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Jan 12, 2024 384 likes •699 views

t rsr s Pt rs strt Ptrs rt r tr

SLIDE 1

❖❜❥❡❝t ❚r❛♥s❢❡r ✉s✐♥❣ P❛t❤ ❘❡✈❡rs❛❧✿ ❉✐str✐❜✉t❡❞ P❛t❤✲❘❡✈❡rs❛❧ ❆❧❣♦r✐t❤♠

❙❤❛♥❦❛r ❙❡♣t❡♠❜❡r ✶✽✱ ✷✵✶✹

SLIDE 2

❖✉t❧✐♥❡

♣r ❛❧❣

P❛t❤ r❡✈❡rs❛❧✿ ❛❧❣♦r✐t❤♠ P❛t❤ r❡✈❡rs❛❧✿ s❛❢❡t② ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ ♣r♦❣r❡ss ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ s❡r✐❛❧✐③❛❜✐❧✐t② ❛♥❛❧②s✐s

SLIDE 3

P❛t❤ r❡✈❡rs❛❧ ❛❧❣♦r✐t❤♠

♣r ❛❧❣

❙②st❡♠s ❛tt❛❝❤❡❞ t♦ ❛ ✜❢♦ ❝❤❛♥♥❡❧❀ ♦❜❥ ✐♥✐t✐❛❧❧② ❛t a0 ▼❡ss❛❣❡s

[REQ,j]✿ r❡q✉❡st ♠s❣ // j ✐s ✐ss✉❡r ✭♥♦t ❢♦r✇❛r❞❡r✮ [OBJ]✿ ♦❜❥❡❝t✲❝❛rr②✐♥❣ ♠s❣ // ✐❣♥♦r❡ ✈❛❧✉❡ ❢♦r ♥♦✇

❙②st❡♠ j✿ ❡❛t✐♥❣ ✭❤❛s ♦❜❥✮✱ ❤✉♥❣r② ✭✇❛♥ts ♦❜❥✮✱ t❤✐♥❦✐♥❣ ✭♦✴✇✮ ❙②st❡♠ j ❤❛s ❛ ✏❧❛st✑ ♣♦✐♥t❡r ❛❞❞r ✐♥ t❤❡ ❧❛st r❡q ♠s❣ r❝✈❞ ❜② j ❛❢t❡r ❧❛st ❜❡❝♦♠✐♥❣ ❤✉♥❣r②✳ ♥✐❧ ✐❢ ♥♦ s✉❝❤ ♠s❣ ✐♥✐t✐❛❧❧② nil ❛t a0✱ ❛♥❞ a0 ❡❧s❡✇❤❡r❡ ❙②st❡♠ j ❤❛s ❛ ✏♥❡①t✑ ♣♦✐♥t❡r ♥✐❧ ✐❢ j t❤✐♥❦✐♥❣ ♦r ♥♦t r❝✈❞ r❡q s✐♥❝❡ ♥♦♥✲t❤✐♥❦✐♥❣ ♦✴✇ ❡q✉❛❧s ❛❞❞r ✐♥ t❤❡ ✜rst r❡q ♠s❣ r❝✈❞ s✐♥❝❡ ♥♦♥✲t❤✐♥❦✐♥❣ ✐♥✐t✐❛❧❧② ♥✐❧

SLIDE 4

❙②st❡♠ j r✉❧❡s

♣r ❛❧❣

❜❡❝♦♠❡ ❤✉♥❣r② ♦♥❧② ✐❢ t❤✐♥❦✐♥❣✿ ✴✴ H(j) s❡♥❞ [REQ,j] t♦ lst s❡t lst t♦ ♥✐❧ r❝✈ [OBJ]✿ ✴✴ E(j) ❜❡❝♦♠❡ ❡❛t✐♥❣ ❜❡❝♦♠❡ t❤✐♥❦✐♥❣ ♦♥❧② ✐❢ ❡❛t✐♥❣ ❛♥❞ nxt ♥♦♥✲♥✐❧✿ ✴✴ T(j) s❡♥❞ ♦❜❥❡❝t t♦ nxt s❡t nxt t♦ ♥✐❧ r❝✈ [REQ,k]✿ ✴✴ R(j,k) ✐❢ lst ♥♦t ♥✐❧ s❡♥❞ [REQ,k] t♦ lst s❡t lst t♦ k ❡❧s❡ s❡t nxt ❛♥❞ lst t♦ k

SLIDE 5

❈♦♥✈❡♥t✐♦♥s

♣r ❛❧❣

j✕k ✐s ❛ ❧❛st ❡❞❣❡✿ j.lst ✐s ♥♦t ♥✐❧ ❛♥❞ ❡q✉❛❧s k j✕k ✐s ❛ ♥❡①t ❡❞❣❡✿ j.nxt ✐s ♥♦t ♥✐❧ ❛♥❞ ❡q✉❛❧s k j✕k ✐s ❛ r❡q✉❡st ❡❞❣❡✿ ♠❡ss❛❣❡ [REQ,j] ✐s ✐♥ tr❛♥s✐t t♦ k

❞✐❣r❛♣❤✿ ❞✐r❡❝t❡❞ ♠✉❧t✐✲❣r❛♣❤ LNR✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ ❧❛st✴♥❡①t✴r❡q✉❡st ❡❞❣❡s ❪ L✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ ❧❛st ❡❞❣❡s ❪ LR✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ ❧❛st✴r❡q✉❡st ❡❞❣❡s ❪ ❉r❛✇✐♥❣ ❝♦♥✈❡♥t✐♦♥s ❧❛st ❡❞❣❡s✿ − − − − − ♥❡①t ❡❞❣❡s✿ ✲ ✲ ✲ ✲ ✲ r❡q✉❡st ❡❞❣❡s✿ · · · · · ·

SLIDE 6

❙❡r✐❛❧ ❡✈♦❧✉t✐♦♥✿ ❛t ♠♦st ✶ ❤✉♥❣r② ❛t ❛ t✐♠❡

♣r ❛❧❣

3 1 2 4 6 3 1 2 6 3 1 4 6 2 3 4 6 2 4 1 sends [REQ,4] to 2 5 5 5 5 4 turns hungry, 1 rcvs 2 forwards 3 1 4 6 2 5 turns eating 3 4 6 2 1 5 1 turns thinking sends obj to 4 initially 4 rcvs object, 1 eating [REQ,4] to 1 [REQ,4]

L ✐s ❛♥ ✐♥✲tr❡❡ ✇❤❡♥ ♥♦ r❡q ♠s❣ ✐♥ tr❛♥s✐t ❡❛❝❤ r❡q✉❡st ❡✛❡❝ts ❛ ♣❛t❤ r❡✈❡rs❛❧

j✬s r❡q tr❛✈❡❧s ❢r♦♠ j t♦ r♦♦t

❛❧❧ ♥♦❞❡s ♦♥ ♣❛t❤ ♥♦✇ ♣♦✐♥t t♦ j ❛♠♦rt✐③❡❞ ❝♦st ♦❢ ❧♦❣ N

SLIDE 7

❙❡r✐❛❧ ❡✈♦❧✉t✐♦♥✿ ♠♦r❡ t❤❛♥ ✶ ❤✉♥❣r② ❛t ❛ t✐♠❡

♣r ❛❧❣ 1 1 2 1 2 4 4 5 3 5 1 3 3 2 2 3 1 2 3 4 5 4 5 4 5 1 rcvs [REQ,3]. 1 turns thinking, 3 turns hungry, 2 turns hungry, sends object. sends [REQ,2] 3 turns eating sends [REQ,3]. 1 frwrds [REQ,2]. 3 rcvs [REQ,2] initially 1 eating

L ❡✈♦❧✈❡s ❛s ❜❡❢♦r❡✱ s♦ ❛♠♦rt✐③❡❞ ❝♦st s❛♠❡ ♥❡①t ♣trs ❢♦r♠ q✉❡✉❡

SLIDE 8

◆♦♥✲s❡r✐❛❧ ❡✈♦❧✉t✐♦♥

♣r ❛❧❣

1 2 1 1 1 2 4 5 4 4 5 3 5 3 1 4 3 3 5 2 2 rcvs 4 2 2 3 5 1 forwards 1 5 4 3 2 1 rcvs [REQ,3] 3 turns hungry. 1 turns thinking. 3 turns eating 3 rcvs [REQ,2]. initially 1 eating 2, 4 turn hungry [REQ,2]. [REQ,4]

L ♠❛② ♥❡✈❡r ❜❡ ❛♥ ✐♥✲tr❡❡ ❝❛♥ ❜❡ s❡✈❡r❛❧ ♥❡①t✲♣tr q✉❡✉❡s ♣r♦❣r❡ss❄ ❛♠♦rt✐③❡❞ ❝♦st❄

SLIDE 9

❉♦❡s ❤✉♥❣r② j ❡✈❡♥t✉❛❧❧② ❡❛t❄

♣r ❛❧❣

P♦ss✐❜✐❧✐t✐❡s ✇❤❡♥ j✬s r❡q✉❡st ♠s❣ ❞✐❡s

(a) j (b) j (c) j (d) j k k p k p k k p r q j (e)

❆❜♦✈❡ ❤❛s ✐♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥s L r❡♠❛✐♥s ❛❝②❝❧✐❝

j.nxt ♥❡✈❡r ♣♦✐♥ts t♦ j

· · · ◆♦✇ t♦ ♠❛❦❡ ❛r❣✉♠❡♥t r✐❣♦r♦✉s

SLIDE 10

❖✉t❧✐♥❡

♣r s❛❢❡t②

P❛t❤ r❡✈❡rs❛❧✿ ❛❧❣♦r✐t❤♠ P❛t❤ r❡✈❡rs❛❧✿ s❛❢❡t② ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ ♣r♦❣r❡ss ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ s❡r✐❛❧✐③❛❜✐❧✐t② ❛♥❛❧②s✐s

SLIDE 11

❇❛s✐❝ s❛❢❡t② ♣r♦♣❡rt✐❡s ✕ ✶

♣r s❛❢❡t②

■♥✈ A✶✕A✸ ❤♦❧❞s

// ✈✐❛ ✐♥✈ r✉❧❡

A✶ : ❢♦r♦♥❡✭j✿ ❡✐t❤❡r ✭j ❡❛t✐♥❣✮ ♦r ✭♦❜❥ ✐♥ tr❛♥s✐t t♦ j✮✮ A✷ : j.nxt = ♥✐❧ ⇒ ✭j.lst = ♥✐❧ ❛♥❞ ✭j ♥♦t t❤✐♥❦✐♥❣✮✮ A✸ : ✭j t❤✐♥❦✐♥❣✮ ⇒ j.lst = ♥✐❧ ■♥✈ B✶✕B✷ ❤♦❧❞s

// ✈✐❛ ✐♥✈ r✉❧❡ ❛ss✉♠✐♥❣ ■♥✈ A✶✕A✸

B✶ : LR ❤❛s ❡①❛❝t❧② ✶ ✉♥❞✐r❡❝t❡❞ ♣❛t❤ ❜❡t✇❡❡♥ ❡✈❡r② t✇♦ ♥♦❞❡s B✷ : ❢♦r❛❧❧✭j: j.lst = j✮

SLIDE 12

❇❛s✐❝ s❛❢❡t② ♣r♦♣❡rt✐❡s ✕ ✷

♣r s❛❢❡t②

■♥✈ B✸✕B✺ ❤♦❧❞s

// ✈✐❛ ✐♥✈ r✉❧❡ ❛ss✉♠✐♥❣ ■♥✈ A✶✕A✸, B✶✕B✸

B✸ : ❢♦r❛❧❧ j✿ ❡①❛❝t❧② ✶ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s

j t❤✐♥❦✐♥❣ ♦r [REQ,j] ✐♥ tr❛♥s✐t ♦r

❢♦rs♦♠❡ ✭k✿ k.nxt = j✮ ♦r

[OBJ] ✐♥ tr❛♥s✐t t♦ j ♦r j ❡❛t✐♥❣

B✹ : ❢♦r❛❧❧ ✭j✿ ❛t ♠♦st ♦♥❡ [REQ,j] ✐♥ tr❛♥s✐t✮ B✺ : ❢♦r❛❧❧ ✭j✿ j.nxt = j ❛♥❞ ♥✉♠✭k✿ k.nxt = j✮ ≤ ✶✮

SLIDE 13

Pr✐♦r✐t② ❞✐❣r❛♣❤ Pr

♣r s❛❢❡t②

❲❛♥t ❛ ❞✐❣r❛♣❤ Pr t❤❛t ❝❛♣t✉r❡s r❡❧❛t✐✈❡ ♣r✐♦r✐t✐❡s ♦❢ ♥♦❞❡s ❲❛♥t j✕k ✐♥ Pr t♦ ♠❡❛♥ j ❤❛s ❧♦✇❡r ♣r✐♦r✐t② t❤❛♥ k

j✕k ✐s ❛ ♣r✲♥❡①t ❡❞❣❡✿ k✕j ✐s ❛ ♥❡①t ❡❞❣❡ j✕k ✐s ❛ ♣r✲❧❛st ❡❞❣❡✿ j✕k ✐s ❛ ❧❛st ❡❞❣❡ ❛♥❞ j t❤✐♥❦✐♥❣ j✕k ✐s ❛ ♣r✲r❡q✉❡st ❡❞❣❡✿ j✕k ✐s ❛ r❡q✉❡st ❡❞❣❡

Pr✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ r❡q✴♣r✲♥❡①t✴♣r✲❧❛st ❡❞❣❡s ❪ ❉❡✜♥❡ ♣r✲♣❛t❤✿ ❞✐r❡❝t❡❞ ♣❛t❤ ✐♥ Pr

j ♣r✲r❡❛❝❤❛❜❧❡ ❢r♦♠ k✿ ♣r✲♣❛t❤ ❢r♦♠ k t♦ j

❧r✲♣❛t❤✿ ✉♥❞✐r❡❝t❡❞ ♣❛t❤ ✐♥ LR

SLIDE 14

❙♦♠❡ s❛❢❡t② ♣r♦♣❡rt✐❡s ♦❢ Pr

♣r s❛❢❡t②

■♥✈ C✶✕C✸

// ✈✐❛ ✐♥✈ r✉❧❡ ❛ss✉♠✐♥❣ ■♥✈ A✶✕A✸, B✶✕B✺

C✶ : ✭Pr ✐♥✲tr❡❡✮ ❛♥❞ ✭Pr✬s r♦♦t ❡❛t✐♥❣ ♦r ♦❜❥ ✐♥ tr❛♥s✐t t♦ ✐t✮ C✷ : ♣r✲♣❛t❤ ❢r♦♠ k t♦ j ⇒ ❢♦r❛❧❧ x ♦♥ t❤❡ ❧r✲♣❛t❤ ❜❡t✇❡❡♥ j ❛♥❞ k✿ ♣r✲♣❛t❤ ❢r♦♠ x t♦ j C✸ : ✭j ♥♦t t❤✐♥❦✐♥❣✮ ❛♥❞ j.lst = k = nil ⇒ ✭✭♣r✲♣❛t❤ ❢r♦♠ k t♦ j✮ ❛♥❞ ✭k ❤✉♥❣r②✮✮

SLIDE 15

Pr♦♦❢ ♦❢ ■♥✈ C✶ ✕ ✶

♣r s❛❢❡t②

■♥✐t✐❛❧❧② Pr ✐s t❤❡ s❛♠❡ ❛s LR✱ s♦ C✶✱ C✷✱ C✸ ❤♦❧❞✳

j st❛rts ❡❛t✐♥❣✿ Pr ♥♦t ❛✛❡❝t❡❞✱ s♦ C✶ ♣r❡s❡r✈❡❞ j ✐ss✉❡s r❡q ✇❤❡♥ j.lst = w✿ j✕w ❣♦❡s ❢r♦♠ ♣r✲❧❛st ❡❞❣❡ t♦ ♣r✲r❡q ❡❞❣❡✳ C✶ ♣r❡s❡r✈❡❞ j r❝✈s r❡q k ✇❤❡♥ t❤✐♥❦✐♥❣✿ j✕w✱ k✕j −

→ k✕w✱ j✕k✳ C✶ ♣r❡s❡r✈❡❞ ✭# ❡❞❣❡s✱ ❝♦♥♥❡❝t✐✈✐t② ♣r❡s❡r✈❡❞✮

j thinking w w j thinking k k j r❝✈s r❡q k ✇❤❡♥ ♥♦t t❤✐♥❦✐♥❣✱ j.lst = ♥✐❧✿ k✕j ❣♦❡s ❢r♦♠ ♣r✲r❡q ❡❞❣❡ t♦ ♣r✲♥❡①t ❡❞❣❡✳ C✶ ✐s ♣r❡s❡r✈❡❞

SLIDE 16

Pr♦♦❢ ♦❢ ■♥✈ C✶ ✕ ✷

♣r s❛❢❡t②

j r❝✈s r❡q k ✇❤❡♥ ♥♦t t❤✐♥❦✐♥❣✱ j.lst = w = ♥✐❧✿ k✕j r❡♣❧❛❝❡❞ ❜② k✕w✳

✭j✕w✱ j✕k ❛r❡ ♥♦t ✐♥ Pr✮

w w k k not thinking, lst = w not thinking, lst = k j j # ❡❞❣❡s ♣r❡s❡r✈❡❞✱ s♦ s✉✛ t♦ s❤♦✇ ❝♦♥♥❡❝t✐✈✐t② ♣r❡s❡r✈❡❞

♦❧❞ Pr ❤❛s ♣r✲♣❛t❤(w,j)

// ❢r♦♠ C✸

s✉✛ ✐❢ ♦❧❞ Pr ❤❛s ♥♦ ♣r✲♣❛t❤(w,k) ❛ss✉♠❡ ♦❧❞ Pr ❤❛s ♣r✲♣❛t❤(w,k) ❛❧❧ ♥♦❞❡s ♦♥ ❧r✲♣❛t❤(k,w) ❤❛✈❡ ♣r✲♣❛t❤ t♦ k

// ❢r♦♠ C✷

s♦ ❧r✲♣❛t❤(k,w) ❛✈♦✐❞s ❧r✲❡❞❣❡s k✕j ❛♥❞ j✕w

// ❢r♦♠ C✶✱ C✷

s♦ ✉♥❞✐r❡❝t❡❞ ❝②❝❧❡ ✐♥ ♦❧❞ LR

// ♥❡❣❛t❡s B✶

SLIDE 17

Pr♦♦❢ ♦❢ ■♥✈ C✶ ✕ ✸

♣r s❛❢❡t②

j st♦♣s ❡❛t✐♥❣ ✇❤❡♥ j.nxt =x✱ j.lst =w✿ x✕j r❡♣❧❛❝❡❞ ❜② j✕w w w x x thinking, lst = w j j y eating, lst = w nxt = x

♦❧❞ Pr ✐♥✲tr❡❡✴r♦♦t j❀ t♦ s❤♦✇ ♥❡✇ Pr ✐♥✲tr❡❡✴r♦♦t x s✉✛ ✐❢ ♦❧❞ Pr ❤❛s ♣r✲♣❛t❤(w,x)❀ ❛ss✉♠❡ ♥♦t s♦ s♦ ♦❧❞ Pr ❤❛s ♣r✲♣❛t❤(w,y) ❛♥❞ ♣r✲❡❞❣❡ y✕j✱ ✇❤❡r❡ y = x

y✕j ✐s ❛❧s♦ ❛ ❧r✲❡❞❣❡

✐❢ y = w✱ t❤❡♥ ♦❧❞ LR ❤❛s ❝②❝❧❡ [y,j,y]

// ♥❡❣❛t❡s B✶

✐❢ y = w✱ t❤❡♥ ❧r✲♣❛t❤(y,w) ❛✈♦✐❞s j

// C✶, C✷

❧r✲♣❛t❤ ❛♥❞ ❧r✲❡❞❣❡s y✕j✱ j✕w ❢♦r♠ ❧r✲❝②❝❧❡

// ♥❡❣❛t❡s B✶

SLIDE 18

❖✉t❧✐♥❡

♣r ♣r♦❣r❡ss

P❛t❤ r❡✈❡rs❛❧✿ ❛❧❣♦r✐t❤♠ P❛t❤ r❡✈❡rs❛❧✿ s❛❢❡t② ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ ♣r♦❣r❡ss ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ s❡r✐❛❧✐③❛❜✐❧✐t② ❛♥❛❧②s✐s

SLIDE 19

Pr♦❣r❡ss ♠❡tr✐❝ ✐♥ Pr

♣r ♣r♦❣r❡ss

Pr ❝❛♥ ❤❛✈❡ s❡✈❡r❛❧ ♥❡①t✲❡❞❣❡ ♣❛t❤s ♥❡①t✲q✉❡✉❡✿ ❛ ♠❛①✐♠❛❧ ♥❡①t✲❡❞❣❡ ♣❛t❤

// ✇✛ ❝♦③ Pr ✐♥✲tr❡❡

t❛✐❧ • ← · · · ← j ← · · · ← • hdj hdj✿ ❤❡❛❞ ♦❢ j✬s ♥❡①t✲q✉❡✉❡

j

✐❢ j ❤❛s ♥♦ ✐♥❝♦♠✐♥❣ ♥❡①t ❡❞❣❡

♦❛❧✿ ❢♥ F(j) st

✐♥❝r❡❛s❡s ✇❤✐❧❡ r❡q hdj ✐♥ tr❛♥s✐t ❤❛s ✉♣♣❡r ❜♦✉♥❞ ❛t ✇❤✐❝❤ hdj ❤❛s ♦❜❥ ✭❛♥❞ ♥♦ r❡q ♠s❣✮

SLIDE 20

Pr♦❣r❡ss✿ αj

♣r ♣r♦❣r❡ss

❈♦♥s✐❞❡r αj✿ s❡t ♦❢ ♥♦❞❡s ✇✐t❤ ♣r✲♣❛t❤s t♦ hdj ❋♦❧❧♦✇✐♥❣ ❤♦❧❞ D✶ : αj ✐♥❝r❡❛s❡s ✇❤❡♥ r❡q hdj ✐s r❝✈❞ ❜② ❛ s②st❡♠ t❤❛t ✐s t❤✐♥❦✐♥❣ ♦r ✇❤♦s❡ ❧❛st ♣♦✐♥t❡r ✐s ♥✐❧ D✷ : αj ❞♦❡s ♥♦t ❞❡❝r❡❛s❡ ✇❤✐❧❡ j ✐s ❤✉♥❣r②

SLIDE 21

Pr♦♦❢ ♦❢ D✶

♣r ♣r♦❣r❡ss

▲❡t r❡q hdj ❜❡ r❝✈❞ ❜② k Pr✐♦r t♦ r❝✈✱ k / ∈ αj

// Pr ✐♥✲tr❡❡✱ ❤❛s r❡q ❡❞❣❡ [hdj, k]

❉✐✛❡r❡♥t ❝❛s❡s ♦❢ k

k t❤✐♥❦✐♥❣✿

♣r✲r❡q hdj✕k → ♣r✲❧❛st k✕hdj αj ↑ ❜② k+

j hd j hd k k

k ♥♦t t❤✐♥❦✐♥❣✱ k.lst ♥✐❧✿

♣r✲r❡q hdj✕k → ♣r✲♥❡①t k✕hdj

// ❛s ✐♥ ❛❜♦✈❡ ✜❣✉r❡ k ❜❡❝♦♠❡s hdj✱ αj ↑ ❜② k+ k ♥♦t t❤✐♥❦✐♥❣✱ k.lst = x = ♥✐❧✿

♣r✲r❡q hdj✕k → ♣r✲r❡q hdj✕x αj ♥♦ ❝❤❛♥❣❡

j hd j hd k k x x

SLIDE 22

Pr♦♦❢ ♦❢ D✷

♣r ♣r♦❣r❡ss

❈♦♥s✐❞❡r st❡♣s ♦t❤❡r t❤❛♥ r① ♦❢ r❡q hdj

z st❛rts ❡❛t✐♥❣✿ ♥❡✐t❤❡r Pr ♥♦r αj ❝❤❛♥❣❡ z ✐ss✉❡s ❛ r❡q✉❡st✿ ♣r✲❧❛st z✕. → ♣r✲r❡q // αj s❛♠❡ z ♥♦t ✐♥ αj✿ ❞♦❡s ♥♦t ❞❡❝r❡❛s❡ αj z ✐♥ αj s❡♥❞s ♦❜❥❡❝t t♦ y✿

♦❧❞ Pr✿ z ✐s hdj✱ Pr✲r♦♦t ♥❡✇ Pr✿ y ✐s hdj✱ Pr✲r♦♦t

// αj s❛♠❡ ✭❛t ♠❛①✮ z ✐♥ αj r❝✈s r❡q k ✇❤❡♥ z.lst = x = ♥✐❧✿

♣r✲r❡q k✕j → ♣r✲r❡q k✕x✱ ♦❧❞ Pr ❤❛s ♣r✲♣❛t❤(z,x) ✭❢r♦♠ C✸✮

// α(j) ❞♦❡s ♥♦t ❞❡❝r❡❛s❡ z ✐♥ αj r❝✈s r❡q k ✇❤❡♥ z.lst ♥✐❧✿ ❁❞♦ ✐t❃

SLIDE 23

Pr♦❣r❡ss✿ βj

♣r ♣r♦❣r❡ss

❚♦ ❝♦♠♣❡♥s❛t❡ ❢♦r αj✱ ✇❛♥t ❢♥ βj t❤❛t X✶ : ↑ ✇❤❡♥ r❡q hdj r❝✈❞ ❜② ♥♦♥✲t❤✐♥❦✐♥❣ k ✇✐t❤ ♥♦♥✲♥✐❧ lst X✷ : ↓ ♦♥❧② ✐❢ αj ↑ s✐♠✉❧t❛♥❡♦✉s❧② ❈♦♥s✐❞❡r βj✿ s❡t ♦❢ ♥♦♥✲t❤✐♥❦✐♥❣ ♥♦❞❡s ✇❤♦s❡ lst ❡q✉❛❧s hdj βj ❛♥❞ αj ❛r❡ ❞✐s❥♦✐♥t

// ♣r✲♣❛t❤ ❢r♦♠ hdj t♦ βj ✭❢r♦♠ C✷✮

X✶ ❤♦❧❞s ❜❡❝❛✉s❡ r❝✈ ❛❞❞s k t♦ βj X✷ ❤♦❧❞s✳ x ❧❡❛✈❡s βj ✐♥ ♦♥❧② t✇♦ ✇❛②s✿

x st❛rts t❤✐♥❦✐♥❣✿ ❝r❡❛t❡s ♣r✲❧❛st x✕hdj✱ s♦ αj ↑ ❜② x x r❝✈s r❡q k✿

♣r✲r❡q k✕x → k✕hd(j) s♦ αj ↑ k ✭k / ∈ ♦❧❞ αj ✭C✶✕C✷✮✮ j hd j hd k k x x

SLIDE 24

Pr♦❣r❡ss

♣r ♣r♦❣r❡ss

❙♦ Fj = [αj.size, βj.size] ✉♥❞❡r ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ✇♦r❦s ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ✭D✺ ✉s❡❞ ✐♥ s❡r✐❛❧✐③❛❜✐❧✐t②✮✿ D✸ : (j eating

and k hungry)

❧❡❛❞s✲t♦

j.nxt = nil

D✹ : ((j eating

and j.nxt = nil)) ❧❡❛❞s✲t♦ j.nxt = nil)

⇒

(k hungry

❧❡❛❞s✲t♦

k eating)

D✺ : ((j and k are hungry) and (j pr-reachable from k)) ✉♥❧❡ss

((j eating) and (k hungry))

SLIDE 25

❖✉t❧✐♥❡

♣r s❡r✐❛❧✐③❛❜✐❧t②

P❛t❤ r❡✈❡rs❛❧✿ ❛❧❣♦r✐t❤♠ P❛t❤ r❡✈❡rs❛❧✿ s❛❢❡t② ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ ♣r♦❣r❡ss ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ s❡r✐❛❧✐③❛❜✐❧✐t② ❛♥❛❧②s✐s

SLIDE 26

❙❡r✐❛❧✐③❛❜✐❧✐t② ✕ ✶

♣r s❡r✐❛❧✐③❛❜✐❧t②

♦❛❧✿ ❚r❛♥s❢♦r♠ ❛♥② ✜♥✐t❡ ❡✈♦❧✉t✐♦♥ x ✈✐❛ ❝♦♠♠✉t❛t✐♦♥s

t♦ ❛ s❡r✐❛❧ ❡✈♦❧✉t✐♦♥ y ✇✐t❤ t❤❡ s❛♠❡ s❡t ♦❢ s❡♥❞s ❛♥❞ r❝✈s ▲❡t p ❞♦ t❤❡ it❤ ❡❛t✐♥❣ st❡♣ ✐♥ x✱ ❛♥❞ q ❞♦ t❤❡ ♣r❡❝❡❞✐♥❣ ♦♥❡✳ ❚❤❡ i t❤ ❡❛t✐♥❣ st❡♣ ✐s t❤❡ ❝✉❧♠✐♥❛t✐♦♥ ♦❢ ♦♥❡ H(p) st❡♣ ✭p ❜❡❝♦♠❡s ❤✉♥❣r②✮ ♦♥❡ ♦r ♠♦r❡ R(., p) st❡♣s ✭r❝✈ r❡q p✮ ♦♥❡ T(q) st❡♣ ✭q st❛rts t❤✐♥❦✐♥❣✮ ♦♥❡ E(p) st❡♣ ✭p st❛rts ❡❛t✐♥❣✮ ▲❡t vi ❜❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ t❤❡ ❛❜♦✈❡ st❡♣s ▲❡t w ❜❡ t❤❡ s❡q✉❡♥❝❡ ♦❢ x✲st❡♣s ♥♦t ✐♥ ❛♥② vi x ✐s ❛ ♠❡r❣❡ ♦❢ v✶✱ v✷✱ · · · ✱ w ❲✐❧❧ s❤♦✇ t❤❛t y ✐s v✶ ◦ v✷ ◦ · · · ◦ w

// ❤❡♥❝❡ s❛♠❡ ❝♦st

SLIDE 27

❙❡r✐❛❧✐③❛❜✐❧✐t② ✕ ✷

♣r s❡r✐❛❧✐③❛❜✐❧t②

▲❡♠♠❛ ✶✻✳✶✿ ▲❡t f ❛♥❞ g ❜❡ t✇♦ s✉❝❝❡ss✐✈❡ st❡♣s ✐♥ x st f ❜❡❧♦♥❣s t♦ vi ❛♥❞ g ❜❡❧♦♥❣s t♦ vj✱ j > i✱ ♦r t♦ w ❚❤❡♥ f ❛♥❞ g ❝♦♠♠✉t❡ ✇rt t❤❡ ♠s❣s s❡♥t ❛♥❞ r❝✈❞ x ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ t♦ y ❜② r❡♣❡❛t❡❞❧② ❛♣♣❧②✐♥❣ ❧❡♠♠❛ ✶✻✳✶ Pr♦♦❢ ♦❢ ❧❡♠♠❛ ❢♦❧❧♦✇s

SLIDE 28

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✻✳✶ ✕ ✶

♣r s❡r✐❛❧✐③❛❜✐❧t②

▲❡t g ❜❡ H ♦r R ♦❢ vi✱ ✐♥✈♦❧✈✐♥❣ r❡q p✳ ▲❡t f ❜❡ H ♦r R ♦❢ vj✱ ✐♥✈♦❧✈✐♥❣ r❡q q✳ ▲❡t g r❝✈ ♠s❣ s❡♥t ❜② f ✳ ❚❤❡♥ g ✐♥ vj

// ❝♦♥tr❛❞✐❝t✐♦♥

▲❡t f ❛♥❞ g ❜❡ ♦❢ s❛♠❡ ♥♦❞❡ x✳ ❚❤❡♥ ♣r✲♣❛t❤(q,p) ❥✉st ❛❢t❡r f ✳ ❙♦ p ❡❛ts ❜❡❢♦r❡ q ✭❢r♦♠ D✺✮✳

// ❝♦♥tr❛❞✐❝t✐♦♥

❍❡♥❝❡ f ❛♥❞ g ❝♦♠♠✉t❡✱ ♣r❡s❡r✈✐♥❣ s❡♥❞s ❛♥❞ r❝✈s ▲❡t g ❜❡ H ♦r R ♦❢ vi✱ ✐♥✈♦❧✈✐♥❣ r❡q p✳ ▲❡t f ❜❡ H ♦r R ♦❢ w✳ s❛♠❡ ❛s ❛❜♦✈❡ ❝❛s❡

SLIDE 29

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✻✳✶ ✕ ✷

♣r s❡r✐❛❧✐③❛❜✐❧t②

▲❡t g ❜❡ E ♦❢ vi✱ ✐❡✱ r❝✈ ♦❜❥✳ ▲❡t f ❜❡ H ♦r R ♦❢ vj ♦r w✱ ✐❡✱ r❝✈ r❡q✳ g r❝✈s ♦❜❥ ❛♥❞ f s❡♥❞s r❡q✳ ❙♦ g ❞♦❡s ♥♦t r❝✈ ❢r♦♠ f ✳ ▲❡t f ❛♥❞ g ❜❡ ❛t t❤❡ s❛♠❡ s②st❡♠✳ ❘❡q r❝✈ st❡♣ ✭f ✮ ✐s s❛♠❡ ✇❤❡t❤❡r ❤✉♥❣r② ✭f , g✮ ♦r ❡❛t✐♥❣ ✭g, f ✮✳ ❙♦ f ❛♥❞ g ❝❛♥ ❜❡ ✐♥t❡r❝❤❛♥❣❡❞✳ ❍❡♥❝❡ f ❛♥❞ g ❝♦♠♠✉t❡✱ ♣r❡s❡r✈✐♥❣ s❡♥❞s ❛♥❞ r❝✈s

SLIDE 30

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✶✻✳✶ ✕ ✸

♣r s❡r✐❛❧✐③❛❜✐❧t②

▲❡t g ❜❡ T ♦❢ vi✱ ✐❡✱ s❡♥❞ ♦❜❥✳ ▲❡t f ❜❡ H ♦r R ♦❢ vj ♦r w✳ g✱ ❜❡✐♥❣ ❛ T✱ ❞♦❡s ♥♦t r❝✈ ❢r♦♠ f ▲❡t f ❛♥❞ g ❜❡ ❛t t❤❡ s❛♠❡ s②st❡♠✱ s❛② x✳ ❚❤❡♥ f ❝❛♥♥♦t ❜❡ H ✭♦✴✇ g ❝♦✉❧❞ ♥♦t ❜❡ T✮ ❚❤✉s f ✐s ❛ R st❡♣✳ ❙✉♣♣♦s❡ x.lst ✇❛s ♥✐❧ ♣r✐♦r t♦ f ✳ ❚❤❡♥ g ✇♦✉❧❞ s❡♥❞ ♦❜❥ ✐♥ r❡s♣♦♥s❡ t♦ f ✱ s♦ f ❜❡❧♦♥❣s t♦ vi

// ❝♦♥tr❛❞✐❝ts j > i✳

❙✉♣♣♦s❡ x.lst ✇❛s ♥♦♥✲♥✐❧ ♣r✐♦r t♦ f ✳ ❚❤❡♥ f ❛♥❞ g ❝♦♠♠✉t❡ ❜❡❝❛✉s❡ r❡q r❝✈ ✭f ✮ s❛♠❡ ✇❤❡t❤❡r ❡❛t✐♥❣ ✭f , g✮ ♦r t❤✐♥❦✐♥❣ ✭g, f ✮✳