡♠❡♥t ❢♦r ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s✿ ❧✐♥❡❛r ♦r ♥❡❛r ❧✐♥❡❛r t✐♠❡ ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❞❡❝♦♠♣r❡ss✐♦♥✱ s✉✐t❛❜✐❧✐t② ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❞✐r❡❝t❧② ♦♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❞❛t❛✳ ⇒ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❝♦♠♣r❡ss❡❞ str✐♥❣s

  • r❛♠♠❛rs

❝♦✈❡r ♠❛♥② ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s ❢r♦♠ ♣r❛❝t✐❝❡ ✭❡✳ ❣✳✱ ▲❡♠♣❡❧✲❩✐✈✮✱ ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ❡❛s② t♦ ❤❛♥❞❧❡✱ ❛❧❧♦✇ s♦❧✈✐♥❣ ♦❢ ❜❛s✐❝ ♣r♦❜❧❡♠s ✭❝♦♠♣❛r✐s♦♥✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣✱ ♠❡♠❜❡rs❤✐♣ ✐♥ ❛ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡✱ r❡tr✐❡✈✐♥❣ s✉❜✇♦r❞s✮ ❡✣❝✐❡♥t❧②✳

  • r❛♠♠❛r✲❜❛s❡❞ ❝♦♠♣r❡ss✐♦♥ ❤❛s

❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✱ ❝♦♠♣✉t✳ t♦♣♦❧♦❣②✱ ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♦❜❥❡❝ts✱ ❡✳ ❣✳✱ tr❡❡s✱ ✇♦r❞s✳

slide-8
SLIDE 8

❆❧❣♦r✐t❤♠✐❝s ♦♥ ❈♦♠♣r❡ss❡❞ ❙tr✐♥❣s

❘❡q✉✐r❡♠❡♥t ❢♦r ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s✿ ❧✐♥❡❛r ♦r ♥❡❛r ❧✐♥❡❛r t✐♠❡ ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❞❡❝♦♠♣r❡ss✐♦♥✱ s✉✐t❛❜✐❧✐t② ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❞✐r❡❝t❧② ♦♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❞❛t❛✳ ⇒ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❝♦♠♣r❡ss❡❞ str✐♥❣s

  • r❛♠♠❛rs

❝♦✈❡r ♠❛♥② ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s ❢r♦♠ ♣r❛❝t✐❝❡ ✭❡✳ ❣✳✱ ▲❡♠♣❡❧✲❩✐✈✮✱ ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ❡❛s② t♦ ❤❛♥❞❧❡✱ ❛❧❧♦✇ s♦❧✈✐♥❣ ♦❢ ❜❛s✐❝ ♣r♦❜❧❡♠s ✭❝♦♠♣❛r✐s♦♥✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣✱ ♠❡♠❜❡rs❤✐♣ ✐♥ ❛ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡✱ r❡tr✐❡✈✐♥❣ s✉❜✇♦r❞s✮ ❡✣❝✐❡♥t❧②✳

  • r❛♠♠❛r✲❜❛s❡❞ ❝♦♠♣r❡ss✐♦♥ ❤❛s

❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✱ ❝♦♠♣✉t✳ t♦♣♦❧♦❣②✱ ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♦❜❥❡❝ts✱ ❡✳ ❣✳✱ tr❡❡s✱ ✇♦r❞s✳

slide-9
SLIDE 9

❆❧❣♦r✐t❤♠✐❝s ♦♥ ❈♦♠♣r❡ss❡❞ ❙tr✐♥❣s

❘❡q✉✐r❡♠❡♥t ❢♦r ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s✿ ❧✐♥❡❛r ♦r ♥❡❛r ❧✐♥❡❛r t✐♠❡ ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❞❡❝♦♠♣r❡ss✐♦♥✱ s✉✐t❛❜✐❧✐t② ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❞✐r❡❝t❧② ♦♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❞❛t❛✳ ⇒ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❝♦♠♣r❡ss❡❞ str✐♥❣s

  • r❛♠♠❛rs

❝♦✈❡r ♠❛♥② ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s ❢r♦♠ ♣r❛❝t✐❝❡ ✭❡✳ ❣✳✱ ▲❡♠♣❡❧✲❩✐✈✮✱ ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ❡❛s② t♦ ❤❛♥❞❧❡✱ ❛❧❧♦✇ s♦❧✈✐♥❣ ♦❢ ❜❛s✐❝ ♣r♦❜❧❡♠s ✭❝♦♠♣❛r✐s♦♥✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣✱ ♠❡♠❜❡rs❤✐♣ ✐♥ ❛ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡✱ r❡tr✐❡✈✐♥❣ s✉❜✇♦r❞s✮ ❡✣❝✐❡♥t❧②✳

  • r❛♠♠❛r✲❜❛s❡❞ ❝♦♠♣r❡ss✐♦♥ ❤❛s

❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✱ ❝♦♠♣✉t✳ t♦♣♦❧♦❣②✱ ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♦❜❥❡❝ts✱ ❡✳ ❣✳✱ tr❡❡s✱ 2D ✇♦r❞s✳

slide-10
SLIDE 10

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

10 100 1000 10000 100000 1000000 10000000 100000000 . . . ❣r❛♠♠❛r ✇✐t❤

slide-11
SLIDE 11

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

10 100 1000 10000 100000 1000000 10000000 100000000 . . . ❣r❛♠♠❛r ✇✐t❤

slide-12
SLIDE 12

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 100 1000 10000 100000 1000000 10000000 100000000 . . . A1 → 10 ❣r❛♠♠❛r ✇✐t❤

slide-13
SLIDE 13

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 100 1000 10000 100000 1000000 10000000 100000000 . . . A1 → 10 ❣r❛♠♠❛r ✇✐t❤

slide-14
SLIDE 14

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 1000 10000 100000 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 ❣r❛♠♠❛r ✇✐t❤

slide-15
SLIDE 15

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 1000 10000 100000 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 ❣r❛♠♠❛r ✇✐t❤

slide-16
SLIDE 16

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 A3 10000 100000 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 A3 → A20 ❣r❛♠♠❛r ✇✐t❤

slide-17
SLIDE 17

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 A3 10000 100000 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 A3 → A20 ❣r❛♠♠❛r ✇✐t❤

slide-18
SLIDE 18

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 A3 A4 100000 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 A3 → A20 A4 → A30 ❣r❛♠♠❛r ✇✐t❤

slide-19
SLIDE 19

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 A3 A4 100000 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 A3 → A20 A4 → A30 ❣r❛♠♠❛r ✇✐t❤

slide-20
SLIDE 20

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 A3 A4 A5 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 A3 → A20 A4 → A30 A5 → A40 ❣r❛♠♠❛r ✇✐t❤

slide-21
SLIDE 21

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 A3 A4 A5 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 A3 → A20 A4 → A30 A5 → A40 . . . ❣r❛♠♠❛r ✇✐t❤

slide-22
SLIDE 22

❊①❛♠♣❧❡s ✭1/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

A1 A2 A3 A4 A5 1000000 10000000 100000000 . . . A1 → 10 A2 → A10 A3 → A20 A4 → A30 A5 → A40 . . . ⇒ ❣r❛♠♠❛r G ✇✐t❤ |G| = 3n − 1

slide-23
SLIDE 23

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

10 100 1000 10000 100000 1000000 10000000 100000000 . . . ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-24
SLIDE 24

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

10 100 1000 10000 100000 1000000 10000000 100000000 . . . ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-25
SLIDE 25

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A100 10000 100000 1000000 10000000 100000000 . . . A1 → 010 ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-26
SLIDE 26

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A100 10000 100000 1000000 10000000 100000000 . . . A1 → 010 ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-27
SLIDE 27

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A1A2A2000 1000000 10000000 100000000 . . . A1 → 010 A2 → 0A10 ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-28
SLIDE 28

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A1A2A2000 1000000 10000000 100000000 . . . A1 → 010 A2 → 0A10 ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-29
SLIDE 29

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A1A2A2A3A30000 100000000 . . . A1 → 010 A2 → 0A10 A3 → 0A20 ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-30
SLIDE 30

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A1A2A2A3A30000 100000000 . . . A1 → 010 A2 → 0A10 A3 → 0A20 . . . ❣r❛♠♠❛r ✇✐t❤ ❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-31
SLIDE 31

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A1A2A2A3A30000 100000000 . . . A1 → 010 A2 → 0A10 A3 → 0A20 . . . ⇒ ❣r❛♠♠❛r G ✇✐t❤ |G| = 5n

2 + 2k − 3

❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿

slide-32
SLIDE 32

❊①❛♠♣❧❡s ✭2/2✮

w = n

i=1 10i✱ ✇✐t❤ n = 2k✳

1A1A1A2A2A3A30000 100000000 . . . A1 → 010 A2 → 0A10 A3 → 0A20 . . . ⇒ ❣r❛♠♠❛r G ✇✐t❤ |G| = 5n

2 + 2k − 3

❇❡st ❣r❛♠♠❛r ✇❡ ❢♦✉♥❞✿ |G| = 9n

4 + 2k − 2

slide-33
SLIDE 33

❉❡r✐✈❛t✐♦♥ ❚r❡❡s

S → ABCAC , A → Bb , B → aC , C → ab .

slide-34
SLIDE 34

❉❡r✐✈❛t✐♦♥ ❚r❡❡s

S → ABCAC , A → Bb , B → aC , C → ab . S A A B C C B b B b a C a b a b a C a C a b a b a b

slide-35
SLIDE 35

❉❡r✐✈❛t✐♦♥ ❚r❡❡s

S → ABCAC , A → Bb , B → aC , C → ab . S A A B C C B b B b a C a b a b a C a C a b a b a b

slide-36
SLIDE 36

1✲▲❡✈❡❧ ●r❛♠♠❛rs

S → ABCAC , A → aabb , B → aab , C → ab .

slide-37
SLIDE 37

1✲▲❡✈❡❧ ●r❛♠♠❛rs

S → ABCAC , A → aabb , B → aab , C → ab . A A B C C aabb aabb aab ab ab

slide-38
SLIDE 38

❙❤♦rt❡st ●r❛♠♠❛r Pr♦❜❧❡♠

❙❤♦rt❡st ●r❛♠♠❛r Pr♦❜❧❡♠ ❙●P

■♥st❛♥❝❡✿ ❆ ✇♦r❞ w ❛♥❞ ❛ k ∈ N✳ ◗✉❡st✐♦♥✿ ∃ ❣r❛♠♠❛r G ✇✐t❤ L(G) = {w} ❛♥❞ |G| ≤ k❄

slide-39
SLIDE 39

❙❤♦rt❡st ●r❛♠♠❛r Pr♦❜❧❡♠

❙❤♦rt❡st ✭1✲▲❡✈❡❧✮ ●r❛♠♠❛r Pr♦❜❧❡♠ ✭1−✮❙●P

■♥st❛♥❝❡✿ ❆ ✇♦r❞ w ❛♥❞ ❛ k ∈ N✳ ◗✉❡st✐♦♥✿ ∃ ✭1✲❧❡✈❡❧✮ ❣r❛♠♠❛r G ✇✐t❤ L(G) = {w} ❛♥❞ |G| ≤ k❄

slide-40
SLIDE 40

❑♥♦✇♥ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

NP✲❝♦♠♣❧❡t❡ ✭❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ▼❛♥② s✐♠♣❧❡ ❛♥❞ ❢❛st ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❡①✐st✳ ❇❡st ❦♥♦✇♥ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✐s ✭✇❤❡r❡ ✐s t❤❡ s✐③❡ ♦❢ ❛ s♠❛❧❧❡st ❣r❛♠♠❛r✮✳ ◆♦ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✭❛ss✉♠✐♥❣ ❛♥❞ ❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ❙♦❧✐❞ t❤❡♦r❡t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s ✭♦r ❤❡✉r✐st✐❝s✮ ♠✐ss✐♥❣✳

slide-41
SLIDE 41

❑♥♦✇♥ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

NP✲❝♦♠♣❧❡t❡ ✭❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ▼❛♥② s✐♠♣❧❡ ❛♥❞ ❢❛st ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❡①✐st✳ ❇❡st ❦♥♦✇♥ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✐s ✭✇❤❡r❡ ✐s t❤❡ s✐③❡ ♦❢ ❛ s♠❛❧❧❡st ❣r❛♠♠❛r✮✳ ◆♦ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✭❛ss✉♠✐♥❣ ❛♥❞ ❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ❙♦❧✐❞ t❤❡♦r❡t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s ✭♦r ❤❡✉r✐st✐❝s✮ ♠✐ss✐♥❣✳

slide-42
SLIDE 42

❑♥♦✇♥ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

NP✲❝♦♠♣❧❡t❡ ✭❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ▼❛♥② s✐♠♣❧❡ ❛♥❞ ❢❛st ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❡①✐st✳ ❇❡st ❦♥♦✇♥ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✐s O(log( |w|

m∗ )) ✭✇❤❡r❡ m∗ ✐s t❤❡

s✐③❡ ♦❢ ❛ s♠❛❧❧❡st ❣r❛♠♠❛r✮✳ ◆♦ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✭❛ss✉♠✐♥❣ ❛♥❞ ❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ❙♦❧✐❞ t❤❡♦r❡t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s ✭♦r ❤❡✉r✐st✐❝s✮ ♠✐ss✐♥❣✳

slide-43
SLIDE 43

❑♥♦✇♥ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

NP✲❝♦♠♣❧❡t❡ ✭❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ▼❛♥② s✐♠♣❧❡ ❛♥❞ ❢❛st ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❡①✐st✳ ❇❡st ❦♥♦✇♥ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✐s O(log( |w|

m∗ )) ✭✇❤❡r❡ m∗ ✐s t❤❡

s✐③❡ ♦❢ ❛ s♠❛❧❧❡st ❣r❛♠♠❛r✮✳ ◆♦ 8569

8568 ≈ 1.0001 ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✭❛ss✉♠✐♥❣ P = NP ❛♥❞ ❢♦r

✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ❙♦❧✐❞ t❤❡♦r❡t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s ✭♦r ❤❡✉r✐st✐❝s✮ ♠✐ss✐♥❣✳

slide-44
SLIDE 44

❑♥♦✇♥ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

NP✲❝♦♠♣❧❡t❡ ✭❢♦r ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ▼❛♥② s✐♠♣❧❡ ❛♥❞ ❢❛st ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❡①✐st✳ ❇❡st ❦♥♦✇♥ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✐s O(log( |w|

m∗ )) ✭✇❤❡r❡ m∗ ✐s t❤❡

s✐③❡ ♦❢ ❛ s♠❛❧❧❡st ❣r❛♠♠❛r✮✳ ◆♦ 8569

8568 ≈ 1.0001 ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✭❛ss✉♠✐♥❣ P = NP ❛♥❞ ❢♦r

✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✳ ⇒ ❙♦❧✐❞ t❤❡♦r❡t✐❝❛❧ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥s ✭♦r ❤❡✉r✐st✐❝s✮ ♠✐ss✐♥❣✳

slide-45
SLIDE 45

◆P✲❈♦♠♣❧❡t❡♥❡ss ♦❢ t❤❡ ❙❤♦rt❡st ●r❛♠♠❛r Pr♦❜❧❡♠

❚❤❡♦r❡♠

1✲❙●P ✐s NP✲❝♦♠♣❧❡t❡✱ ❡✈❡♥ ❢♦r ❛❧♣❤❛❜❡ts ♦❢ s✐③❡ 5✳

❚❤❡♦r❡♠

❙●P ✐s ✲❝♦♠♣❧❡t❡✱ ❡✈❡♥ ❢♦r ❛❧♣❤❛❜❡ts ♦❢ s✐③❡ ✳

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SLIDE 46

◆P✲❈♦♠♣❧❡t❡♥❡ss ♦❢ t❤❡ ❙❤♦rt❡st ●r❛♠♠❛r Pr♦❜❧❡♠

❚❤❡♦r❡♠

1✲❙●P ✐s NP✲❝♦♠♣❧❡t❡✱ ❡✈❡♥ ❢♦r ❛❧♣❤❛❜❡ts ♦❢ s✐③❡ 5✳

❚❤❡♦r❡♠

❙●P ✐s NP✲❝♦♠♣❧❡t❡✱ ❡✈❡♥ ❢♦r ❛❧♣❤❛❜❡ts ♦❢ s✐③❡ 24✳

slide-47
SLIDE 47

Pr♦♦❢ ■❞❡❛s

❘❡❞✉❝t✐♦♥ ❢r♦♠ ✈❡rt❡① ❝♦✈❡r ✭✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✿

◮ r❡♣r❡s❡♥t ❡❞❣❡s (vi, vj) ❛s ❢❛❝t♦rs #vi#vj#✱ ◮ ♠❛❦❡ s✉r❡ t❤❛t ♦♥❧② #vi✱ vi# ♦r #vi# ❛r❡ ❝♦♠♣r❡ss❡❞✱ ◮ ❣r❛♠♠❛r ✐s s♠❛❧❧❡st ✐❢ ❡✈❡r② #vi#vj# ✐s ❝♦♠♣r❡ss❡❞ ❜② ❛

Ai → #vi# ♦r Aj → #vj#✳

❋✐♥✐t❡ ❛❧♣❤❛❜❡t✿ ❯s✐♥❣ s②♠❜♦❧s ♦r ❝♦♥st❛♥t s✐③❡ ❢❛❝t♦rs ❢♦r r❡♣r❡s❡♥t✐♥❣ ✈❡rt✐❝❡s ✭❡❞❣❡s✮ ♦r ❛s s❡♣❛r❛t♦rs ✐s ♥♦t ♣♦ss✐❜❧❡ ✭ s♦♠❡ ❡♥❝♦❞✐♥❣ ♥❡❡❞❡❞✦✮✳ ✲❧❡✈❡❧ ❝❛s❡✿ ❯s✐♥❣ ✉♥❛r② s❡q✉❡♥❝❡s ❛s s❡♣❛r❛t♦rs ✇♦r❦s✦ ✭✐♥ ✲❧❡✈❡❧ ❝❛s❡✱ s❡♣❛r❛t♦rs ❢✉❧❧② ❞❡t❡r♠✐♥❡ t❤❡ ❝♦♠♣r❡ss❡❞ ❢❛❝t♦rs✳✮ ▼✉❧t✐✲❧❡✈❡❧ ❝❛s❡✿ ❲❡ ❞♦ ♥♦t ❦♥♦✇ ❤♦✇ t❤❡ ❡♥❝♦❞✐♥❣s ✇✐❧❧ ❜❡ ❝♦♠♣r❡ss❡❞ ❜② ❛ s❤♦rt❡st ❣r❛♠♠❛r ✭r❡❝❛❧❧ ✐♥tr♦❞✉❝t♦r② ❡①❛♠♣❧❡ ✮✳

slide-48
SLIDE 48

Pr♦♦❢ ■❞❡❛s

❘❡❞✉❝t✐♦♥ ❢r♦♠ ✈❡rt❡① ❝♦✈❡r ✭✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✿

◮ r❡♣r❡s❡♥t ❡❞❣❡s (vi, vj) ❛s ❢❛❝t♦rs #vi#vj#✱ ◮ ♠❛❦❡ s✉r❡ t❤❛t ♦♥❧② #vi✱ vi# ♦r #vi# ❛r❡ ❝♦♠♣r❡ss❡❞✱ ◮ ❣r❛♠♠❛r ✐s s♠❛❧❧❡st ✐❢ ❡✈❡r② #vi#vj# ✐s ❝♦♠♣r❡ss❡❞ ❜② ❛

Ai → #vi# ♦r Aj → #vj#✳

❋✐♥✐t❡ ❛❧♣❤❛❜❡t✿ ❯s✐♥❣ s②♠❜♦❧s ♦r ❝♦♥st❛♥t s✐③❡ ❢❛❝t♦rs ❢♦r r❡♣r❡s❡♥t✐♥❣ ✈❡rt✐❝❡s ✭❡❞❣❡s✮ ♦r ❛s s❡♣❛r❛t♦rs ✐s ♥♦t ♣♦ss✐❜❧❡ ✭⇒ s♦♠❡ ❡♥❝♦❞✐♥❣ ♥❡❡❞❡❞✦✮✳ ✲❧❡✈❡❧ ❝❛s❡✿ ❯s✐♥❣ ✉♥❛r② s❡q✉❡♥❝❡s ❛s s❡♣❛r❛t♦rs ✇♦r❦s✦ ✭✐♥ ✲❧❡✈❡❧ ❝❛s❡✱ s❡♣❛r❛t♦rs ❢✉❧❧② ❞❡t❡r♠✐♥❡ t❤❡ ❝♦♠♣r❡ss❡❞ ❢❛❝t♦rs✳✮ ▼✉❧t✐✲❧❡✈❡❧ ❝❛s❡✿ ❲❡ ❞♦ ♥♦t ❦♥♦✇ ❤♦✇ t❤❡ ❡♥❝♦❞✐♥❣s ✇✐❧❧ ❜❡ ❝♦♠♣r❡ss❡❞ ❜② ❛ s❤♦rt❡st ❣r❛♠♠❛r ✭r❡❝❛❧❧ ✐♥tr♦❞✉❝t♦r② ❡①❛♠♣❧❡ ✮✳

slide-49
SLIDE 49

Pr♦♦❢ ■❞❡❛s

❘❡❞✉❝t✐♦♥ ❢r♦♠ ✈❡rt❡① ❝♦✈❡r ✭✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✿

◮ r❡♣r❡s❡♥t ❡❞❣❡s (vi, vj) ❛s ❢❛❝t♦rs #vi#vj#✱ ◮ ♠❛❦❡ s✉r❡ t❤❛t ♦♥❧② #vi✱ vi# ♦r #vi# ❛r❡ ❝♦♠♣r❡ss❡❞✱ ◮ ❣r❛♠♠❛r ✐s s♠❛❧❧❡st ✐❢ ❡✈❡r② #vi#vj# ✐s ❝♦♠♣r❡ss❡❞ ❜② ❛

Ai → #vi# ♦r Aj → #vj#✳

❋✐♥✐t❡ ❛❧♣❤❛❜❡t✿ ❯s✐♥❣ s②♠❜♦❧s ♦r ❝♦♥st❛♥t s✐③❡ ❢❛❝t♦rs ❢♦r r❡♣r❡s❡♥t✐♥❣ ✈❡rt✐❝❡s ✭❡❞❣❡s✮ ♦r ❛s s❡♣❛r❛t♦rs ✐s ♥♦t ♣♦ss✐❜❧❡ ✭⇒ s♦♠❡ ❡♥❝♦❞✐♥❣ ♥❡❡❞❡❞✦✮✳ 1✲❧❡✈❡❧ ❝❛s❡✿ ❯s✐♥❣ ✉♥❛r② s❡q✉❡♥❝❡s ❛s s❡♣❛r❛t♦rs ✇♦r❦s✦ ✭✐♥ 1✲❧❡✈❡❧ ❝❛s❡✱ s❡♣❛r❛t♦rs ❢✉❧❧② ❞❡t❡r♠✐♥❡ t❤❡ ❝♦♠♣r❡ss❡❞ ❢❛❝t♦rs✳✮ ▼✉❧t✐✲❧❡✈❡❧ ❝❛s❡✿ ❲❡ ❞♦ ♥♦t ❦♥♦✇ ❤♦✇ t❤❡ ❡♥❝♦❞✐♥❣s ✇✐❧❧ ❜❡ ❝♦♠♣r❡ss❡❞ ❜② ❛ s❤♦rt❡st ❣r❛♠♠❛r ✭r❡❝❛❧❧ ✐♥tr♦❞✉❝t♦r② ❡①❛♠♣❧❡ ✮✳

slide-50
SLIDE 50

Pr♦♦❢ ■❞❡❛s

❘❡❞✉❝t✐♦♥ ❢r♦♠ ✈❡rt❡① ❝♦✈❡r ✭✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡ts✮✿

◮ r❡♣r❡s❡♥t ❡❞❣❡s (vi, vj) ❛s ❢❛❝t♦rs #vi#vj#✱ ◮ ♠❛❦❡ s✉r❡ t❤❛t ♦♥❧② #vi✱ vi# ♦r #vi# ❛r❡ ❝♦♠♣r❡ss❡❞✱ ◮ ❣r❛♠♠❛r ✐s s♠❛❧❧❡st ✐❢ ❡✈❡r② #vi#vj# ✐s ❝♦♠♣r❡ss❡❞ ❜② ❛

Ai → #vi# ♦r Aj → #vj#✳

❋✐♥✐t❡ ❛❧♣❤❛❜❡t✿ ❯s✐♥❣ s②♠❜♦❧s ♦r ❝♦♥st❛♥t s✐③❡ ❢❛❝t♦rs ❢♦r r❡♣r❡s❡♥t✐♥❣ ✈❡rt✐❝❡s ✭❡❞❣❡s✮ ♦r ❛s s❡♣❛r❛t♦rs ✐s ♥♦t ♣♦ss✐❜❧❡ ✭⇒ s♦♠❡ ❡♥❝♦❞✐♥❣ ♥❡❡❞❡❞✦✮✳ 1✲❧❡✈❡❧ ❝❛s❡✿ ❯s✐♥❣ ✉♥❛r② s❡q✉❡♥❝❡s ❛s s❡♣❛r❛t♦rs ✇♦r❦s✦ ✭✐♥ 1✲❧❡✈❡❧ ❝❛s❡✱ s❡♣❛r❛t♦rs ❢✉❧❧② ❞❡t❡r♠✐♥❡ t❤❡ ❝♦♠♣r❡ss❡❞ ❢❛❝t♦rs✳✮ ▼✉❧t✐✲❧❡✈❡❧ ❝❛s❡✿ ❲❡ ❞♦ ♥♦t ❦♥♦✇ ❤♦✇ t❤❡ ❡♥❝♦❞✐♥❣s ✇✐❧❧ ❜❡ ❝♦♠♣r❡ss❡❞ ❜② ❛ s❤♦rt❡st ❣r❛♠♠❛r ✭r❡❝❛❧❧ ✐♥tr♦❞✉❝t♦r② ❡①❛♠♣❧❡ 10100100010000 . . .✮✳

slide-51
SLIDE 51

Pr♦♦❢ ■❞❡❛s ▼✉❧t✐✲▲❡✈❡❧ ❈❛s❡

♣❛❧✐♥❞r♦♠✐❝ ❝♦❞❡✇♦r❞s✿ u ⋆ uR✱ ⋆ ∈ Σ✱ u ✐s 7✲❛r② ♥✉♠❜❡r✳ ❈r✉❝✐❛❧ ♣r♦♣❡rt✐❡s✿

♦✈❡r❧❛♣♣✐♥❣ ❜❡t✇❡❡♥ ♥❡✐❣❤❜♦✉r✐♥❣ ❝♦❞❡✇♦r❞s ❛r❡ ♥♦t r❡♣❡❛t❡❞ ❝♦❞❡✇♦r❞s ❛r❡ ❝♦♠♣r❡ss❡❞ ✐♥❞✐✈✐❞✉❛❧❧②✳ ❝♦❞❡✇♦r❞s ❛r❡ ♣r♦❞✉❝❡❞ ❜❡st ✏❢r♦♠ t❤❡ ♠✐❞❞❧❡✑✿ ✱ ✱ ✱ ✳

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SLIDE 52

Pr♦♦❢ ■❞❡❛s ▼✉❧t✐✲▲❡✈❡❧ ❈❛s❡

♣❛❧✐♥❞r♦♠✐❝ ❝♦❞❡✇♦r❞s✿ u ⋆ uR✱ ⋆ ∈ Σ✱ u ✐s 7✲❛r② ♥✉♠❜❡r✳ ❈r✉❝✐❛❧ ♣r♦♣❡rt✐❡s✿

◮ ♦✈❡r❧❛♣♣✐♥❣ ❜❡t✇❡❡♥ ♥❡✐❣❤❜♦✉r✐♥❣ ❝♦❞❡✇♦r❞s ❛r❡ ♥♦t r❡♣❡❛t❡❞

⇒ ❝♦❞❡✇♦r❞s ❛r❡ ❝♦♠♣r❡ss❡❞ ✐♥❞✐✈✐❞✉❛❧❧②✳

◮ ❝♦❞❡✇♦r❞s ❛r❡ ♣r♦❞✉❝❡❞ ❜❡st ✏❢r♦♠ t❤❡ ♠✐❞❞❧❡✑✿ A → a ⋆ a✱

B → bAb✱ C → cBc✱ . . .✳

slide-53
SLIDE 53

❘❡❞✉❝t✐♦♥

  • r❛♣❤ G ⇒ ✇♦r❞ uvw

➣ ➣ ➣ ➣

slide-54
SLIDE 54

❘❡❞✉❝t✐♦♥

  • r❛♣❤ G ⇒ ✇♦r❞ uvw

u =

6

  • j=0

14n

  • i=1

(i⋄ M(i + j, 14n)v)

  • $1

v =

n

  • i=1

(# 7i + Cv(i)v ➣1 7i − 1⋄) $2

n

  • i=1

(# 7i + Cv(i)v ➣2 7i − 2⋄) $3

n

  • i=1

(7i + Cv(i)v # 7i − 2⋄ ➣1) $4

n

  • i=1

(7i + Cv(i)v # 7i − 1⋄ ➣2) $5

n

  • i=1

(# 7i + Cv(i)v # 7i⋄) $6 w =

m−1

  • i=1

(# 7j2i−1 + Cv(j2i−1)v # 7j2i + Cv(j2i)v # 7i + Ce(vj2i, vj2i+1)⋄) # 7j2m−1 + Cv(j2m−1)v # 7j2m + Cv(j2m)v #

slide-55
SLIDE 55

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s

❚❤❡♦r❡♠

▲❡t w ∈ Σ∗ ❛♥❞ k ∈ N✱ ❛ ❣r❛♠♠❛r t❤❛t ✐s ♠✐♥✐♠❛❧ ❛♠♦♥❣ ❛❧❧ ❣r❛♠♠❛rs ✇✐t❤ ❛t ♠♦st k r✉❧❡s✱ ❛ 1✲❧❡✈❡❧ ❣r❛♠♠❛r t❤❛t ✐s ♠✐♥✐♠❛❧ ❛♠♦♥❣ ❛❧❧ 1✲❧❡✈❡❧ ❣r❛♠♠❛rs ✇✐t❤ ❛t ♠♦st k r✉❧❡s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

slide-56
SLIDE 56

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s

  • ❡♥❡r❛❧ ✐❞❡❛✿

❚r❛♥s❢♦r♠ ✇♦r❞ w ✐♥t♦ ❛♥ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ Gw✱ s✉❝❤ t❤❛t ✐♥❞❡♣❡♥❞❡♥t ❞♦♠✐♥❛t✐♥❣ s❡ts ♦❢ Gw ❝♦rr❡s♣♦♥❞ t♦ ❣r❛♠♠❛rs ❢♦r w✳ ✏❈♦♠♣✉t❡ ♠✐♥✐♠❛❧ ✐♥❞❡♣❡♥❞❡♥t ❞♦♠✐♥❛t✐♥❣ s❡ts ♦❢ Gw✑✳

slide-57
SLIDE 57

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s 1✲▲❡✈❡❧ ❈❛s❡

w = abbababab

slide-58
SLIDE 58

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s 1✲▲❡✈❡❧ ❈❛s❡

w = abbababab

(1, 1) (2, 2) (3, 3) . . . . . . . . . (9, 9) (1, 2) (2, 3) (3, 4) . . . (8, 9) (1, 3) (2, 4) . . . (7, 9) ✳ ✳ ✳ ✳ ✳ ✳

slide-59
SLIDE 59

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s 1✲▲❡✈❡❧ ❈❛s❡

w = abbababab

(1, 1) (2, 2) (3, 3) . . . . . . . . . (9, 9) (1, 2) (2, 3) (3, 4) . . . (8, 9) (1, 3) (2, 4) . . . (7, 9) ✳ ✳ ✳ ✳ ✳ ✳ ab ba bb abb bba . . .

slide-60
SLIDE 60

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s 1✲▲❡✈❡❧ ❈❛s❡

w = abbababab

(1, 1) (2, 2) (3, 3) . . . . . . . . . (9, 9) (1, 2) (2, 3) (3, 4) . . . (8, 9) (1, 3) (2, 4) . . . (7, 9) ✳ ✳ ✳ ✳ ✳ ✳ ab ba bb abb bba . . . . . .

slide-61
SLIDE 61

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s 1✲▲❡✈❡❧ ❈❛s❡

w = abbababab

(1, 1) (2, 2) (3, 3) . . . . . . . . . (9, 9) (1, 2) (2, 3) (3, 4) . . . (8, 9) (1, 3) (2, 4) . . . (7, 9) ✳ ✳ ✳ ✳ ✳ ✳ ab ba bb abb bba . . . . . .

slide-62
SLIDE 62

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s 1✲▲❡✈❡❧ ❈❛s❡

w = abbababab

(1, 1) (2, 2) (3, 3) . . . . . . . . . (9, 9) (1, 2) (2, 3) (3, 4) . . . (8, 9) (1, 3) (2, 4) . . . (7, 9) ✳ ✳ ✳ ✳ ✳ ✳ ab ba bb abb bba . . . . . .

slide-63
SLIDE 63

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s 1✲▲❡✈❡❧ ❈❛s❡

w = abbababab

(1, 1) (2, 2) (3, 3) . . . . . . . . . (9, 9) (1, 2) (2, 3) (3, 4) . . . (8, 9) (1, 3) (2, 4) . . . (7, 9) ✳ ✳ ✳ ✳ ✳ ✳ ab ba bb abb bba . . . . . .

slide-64
SLIDE 64

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s ▼✉❧t✐✲▲❡✈❡❧ ❈❛s❡

ab ba aa

(ab, 0) (ba, 0) (aa, 0) (ab, 1, 1) (ab, 2, 2) (ba, 1, 1) (ba, 2, 2) (aa, 1, 1) (aa, 2, 2)

aba baa aab

(aba, 0) (baa, 0) (aab, 0) (aba, 1, 1) (aba, 2, 2) (aba, 3, 3) (baa, 1, 1) (baa, 2, 2) (baa, 3, 3) (aab, 1, 1) (aab, 2, 2) (aab, 3, 3) (aba, 1, 2) (aba, 2, 3) (baa, 1, 2) (baa, 2, 3) (aab, 1, 2) (aab, 2, 3)

slide-65
SLIDE 65

❇♦✉♥❞❡❞ ◆✉♠❜❡r ♦❢ ◆♦♥✲❚❡r♠✐♥❛❧s ▼✉❧t✐✲▲❡✈❡❧ ❈❛s❡

❚❤❡♦r❡♠

▲❡t w ∈ Σ∗ ❛♥❞ k ∈ N✳ ❆ 1✲❧❡✈❡❧ ❣r❛♠♠❛r ❢♦r w ✇✐t❤ ❛t ♠♦st k r✉❧❡s t❤❛t ✐s ♠✐♥✐♠❛❧ ❛♠♦♥❣ ❛❧❧ 1✲❧❡✈❡❧ ❣r❛♠♠❛rs ❢♦r w ✇✐t❤ ❛t ♠♦st k r✉❧❡s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ O(|w|2k+4)✳

❚❤❡♦r❡♠

▲❡t w ∈ Σ∗ ❛♥❞ k ∈ N✳ ❆ ❣r❛♠♠❛r ❢♦r w ✇✐t❤ ❛t ♠♦st k r✉❧❡s t❤❛t ✐s ♠✐♥✐♠❛❧ ❛♠♦♥❣ ❛❧❧ ❣r❛♠♠❛rs ❢♦r w ✇✐t❤ ❛t ♠♦st k r✉❧❡s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ O(|w|2k+6)✳

slide-66
SLIDE 66

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

Pr❡✈✐♦✉s ❛♣♣r♦❛❝❤✿ O(2|w|2)✳ ❊♥✉♠❡r❛t✐♥❣ ❛❧❧ ♦r❞❡r❡❞ tr❡❡s ✇✐t❤ |w| ❧❡❛✈❡s✿ O(8|w|)✳

❚❤❡♦r❡♠

❙♠❛❧❧❡st ✲❧❡✈❡❧ ❣r❛♠♠❛rs ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ ✳ Pr♦♦❢ s❦❡t❝❤✿ ❊♥✉♠❡r❛t❡ ❛❧❧ ❢❛❝t♦r✐s❛t✐♦♥s ♦❢ ✇✐t❤♦✉t ❝♦♥s❡❝✉t✐✈❡ ❢❛❝t♦rs ♦❢ ❧❡♥❣t❤ ✳

slide-67
SLIDE 67

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

Pr❡✈✐♦✉s ❛♣♣r♦❛❝❤✿ O(2|w|2)✳ ❊♥✉♠❡r❛t✐♥❣ ❛❧❧ ♦r❞❡r❡❞ tr❡❡s ✇✐t❤ |w| ❧❡❛✈❡s✿ O(8|w|)✳

❚❤❡♦r❡♠

❙♠❛❧❧❡st 1✲❧❡✈❡❧ ❣r❛♠♠❛rs ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ O∗(1.8392|w|)✳ Pr♦♦❢ s❦❡t❝❤✿ ❊♥✉♠❡r❛t❡ ❛❧❧ ❢❛❝t♦r✐s❛t✐♦♥s ♦❢ w ✇✐t❤♦✉t ❝♦♥s❡❝✉t✐✈❡ ❢❛❝t♦rs ♦❢ ❧❡♥❣t❤ 1✳

slide-68
SLIDE 68

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❖❜✈✐♦✉s ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤ ❢♦r ♠✉❧t✐✲❧❡✈❡❧✿ ❈♦♠♣✉t❡ s✐③❡ ♦❢ ❜❡st k✲❧❡✈❡❧ ❣r❛♠♠❛r✱ st♦r❡ ✏❧❛st ❧❡✈❡❧✑ ✭kt❤ ❧❡✈❡❧❄❄✮✱ ❝♦♠♣✉t❡ s✐③❡ ♦❢ ❜❡st k + 1✲❧❡✈❡❧ ❣r❛♠♠❛r ❜② s♦♠❡❤♦✇ ❡①t❡♥❞✐♥❣ t❤❡ ❧❛st ❧❡✈❡❧✱ ❛❣❛✐♥ st♦r❡ ✏❧❛st ❧❡✈❡❧✑✱ . . .✳

slide-69
SLIDE 69

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

a a a b b a b a a a a a a b b a b

slide-70
SLIDE 70

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

a a a b b a b a a a a a a b b a b

slide-71
SLIDE 71

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

A B A B aaabb aaa ab aaabb ab

slide-72
SLIDE 72

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

A B A B aaabb aaa ab aaabb ab

slide-73
SLIDE 73

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

C C A B A B aaabb aaa ab aaabb ab

slide-74
SLIDE 74

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

C C A B A B aaabb aaa ab aaabb ab

slide-75
SLIDE 75

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

C C A B A B a B b aaa ab a B b ab ab ab

slide-76
SLIDE 76

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

C C A B A B aaabb aaa ab aaabb ab

slide-77
SLIDE 77

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

C C A B A B aaabb aaa ab aaabb ab

slide-78
SLIDE 78

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

C C A B A B D D bb aaa aaa aaa ab D bb ab

slide-79
SLIDE 79

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ ✱ ✱ ✱

st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ ♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧

✐♥

st ❧❡✈❡❧✱ r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧

✐♥

♥❞ ❧❡✈❡❧✱

slide-80
SLIDE 80

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ C C A B A B D D bb aaa aaa aaa ab D bb ab ✱ ✱ ✱

st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ ♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧

✐♥

st ❧❡✈❡❧✱ r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧

✐♥

♥❞ ❧❡✈❡❧✱

slide-81
SLIDE 81

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ C C A B A B D D bb aaa aaa aaa ab D bb ab N1 = {B, D}✱ N2 = {A}✱ N3 = {C}✱

st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ ♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧

✐♥

st ❧❡✈❡❧✱ r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧

✐♥

♥❞ ❧❡✈❡❧✱

slide-82
SLIDE 82

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ C C A B A B D D bb aaa aaa aaa ab D bb ab N1 = {B, D}✱ N2 = {A}✱ N3 = {C}✱ 1st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ 2♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk ✐♥ 1st ❧❡✈❡❧✱ 3r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk−1 ✐♥ 2♥❞ ❧❡✈❡❧✱ . . .

slide-83
SLIDE 83

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ C C A B A B D D bb aaa aaa aaa ab D bb ab N1 = {B, D}✱ N2 = {A}✱ N3 = {C}✱ 1st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ CDC 2♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk ✐♥ 1st ❧❡✈❡❧✱ 3r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk−1 ✐♥ 2♥❞ ❧❡✈❡❧✱ . . .

slide-84
SLIDE 84

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ C C A B A B D D bb aaa aaa aaa ab D bb ab N1 = {B, D}✱ N2 = {A}✱ N3 = {C}✱ 1st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ CDC 2♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk ✐♥ 1st ❧❡✈❡❧✱ ABDAB 3r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk−1 ✐♥ 2♥❞ ❧❡✈❡❧✱ . . .

slide-85
SLIDE 85

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ C C A B A B D D bb aaa aaa aaa ab D bb ab N1 = {B, D}✱ N2 = {A}✱ N3 = {C}✱ 1st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ CDC 2♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk ✐♥ 1st ❧❡✈❡❧✱ ABDAB 3r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk−1 ✐♥ 2♥❞ ❧❡✈❡❧✱ DbbBDDbbB . . .

slide-86
SLIDE 86

❊①❛❝t ❊①♣♦♥❡♥t✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠s

❙♦❧✉t✐♦♥✿ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❞❡✜♥✐t✐♦♥ ♦❢ ✏❧❡✈❡❧s✑✳ Ni = {A ∈ N | A ②✐❡❧❞s t❡r♠✐♥❛❧ str✐♥❣ ✐♥ i st❡♣s}✳ C C A B A B D D bb aaa aaa aaa ab D bb ab N1 = {B, D}✱ N2 = {A}✱ N3 = {C}✱ 1st ❧❡✈❡❧✿ ❝♦♠♣r❡ss❡❞ str✐♥❣✱ CDC 2♥❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk ✐♥ 1st ❧❡✈❡❧✱ ABDAB 3r❞ ❧❡✈❡❧✿ ❞❡r✐✈❡ ❛❧❧ Nk−1 ✐♥ 2♥❞ ❧❡✈❡❧✱ DbbBDDbbB . . . aaabbabaaaaaabbab

slide-87
SLIDE 87

❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❆❧❣♦r✐t❤♠

kt❤ ABaCbCBa w1w2 . . . w8

t❤

❙✉✣❝✐❡♥t ❧♦❝❛❧ ✐♥❢♦r♠❛t✐♦♥✿ ❙✐③❡ ♦❢ s♠❛❧❧❡st ✲❧❡✈❡❧ ❣r❛♠♠❛r

❚❤❡♦r❡♠

❙♠❛❧❧❡st ❣r❛♠♠❛rs ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ ✳

slide-88
SLIDE 88

❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❆❧❣♦r✐t❤♠

kt❤ ABaCbCBa w1w2 . . . w8 (k + 1)t❤ ABaDbAbDbABa w1 . . . w4,1w4,2w4,3 . . . w6,1w6,2w6,3 . . . w8 ❙✉✣❝✐❡♥t ❧♦❝❛❧ ✐♥❢♦r♠❛t✐♦♥✿ ❙✐③❡ ♦❢ s♠❛❧❧❡st ✲❧❡✈❡❧ ❣r❛♠♠❛r

❚❤❡♦r❡♠

❙♠❛❧❧❡st ❣r❛♠♠❛rs ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ ✳

slide-89
SLIDE 89

❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❆❧❣♦r✐t❤♠

kt❤ ABaCbCBa w1w2 . . . w8 (k + 1)t❤ ABaDbAbDbABa w1 . . . w4,1w4,2w4,3 . . . w6,1w6,2w6,3 . . . w8 ❙✉✣❝✐❡♥t ❧♦❝❛❧ ✐♥❢♦r♠❛t✐♦♥✿ ❙✐③❡ ♦❢ s♠❛❧❧❡st ✲❧❡✈❡❧ ❣r❛♠♠❛r

❚❤❡♦r❡♠

❙♠❛❧❧❡st ❣r❛♠♠❛rs ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ ✳

slide-90
SLIDE 90

❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❆❧❣♦r✐t❤♠

kt❤ ABaCbCBa w1w2 . . . w8 (k + 1)t❤ ABaDbAbDbABa w1 . . . w4,1w4,2w4,3 . . . w6,1w6,2w6,3 . . . w8 ❙✉✣❝✐❡♥t ❧♦❝❛❧ ✐♥❢♦r♠❛t✐♦♥✿ ❙✐③❡ ♦❢ s♠❛❧❧❡st k✲❧❡✈❡❧ ❣r❛♠♠❛r u . . . v . . . u . . . v (u1 . . . um) . . . (v1 . . . vℓ) . . . (u1 . . . um) . . . (v1 . . . vℓ)

❚❤❡♦r❡♠

❙♠❛❧❧❡st ❣r❛♠♠❛rs ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ ✳

slide-91
SLIDE 91

❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❆❧❣♦r✐t❤♠

kt❤ ABaCbCBa w1w2 . . . w8 (k + 1)t❤ ABaDbAbDbABa w1 . . . w4,1w4,2w4,3 . . . w6,1w6,2w6,3 . . . w8 ❙✉✣❝✐❡♥t ❧♦❝❛❧ ✐♥❢♦r♠❛t✐♦♥✿ ❙✐③❡ ♦❢ s♠❛❧❧❡st k✲❧❡✈❡❧ ❣r❛♠♠❛r u . . . v . . . u . . . v (u1 . . . um) . . . (v1 . . . vℓ) . . . (u1 . . . um) . . . (v1 . . . vℓ)

❚❤❡♦r❡♠

❙♠❛❧❧❡st ❣r❛♠♠❛rs ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ O∗(3|w|)✳

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SLIDE 92

❚❤❛♥❦ ②♦✉ ✈❡r② ♠✉❝❤ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥