Projeto e Análise de Algoritmos Algoritmos Gulosos

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦

Pr♦❥❡t♦ ❡ ❆♥á❧✐s❡ ❞❡ ❆❧❣♦r✐t♠♦s

❆❧❣♦r✐t♠♦s ●✉❧♦s♦s

Pr♦❢✳ ❚✐❛❣♦ ❇✉❛rq✉❡ ❆ss✉♥çã♦ ❞❡ ❈❛r✈❛❧❤♦

❯♥✐❞❛❞❡ ❆❝❛❞ê♠✐❝❛ ❞❡ ●❛r❛♥❤✉♥s ✕ ❯❋❘P❊ ❇❛❝❤❛r❡❧❛❞♦ ❡♠ ❈✐ê♥❝✐❛s ❞❛ ❈♦♠♣✉t❛çã♦

✷✸ ❞❡ ♥♦✈❡♠❜r♦ ❞❡ ✷✵✶✽

(2)

❘♦t❡✐r♦

✶ ■♥tr♦❞✉çã♦

✷ ❯♠ ♣r♦❜❧❡♠❛ ❞❡ s❡❧❡çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s

❙✉❜❡str✉t✉r❛ ót✐♠❛ ❊s❝♦❧❤❛ ❣✉❧♦s❛ ❆❧❣♦r✐t♠♦ r❡❝✉rs✐✈♦ ❆❧❣♦r✐t♠♦ ■t❡r❛t✐✈♦

✸ ❊❧❡♠❡♥t♦s ❞❛ ❡str❛té❣✐❛ ❣✉❧♦s❛

Pr♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛ ❙✉❜❡str✉t✉r❛ ót✐♠❛

✹ ❈ó❞✐❣♦ ❍✉✛♠❛♥

❈ó❞✐❣♦s ❞❡ Pr❡✜①♦

❈♦♥str✉çã♦ ❞❡ ✉♠ ❝ó❞✐❣♦ ❍✉✛♠❛♥ Pr♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛ ❙✉❜❡str✉t✉r❛ ót✐♠❛

❈ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦

✺ ❊①❡r❝í❝✐♦

(3)

❆❧❣♦r✐t♠♦s ●✉❧♦s♦s

• ❯t✐❧✐③❛❞♦s ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦✳

• ▼❛✐s s✐♠♣❧❡s ❡ ❡✜❝✐❡♥t❡s ❞♦ q✉❡ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛✳

• ◆ã♦ ♣r♦❞✉③ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s✳ • ❊s❝♦❧❤❛s ót✐♠❛s ❧♦❝❛✐s ♣♦❞❡♠ ❧❡✈❛r ❛♦ ót✐♠♦ ❣❧♦❜❛❧✳ • ❊①❡♠♣❧♦s✿

• ➪r✈♦r❡ ❣❡r❛❞♦r❛ ♠í♥✐♠❛ ✭ár✈♦r❡ ❝♦♠ ♠í♥✐♠❛ s♦♠❛ ❞❡ ♣❡s♦ q✉❡ ❧✐❣❛ t♦❞♦s ♦s ♥ós ❞❡ ✉♠ ❣r❛❢♦✮✳

• ❉✐❥❦str❛ ✭❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦ ❞❡ ♦r✐❣❡♠ ú♥✐❝❛ ❡♠ ❣r❛❢♦✮✳

(4)

❯♠ ♣r♦❜❧❡♠❛ ❞❡ s❡❧❡çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s

• Pr♦❣r❛♠❛r ♦ ♠á①✐♠♦ ❞❡ ❛t✐✈✐❞❛❞❡s ❞✐st✐♥t❛s q✉❡ ✉t✐❧✐③❛♠

r❡❝✉rs♦ ❝♦♠♣❛rt✐❧❤❛❞♦✳

• ❊①✳✿ ❛❧♦❝❛r ♦ ♠á①✐♠♦ ❞❡ ❛t✐✈✐❞❛❞❡s ❡♠ ✉♠ ❛✉❞✐tór✐♦✱ ❞❛❞♦ ❛

❤♦r❛ ❞❡ ✐♥í❝✐♦ ❡ ✜♠ ❞❡ ❝❛❞❛ ❛t✐✈✐❞❛❞❡✳

• S ={a1, a2, ..., a_n}é ♦ ❝♦♥❥✉♥t♦ ❝♦♠ n ❛t✐✈✐❞❛❞❡s✳

• si é ♦ ✐♥st❛♥t❡ ❞❡ ✐♥í❝✐♦ ❞❡ ai✱ i= 1, . . . , n✳

• fi é ♦ ✐♥st❛♥t❡ ❞❡ ✜♠ ❞❛ ❛t✐✈✐❞❛❞❡✳

• 0≤si < fi <∞✳

• ai ❡aj sã♦ ❝♦♠♣❛tí✈❡✐s s❡ [si, fi) ❡[sj, fj) ♥ã♦ s❡ s♦❜r❡♣õ❡♠ ✭si ≥fj ♦✉sj ≥fi✮✳

(5)

❊①❡♠♣❧♦

❖r❣❛♥✐③❛♥❞♦ ❛s ❛t✐✈✐❞❛❞❡s ❞❡ ❢♦r♠❛ q✉❡✿

f1 ≤f2 ≤. . .≤f_n−1 ≤f_n.

i ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶

si ✶ ✸ ✵ ✺ ✸ ✺ ✻ ✽ ✽ ✷ ✶✷

fi ✹ ✺ ✻ ✼ ✾ ✾ ✶✵ ✶✶ ✶✷ ✶✹ ✶✻

❙ã♦ ❛t✐✈✐❞❛❞❡s ♠✉t✉❛♠❡♥t❡ ❝♦♠♣❛tí✈❡✐s✿

• {a3, a9, a11}

• {a1, a4, a8, a11}

• {a2, a4, a9, a11}

(6)

❙✉❜❡str✉t✉r❛ ót✐♠❛

• Sij é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛t✐✈✐❞❛❞❡s q✉❡ ❝♦♠❡ç❛♠ ❛♣ós fi ❡ t❡r♠✐♥❛♠ ❛♥t❡s ❞❡ sj✳

• Aij é ♦ ❝♦♥❥✉♥t♦ ♠á①✐♠♦ ❞❡ ❛t✐✈✐❞❛❞❡s ♠✉t✉❛♠❡♥t❡ ❝♦♠♣❛tí✈❡✐s ❡♠Sij✳

• ak ∈Aij.

• Aik =Aij ∩Sik ❡ Akj =Aij ∩Skj✳

• Aij =Aik∪ak∪Akj.

• |Aij|=|Aik|+ 1 +|Akj| é ♦ ♠á①✐♠♦ ❞❡ ❛t✐✈✐❞❛❞❡s ♠✉t✉❛♠❡♥t❡ ❝♦♠♣❛tí✈❡✐s ❡♠Sij.

• ❆ s♦❧✉çã♦ ♣❛r❛ Sij ❞❡♣❡♥❞❡ ❞❛s s♦❧✉çõ❡s Sik ❡Skj✳

(7)

❆r❣✉♠❡♥t♦ ✏r❡❝♦rt❛ ❡ ❝♦❧❛✑

• ❆ s♦❧✉çã♦ ♣❛r❛ Sij ❞❡♣❡♥❞❡ ❞❛s s♦❧✉çõ❡s Sik ❡Skj✳

• ❆ ♣r♦✈❛ ♣♦r r❡❝♦rt❛ ❡ ❝♦❧❛ ♣❛rt❡ ❞❡ ✉♠❛ s✉♣♦st❛ s♦❧✉çã♦

♠á①✐♠❛ ❡ ❡♥❝♦♥tr❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

• ❙✉♣♦♥❤❛ q✉❡ Aij s❡❥❛ ❛ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛Sij✳

• |Aij|=|Aik|+ 1 +|Akj| é ♦ ♠á①✐♠♦ ❞❡ ❛t✐✈✐❞❛❞❡s ♠✉t✉❛♠❡♥t❡ ❝♦♠♣❛tí✈❡✐s ❡♠Sij.

• ❙❡ ❡①✐st✐ss❡A′

ik t❛❧ q✉❡ |A

′

ik|>|Aik|✱ ❡♥tã♦

|A′

ik|+|Akj|+ 1>|Aik|+|Akj|+ 1 =|Aij|

♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ ❛ s✉♣♦s✐çã♦ ✐♥✐❝✐❛❧ ❞❡ q✉❡ |Aij| s❡❥❛ ♠á①✐♠♦✳

(8)

❙♦❧✉çã♦ ❝♦♠ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛

• s❡❥❛c[i, k] ♦ t❛♠❛♥❤♦ ❞❛ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛Sij✿

c[i, j] =c[i, k] +ck, j + 1.

• ▼❛s q✉❛❧ ❛t✐✈✐❞❛❞❡ak ✐♥❝❧✉✐r❄ ❆ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛ ❡①❛♠✐♥❛ ❝❛❞❛ ♣♦ss✐❜✐❧✐❞❛❞❡✳

c[i, j] =

0 s❡Sij =∅

maxak∈Sij(c[i, k] +c[k, j] + 1) s❡Sij 6=∅

.

• ❇❛st❛ ❛❣♦r❛ ❝❤❡❝❛r s❡ ❡①✐st❡ s♦❜r❡♣♦s✐çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳ • ❆ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ❝♦♠ ✉♠ ❛❧❣♦r✐t♠♦

r❡❝✉rs✐✈♦ ♠❡♠♦✐③❛❞♦ ♦✉ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❜❛✐①♦ ♣❛r❛ ❝✐♠❛✳

• ▼❛s ✉♠ ❛❧❣♦r✐t♠♦ ❣✉❧♦s♦ ♣♦❞❡ ❡①♣❧♦r❛r ♦✉tr❛s ✈❛♥t❛❣❡♥s✳

(9)

❯♠❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛

• ❋❛③❡r ❛ ❡s❝♦❧❤❛ s❡♠ r❡s♦❧✈❡r ♦s t♦❞♦s ♦s s✉❜♣r♦❜❧❡♠❛s✳ • ❙♦♠❡♥t❡ ✉♠❛ ❡s❝♦❧❤❛✿ ❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛✳

• ❊①✐st❡♠ ✈ár✐❛s ♣♦ssí✈❡❧ ❡s❝♦❧❤❛s ❣✉❧♦s❛s ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✳ • ❱❛♠♦s ✜①❛r ✉♠❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛✿ ❛ ❛t✐✈✐❞❛❞❡ q✉❡ t❡r♠✐♥❛

♠❛✐s ❝❡❞♦✳

• ❆ ❡s❝♦❧❤❛ é ❢❛❝✐❧✐t❛❞❛ s❡ ❛s ❛t✐✈✐❞❛❞❡s ❡st✐✈❡r❡♠ ♦r❞❡♥❛❞❛s

♣♦r ♦r❞❡♠ ❝r❡s❝❡♥t❡ ❞❡ tér♠✐♥♦✳

• ❉❡st❛ ❢♦r♠❛ ❡s❝♦❧❤❡✲s❡a1✳

• ❘❡st❛ ♦t✐♠✐③❛r ♦ s✉❜♣r♦❜❧❡♠❛S1 ✭t♦❞❛s ❛s ❛t✐✈✐❞❛❞❡s q✉❡ ❝♦♠❡ç❛ ❛♣ós ♦ ✜♥❛❧ ❞❡a1✮✳

• ❖❜s❡r✈❛✲s❡ q✉❡ ♥ã♦ ❡①✐st❡♠ ❛t✐✈✐❞❛❞❡s ❝♦♠♣❛tí✈❡✐s ❛♥t❡s ❞❡a1 ♣♦✐s a1 ❛❝❛❜❛ ♣r✐♠❡✐r♦ q✉❡ t♦❞❛s ❛s ❛t✐✈✐❞❛❞❡s✳

• ➱ ✉♠ ♣r♦❥❡t♦ ❝♦♠✉♠ ❞❡ ❛❧❣♦r✐t♠♦s ❣✉❧♦s♦s ❝♦♥str✉✐r ✉♠

❛❧❣♦r✐t♠♦ ❞❡ ❝✐♠❛ ♣❛r❛ ❜❛✐①♦ ❡♠ ❞✉❛s ❡t❛♣❛s✿ ❢❛ç❛ ❛ ❡s❝♦❧❤❛ ❡ ❞❡♣♦✐s r❡s♦❧✈❛ ♦ s✉❜♣r♦❜❧❡♠❛✳

(10)

❊st❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛ é ót✐♠❛✦

• ❙❡❥❛Sk ✉♠ s✉❜♣r♦❜❧❡♠❛ ♥ã♦ ✈❛③✐♦✳

• ❊ am ✉♠❛ ❛t✐✈✐❞❛❞❡ ❞❡ Sk ❝♦♠ ♦ tér♠✐♥♦ ♠❛✐s ❝❡❞♦✳

• ❊♥tã♦ am ❡stá ✐♥❝❧✉í❞❛ ❡♠ ❛❧❣✉♠ s✉❜❝♦♥❥✉♥t♦ ♠á①✐♠♦ ❞❡ ❛t✐✈✐❞❛❞❡s ❝♦♠♣❛tí✈❡✐s✳

Pr♦✈❛

• ❙❡❥❛Ak ✉♠ s✉❜❝♦♥❥✉♥t♦ ♠á①✐♠♦ ❞❡ ❛t✐✈✐❞❛❞❡s ❝♦♠♣❛tí✈❡✐s ❞❡

Sk✳

• ❊ aj ❛ ❛t✐✈✐❞❛❞❡ ❝♦♠ ♦ tér♠✐♥♦ ♠❛✐s ❝❡❞♦ ❡♠Ak.

• ❈❛s♦ ✶✳ aj =am✳

• ❈❛s♦ ✷✳ aj 6=am✳

• ❊♥tã♦fm < fj✳ P♦❞❡✲s❡ tr♦❝❛r aj ♣♦ram ❡♠ Ak✳

• A′

k=Ak−aj∪am.

• A′

k é ♠á①✐♠♦✱ |Ak|=|A′k|✳

(11)

❯♠ ❛❧❣♦r✐t♠♦ ❣✉❧♦s♦ r❡❝✉rs✐✈♦

• ❆ss✉♠❡✲s❡ q✉❡ ❛s ❛t✐✈✐❞❛❞❡s ❡stã♦ ♦r❞❡♥❛❞❛s ♣♦r ♦r❞❡♠

❝r❡s❝❡♥t❡ ❞❡ tér♠✐♥♦✳

• s[] é ♦ ❛rr❛♥❥♦ ❝♦♠ ♦s t❡♠♣♦s ❞❡ ✐♥í❝✐♦ ❞❛s ❛t✐✈✐❞❛❞❡s✳

• f[] é ♦ ❛rr❛♥❥♦ ❝♦♠ ♦s t❡♠♣♦s ❞❡ ❡♥❝❡rr❛♠❡♥t♦ ❞❛s ❛t✐✈✐❞❛❞❡s✳

• k é ❛ ú❧t✐♠❛ ❛t✐✈✐❞❛❞❡ ✐♥s❡r✐❞❛✳ • n é ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❛t✐✈✐❞❛❞❡s✳ • a0 é ✉♠ ❛t✐✈✐❞❛❞❡ ✜❝tí❝✐❛ ❝♦♠f0 = 0

• ❘❊❈❯❘❙■❱❊✲❆❈❚■❱■❚❨✲❙❊▲❊❈❚❖❘✭s, f,0, n✮ é ❛ ❝❤❛♠❛❞❛ ✐♥✐❝✐❛❧✳

(12)

(13)

❯♠ ❛❧❣♦r✐t♠♦ ❣✉❧♦s♦ ✐t❡r❛t✐✈♦

• _{❘❊❈❯❘❙■❱❊✲❆❈❚■❱■❚❨✲❙❊▲❊❈❚❖❘} é q✉❛s❡ r❡❝✉rs✐✈♦ ❞❡ ❝❛❧❞❛✳

• A é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❛t✐✈✐❞❛❞❡s s❡❧❡❝✐♦♥❛❞❛s ❛té k✳ • k é ♦ í♥❞✐❝❡ ❞❛ ú❧t✐♠❛ ❛t✐✈✐❞❛❞❡ ❛❞✐❝✐♦♥❛❞❛✳ • ■♥✐❝✐❛❧ ❛❞✐❝✐♦♥❛❞♦a1✳

• ▲❛ç♦✿ ❡♥❝♦♥tr❡ ❛ ♣ró①✐♠❛ ❛t✐✈✐❞❛❞❡ ♣❛r❛ ❛❞✐❝✐♦♥❛r✿ ❛❞✐❝✐♦♥❡ ❡

❛t✉❛❧✐③❡k✳

(14)

❘♦t❡✐r♦

✶ ■♥tr♦❞✉çã♦

✷ ❯♠ ♣r♦❜❧❡♠❛ ❞❡ s❡❧❡çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s

✸ ❊❧❡♠❡♥t♦s ❞❛ ❡str❛té❣✐❛ ❣✉❧♦s❛

Pr♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛ ❙✉❜❡str✉t✉r❛ ót✐♠❛

✹ ❈ó❞✐❣♦ ❍✉✛♠❛♥

❈ó❞✐❣♦s ❞❡ Pr❡✜①♦

❈ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦

✺ ❊①❡r❝í❝✐♦

(15)

❊❧❡♠❡♥t♦s ❞❛ ❡str❛té❣✐❛ ❣✉❧♦s❛

• ❆ ❡str❛té❣✐❛ ❣✉❧♦s❛ ❡s❝♦❧❤❡ s❡♠♣r❡ ❛q✉❡❧❛ q✉❡ ♣❛r❡❝❡ s❡r ❛

♠❡❧❤♦r s♦❧✉çã♦ ♥♦ ♠♦♠❡♥t♦✳

• ◆❡♠ s❡♠♣r❡ ♣r♦❞✉③ ❛ s♦❧✉çã♦ ót✐♠❛✳

❙❡q✉ê♥❝✐❛ ❞❡ ❡t❛♣❛s ♣❛r❛ ♣r♦❥❡t❛r ✉♠ ❛❧❣♦r✐t♠♦ ❣✉❧♦s♦✿

✶✳ ❈❛❞❛ ❡s❝♦❧❤❛ ❣❡r❛ ✉♠ ú♥✐❝♦ s✉❜♣r♦❜❧❡♠❛✳

✷✳ Pr♦✈❛r q✉❡ ❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛ é s❡♠♣r❡ s❡❣✉r❛✱ ❛❧❣✉♠❛ s♦❧✉çã♦ ót✐♠❛ ✉t✐❧✐③❛ ❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛✳

✸✳ ▼♦str❛r q✉❡ ❡①✐st❡ s✉❜❡str✉t✉r❛ ót✐♠❛✳ ❊❧❡♠❡♥t♦ ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❡str❛té❣✐❛ ❣✉❧♦s❛✿

• Pr♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛✳ • ❙✉❜❡str✉t✉r❛ ót✐♠❛✳

(16)

Pr♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛

• ✏P♦❞❡♠♦s ♠♦♥t❛r ✉♠ s♦❧✉çã♦ ❣❧♦❜❛❧♠❡♥t❡ ót✐♠❛ ✉t✐❧✐③❛♥❞♦

❡s❝♦❧❤❛s ❣✉❧♦s❛s✱ ❧♦❝❛❧♠❡♥t❡ ót✐♠❛s✳✑

• ❉✐❢❡r❡♥t❡ ❞❛ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛ ✭❞❛ ❜❛✐①♦ ♣❛r❛ ❝✐♠❛✱ r❡s♦❧✈❡

♣r✐♠❡✐r♦ ♦s s✉❜♣r♦❜❧❡♠❛s ♣❛r❛ ♣♦❞❡r ❢❛③❡r ❛ ❡s❝♦❧❤❛✮✳

• Pr✐♠❡✐r♦ ❢❛③ ❛ ❡s❝♦❧❤❛ ❞❡♣♦✐s r❡s♦❧✈❡ ♦ s✉❜♣r♦❜❧❡♠❛ ✭❞❡ ❝✐♠❛

♣❛r❛ ❜❛✐①♦✮✳

• ➱ ♣r❡❝✐s♦ ♣r♦✈❛r q✉❡ ❝❛❞❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛ ✈❛✐ ♣r♦❞✉③✐r ✉♠❛

s♦❧✉çã♦ ❣❧♦❜❛❧♠❡♥t❡ ót✐♠❛✳

• ❖ ❛❧❣♦r✐t♠♦ ♣♦❞❡ s❡r ❡✜❝✐❡♥t❡ ♣♦rq✉❡ ❛❧❣✉♠❛s ✈❡③❡s é rá♣✐❞♦

❢❛③❡r ❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛✳

(17)

❙✉❜❡str✉t✉r❛ ót✐♠❛

• ❆ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡♣❡♥❞❡ ❞❛ s♦❧✉çã♦ ót✐♠❛

♣❛r❛ s❡✉s s✉❜♣r♦❜❧❡♠❛s✳

• Út✐❧ t❛♥t♦ ❡♠ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛ q✉❛♥t♦ ❡♠ ❛❧❣♦r✐t♠♦s

❣✉❧♦s♦s✳

• ❊①❡♠♣❧♦ ✭s❡❧❡çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s✮✿ s❡ ❛ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ♦

♣r♦❜❧❡♠❛ Sij ✐♥❝❧✉✐r ❛ ❛t✐✈✐❞❛❞❡ ak✱ é ♣r❡❝✐s♦ ❡♥❝♦♥tr❛r ❛s s♦❧✉çõ❡s ót✐♠❛s ♣❛r❛ ♦s s✉❜♣r♦❜❧❡♠❛sSik ❡Skj✳

• P❛r❛ ❞❡♠♦♥str❛r q✉❡ ❡①✐st❡ s✉❜❡str✉t✉r❛ ót✐♠❛ ❡♠ ❛❧❣♦r✐t♠♦

❣✉❧♦s♦s é ♣r❡❝✐s♦

✏♠♦str❛r q✉❡ ❛ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ♦ s✉❜♣r♦❜❧❡♠❛✱ ❝♦♠❜✐♥❛❞❛ ❝♦♠ ❛ ❡s❝♦❧❤❛ ❣✉❧♦s❛✱ ♣r♦❞✉③ ❛ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✐♥✐❝✐❛❧✳✑

(18)

❊str❛té❣✐❛ ●✉❧♦s❛ ✈s✳ Pr♦❣r❛♠❛çã♦ ❉✐♥â♠✐❝❛

• ◗✉❛♥❞♦ ❛ ❡str❛té❣✐❛ ❣✉❧♦s❛ é s✉✜❝✐❡♥t❡❄ ✭♠❛✐s ❡✜❝✐❡♥t❡✮

• ◗✉❛♥❞♦ é ♥❡❝❡ssár✐♦ ✉t✐❧✐③❛r ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛❄ ✭❣✉❧♦s❛ ♥ã♦

r❡s♦❧✈❡✮✳

• ❉✉❛s ✈❡rsõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ ✭❦♥❛♣s❛❝❦✮✳

• ❖ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ ✵✲✶✿ ❡s❝♦❧❤❡r ❡♥tr❡ n ✐t❡♥s q✉❛✐s

❧❡✈❛r ❛té ♦ ♣❡s♦ ♠á①✐♠♦ W ♣❛r❛ t❡r ❛ ❝❛r❣❛ ♠❛✐s ✈❛❧✐♦s❛✳ • ❖ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ ❢r❛❝✐♦♥ár✐♦✿ ♣r♦❜❧❡♠❛ s❡♠❡❧❤❛♥t❡

♠❛s ❝♦♠ ❛ ❣❛r❣❛ ❞❡ ♣❡s♦ ❢r❛❝✐♦♥ár✐♦ ✭❡①✳ ❣rã♦s✮✳

(19)

❖ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ ❢r❛❝✐♦♥ár✐♦

• ❙✉❜❡str✉t✉r❛ ót✐♠❛✿ r❡♠♦✈❡♥❞♦ ✉♠❛ ❝❛r❣❛w ❞❛ ♠♦❝❤✐❧❛ t❡♠✲s❡

❛ ♠❡❧❤♦r s♦❧✉çã♦ ❝♦♠ ❝❛r❣❛r ❛téW −w✱ r❡♠♦✈❡♥❞♦ ❛ ♠❡s♠❛

❝❛r❣❛w ❞❛s ♦♣çõ❡s ♣❛r❛ ❧❡✈❛r✳ • P♦❞❡ ✉t✐❧✐③❛r ❡str❛té❣✐❛ ❣✉❧♦s❛✳

• P❛r❛ ❝❛❞❛ ✐t❡♠ i ❝❛❧❝✉❧❛✲s❡ ♦ ♣r❡ç♦ ♣♦r q✉✐❧♦ vi/wi✳

• ❊s❝♦❧❤❡ ❛ ♠á①✐♠❛ q✉❛♥t✐❞❛❞❡ ♣♦ssí✈❡❧ ❞♦ ✐t❡♠ ❝♦♠ ♠❛✐♦r ✈❛❧♦r

♣♦r q✉✐❧♦✳ ◗✉❛♥❞♦ s❡ ❡s❣♦t❛ ♣❛ss❛ ♣❛r❛ ♦ ✐t❡♠ s❡❣✉✐♥t❡ ❝♦♠ ♠❛✐♦r ✈❛❧♦r ♣♦r q✉✐❧♦✳

• ➱ ♣r❡❝✐s♦ ♦r❞❡♥❛r ♦s ✐t❡♥s ♣❡❧♦ ✈❛❧♦r ♣♦r q✉✐❧♦✳ • ❈♦♠♣❧❡①✐❞❛❞❡ O(nlgn)✳

(20)

❖ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ ✵✲✶

• ❙✉❜❡str✉t✉r❛ ót✐♠❛✿ r❡♠♦✈❡♥❞♦ ♦ ✐t❡♠j✱ ❞❡ ♣❡s♦ wj✱ ❞❛ ♠♦❝❤✐❧❛ t❡♠✲s❡ ❛ ♠❡❧❤♦r s♦❧✉çã♦ ❝♦♠ ❝❛r❣❛r ❛té W −wj ❞♦s

n−1 ✐t❡♥s ♦r✐❣✐♥❛✐s✳

• ◆ã♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦ ❝♦♠ ❡str❛té❣✐❛ ❣✉❧♦s❛✳

• ❈♦♥tr❛✲❡①❡♠♣❧♦✿ ♦ ✐t❡♠ ✶ t❡♠ ♦ ♠❛✐♦r ✈❛❧♦r ♣♦r ♣❡s♦✱ ♠❛s ♥ã♦

♣♦❞❡ ❡st❛r ♥❛ s♦❧✉çã♦ ót✐♠❛ ♣♦rq✉❡ ♦❝✉♣❛ ❡s♣❛ç♦ ❞❡ ❝♦❧♦❝❛r ♦✉tr♦s ✐t❡♥s ♠❛✐♦r❡s✳

(21)

❘♦t❡✐r♦

✶ ■♥tr♦❞✉çã♦

✷ ❯♠ ♣r♦❜❧❡♠❛ ❞❡ s❡❧❡çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s

✸ ❊❧❡♠❡♥t♦s ❞❛ ❡str❛té❣✐❛ ❣✉❧♦s❛

Pr♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛ ❙✉❜❡str✉t✉r❛ ót✐♠❛

✹ ❈ó❞✐❣♦ ❍✉✛♠❛♥

❈ó❞✐❣♦s ❞❡ Pr❡✜①♦

❈ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦

✺ ❊①❡r❝í❝✐♦

(22)

❈ó❞✐❣♦ ❍✉✛♠❛♥

• ❊①❡♠♣❧♦✿ ❛rq✉✐✈♦ ❞❡ ✶✵✵✳✵✵✵ ❝❛r❛❝t❡r❡s✱ ✻ ❝❛r❛❝t❡r❡s ❞✐st✐♥t♦s✳ • ❯t✐❧✐③❛♥❞♦ ❝ó❞✐❣♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✜①♦✿ ✸✵✵✳✵✵✵ ❜✐ts✳ • ❯t✐❧✐③❛♥❞♦ ❝ó❞✐❣♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✈❛r✐á✈❡❧✿ ✷✷✹✳✵✵✵ ❜✐ts✳

✭❡❝♦♥♦♠✐❛ ❞❡ ✷✺✪✮✳

(23)

❈ó❞✐❣♦s ❞❡ Pr❡✜①♦

• ❈ó❞✐❣♦s ❞❡ Pr❡✜①♦✿ ♥❡♥❤✉♠ ❝ó❞✐❣♦ é ♣r❡✜①♦ ❞❡ ♦✉tr♦✳ • P♦❞❡♠ s❡r ❧✐❞♦s ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❞✐r❡✐t❛ s❡♠ ❛♠❜✐❣✉✐❞❛❞❡✳ • ❊①❡♠♣❧♦✿ ❜❜❝ ❂ ✶✵✶✶✵✶✶✵✵✳

• ◗✉❛❧ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❝❛r❛❝t❡r❡s ♣❛r❛ ✶✶✵✵✶✶✵✶✵✵✶✶✶✵✵❄

• P♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ár✈♦r❡ ✭♦ ♥ó ❝♦♥té♠ ❛ ❢r❡q✉ê♥❝✐❛

❞❛ s✉❜ár✈♦r❡✮✳

(24)

❈♦♥str✉çã♦ ❞❡ ✉♠ ❝ó❞✐❣♦ ❍✉✛♠❛♥

• Qé ✉♠❛ ✜❧❛ ❞❡ ♣r✐♦r✐❞❛❞❡ ♠í♥✐♠❛✳

• ❊❳❚❘❆❈❚✲▼■◆✭Q✮ s❡❧❡❝✐♦♥❛ ♦ ✐t❡♠ ❞❡ ♠❡♥♦r ❢r❡q✉❡♥❝✐❛ ❡♠ Q✳ • ❏✉♥t❛ s✉❝❡ss✐✈❛♠❡♥t❡ ♦s ✐t❡♥s ❞❡ ♠❡♥♦r ❢r❡q✉ê♥❝✐❛✳

(25)

(26)

❚❡♠♣♦ ♣❛r❛ ❝♦♥str✉çã♦ ❞♦ ❝ó❞✐❣♦ ❍✉✛♠❛♥

• ❙❡Q é ✉♠ ❤❡❛♣ ♠í♥✐♠♦ ❜✐♥ár✐♦✳ • ■♥✐❝✐❛❧✐③❛çã♦O(n)✳

• ❖ ❧❛ç♦ é r❡♣❡t✐❞♦ n−1 ✈❡③❡s✳ • ❊❳❚❘❆❈❚✲▼■◆✭Q✮ ❝✉st❛ O(lgn)✳

• P♦rt❛♥t♦ ♦ ❝✉st♦ ❞♦ ❛❧❣♦r✐t♠♦ éO(nlgn)✳

• ❙❡Q é ✉♠❛ ár✈♦r❡ ❞❡ ✈❛♥ ❊♠❞❡ ❇♦❛s✱ ♦ ❝✉st♦ ❝❛✐ ♣❛r❛ O(nlg lgn).

(27)

❈✉st♦ ❞❛ ár✈♦r❡

B(T) =X c∈C

c.f req×dT(c),

• ◗✉❛♥t✐❞❛❞❡ ❞❡ ❜✐ts ♣❛r❛ ❝♦❞✐✜❝❛r ✉♠ ❛rq✉✐✈♦✳

• cé ✉♠ ❝❛r❛❝t❡r❡ ❞♦ ❛❧❢❛❜❡t♦ C✱ ❝♦♠ ❢r❡q✉ê♥❝✐❛ c.f req✱ • dT(c) é ❛ ♣r♦❢✉♥❞✐❞❛❞❡ ❞❛ ❢♦❧❤❛c ♥❛ ár✈♦r❡ T✳

(28)

Pr♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛

• x ❡y sã♦ ♦s ❞♦✐s ❝❛r❛❝t❡r❡s ❞♦ ❛❧❢❛❜❡t♦ C ❝♦♠ ❢r❡q✉ê♥❝✐❛ ♠❛✐s

❜❛✐①❛ ❡x.f req ≤y.f req✳

• ▲❡♠❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❡s❝♦❧❤❛ ❣✉❧♦s❛✳ ❊①✐st❡ ✭i✮ ❝ó❞✐❣♦

❞❡ ♣r❡✜①♦ ót✐♠♦ ♣❛r❛ C ♥♦ q✉❛❧ ♦s ❝ó❞✐❣♦s ♣❛r❛ x ❡ y ✭ii✮ t❡♠

♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦ ❡ ✭iii✮ ❞✐❢❡r❡♠ ❛♣❡♥❛s ♥♦ ú❧t✐♠♦ ❜✐t✳ • Pr♦✈❛✳ ▼♦❞✐✜❝❛✲s❡ ✉♠❛ ár✈♦r❡ ót✐♠❛ ♣❛r❛ q✉❡ t❡♥❤❛ ❛ ❢♦r♠❛

❞❡s❡❥❛❞❛ ❡ ✈❡r✐✜❝❛ q✉❡ ❛ ár✈♦r❡ ♠♦❞✐✜❝❛❞❛ ♥ã♦ t❡♠ ❝✉st♦ ♠❛✐♦r✳

• P❛r❛ ❞♦✐s ❝❛r❛❝t❡r❡s q✉❛✐sq✉❡ra❡ b✱ ❝♦♠a.f req≤b.f req.

• x.f req≤a.f req❡ y.f req≤b.f req✳

• ❚r♦❝❛✲s❡a♣♦rx ❡ b♣♦r y✳

(29)

❈✉st♦ ❞❛ ár✈♦r❡ ♠♦❞✐✜❝❛❞❛

❖ ❝✉st♦ ❞❛ ár✈♦r❡ ♠♦❞✐✜❝❛❞❛T′ ♥ã♦ é ♠❛✐♦r q✉❡ ♦ ❝✉st♦ ❞❛ ár✈♦r❡

♦r✐❣✐♥❛❧T✳

(30)

❙✉❜❡str✉t✉r❛ ót✐♠❛

• x ❡y sã♦ ♦s ❝❛r❛❝t❡r❡s ❞♦ ❛❧❢❛❜❡t♦ C ❝♦♠ ♠❡♥♦r ❢r❡q✉ê♥❝✐❛✳ • C′

=C− {x, y} ∪ {z}✱ ❝♦♠ z.f req=x.f req+y.f req. • T′ é ✉♠❛ ár✈♦r❡ ❝♦♠ ❝ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦ ♣❛r❛

C′✳ • ▲❡♠❛ ❞❛ s✉❜❡str✉t✉r❛ ót✐♠❛✳ ❆ ár✈♦r❡ T r❡♣r❡s❡♥t❛ ✉♠

❝ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦ ❡ ♣♦❞❡ s❡r ❣❡r❛❞❛ s✉❜st✐t✉✐♥❞♦ ❛ ❢♦❧❤❛ z

❞❡T′ ♣❡❧❛ ♥♦ ❝♦♠ ❛s ❢♦❧❤❛s x ❡y✳ • dT(x) =dT(y) = 1 +dT(z).

(31)

❙✉❜❡str✉t✉r❛ ót✐♠❛ ✭Pr♦✈❛✮

• Pr♦✈❛✳ ❙✉♣♦♥❤❛ q✉❡ T ♥ã♦ s❡❥❛ ✉♠ ❝ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦

✈❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

• ❙❡T ♥ã♦ é ót✐♠♦ ❡♥tã♦ ❡①✐t❡T′′ t❛❧ q✉❡ B(T′′

)< B(T)✳

• ❚r❛♥s❢♦r♠❛✲s❡T′′ ❡♠

T′′′ q✉❡ é ✉♠ ❝ó❞✐❣♦ ót✐♠♦ ♣❛r❛ C′✳ B(T′′′_{) =}_B₍_T′_{) =}_B₍_T′′₎₋_{x.f req}₋_{y.f req}

< B(T)−x.f req−y.f req ✳ • T′ é ✉♠ ❝ó❞✐❣♦ ót✐♠♦ ♣❛r❛

C′✱ ❝♦♠♦

T′′′✱ ❧♦❣♦ B(T′′′

) =B(T′)✳ • B(T′

) =B(T)−x.f req−y.f req✱ ♣♦rt❛♥t♦ B(T′

)< B(T)−x.f req−y.f req é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

• ▲♦❣♦T t❡♠ q✉❡ r❡♣r❡s❡♥t❛r ✉♠ ❝ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦ ♣❛r❛C

✉♠❛ ✈❡③ q✉❡ T′ ✭s✉❜♣r♦❜❧❡♠❛✮ é ✉♠ ❝ó❞✐❣♦ ❞❡ ❞❡ ♣r❡✜①♦ ót✐♠♦

♣❛r❛ C′✳

(32)

❈ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦

• ❆ ❡s❝♦❧❤❛ ❣✉❧♦s❛ ❞❡ ❝♦❧♦❝❛r ♦s ❞♦✐s ❝❛r❛❝t❡r❡s ❞❡ ♠❡♥♦r

❢r❡q✉ê♥❝✐❛ ❝♦♠♦ ❢♦❧❤❛s ❞❡ ✉♠ ♠❡s♠♦ ♥ó ❣❛r❛♥t❡ q✉❡ s❡❥❛ ❣❡r❛❞♦ ✉♠ ❝ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦✳

• ❙✉❜st✐t✉✐r ♦s ❞♦✐s ♥ós ❞❡ ♠❡♥♦r ❢r❡q✉ê♥❝✐❛ ❝♦♠♦ ✉♠ ♥♦✈♦ ♥ó

✭s♦♠❛♥❞♦ ❛s ❞✉❛s ❢r❡q✉ê♥❝✐❛s✮ é ✉♠❛ s✉❜❡str✉t✉r❛ ót✐♠❛✳

• ❙❡ é ❡♥❝♦♥tr❛❞♦ ✉♠ ❝ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦ ♣❛r❛ ♦

s✉❜♣r♦❜❧❡♠❛ ❡♥tã♦ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ❝ó❞✐❣♦ ❞❡ ♣r❡✜①♦ ót✐♠♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳

(33)

❊①❡r❝í❝✐♦✿ ❯♠ ❜✉r❛❝♦ ♣❛r❛ ❝❛❞❛ r❛t♦

• ❙ã♦n r❛t♦s ❡ n ❜✉r❛❝♦s ❛❜❡rt♦s✳ ❈❛❞❛ r❛t♦ ❡ ❝❛❞❛ ❜✉r❛❝♦s ❡stá

❡♠ ✉♠❛ ♣♦s✐çã♦ ♥✉♠❡r❛❞❛✳

• ◗✉❛❧ ❛ ❞✐stâ♥❝✐❛ ♠á①✐♠❛ q✉❡ ✉♠ r❛t♦ ♣r❡❝✐s❛ ♣❡r❝♦rr❡r ♣❛r❛ q✉❡

❝❛❞❛ r❛t♦ ❡♥tr❡ ❡♠ ✉♠ ❜✉r❛❝♦❄

• ❊①❡♠♣❧♦✿

• ❊♥tr❛❞❛ ❞✉❛s ❧✐st❛s✿ ♣♦s✐çõ❡s ❞♦s ❜✉r❛❝♦s{−1,2,3} ❡ ❞♦s r❛t♦s {−2,2,6}✳

• ❙❡ ♦ r❛t♦−2❡♥tr❛ ♥♦ ❜✉r❛❝♦ −1✱ ❛♥❞❛ ✶ ❝❛s❛❀ s❡ ♦ r❛t♦ ✷ ❡♥tr❛ ♥♦ ❜✉r❛❝♦ ✷✱ ❛♥❞❛ ✵❀ s❡ ♦ r❛t♦ ✻ ❡♥tr❛ ♥♦ ❜✉r❛❝♦ ✸✱ ❛♥❞❛ ✸✳ • ❙❛í❞❛✿ ✸ ✭♦ r❛t♦ q✉❡ ♠❛✐s ❛♥❞♦✉ ♣❡r❝♦rr❡✉ ❡st❛ ❞✐stâ♥❝✐❛✮✳

(34)

❘❡❢❡rê♥❝✐❛

▲✐✈r♦✿

❚✳❍✳ ❈♦r♠❡♥✱ ❈✳❊✳ ▲❡✐s❡rs♦♥✱ ❘✳▲✳ ❘✐✈❡st✱ ❈✳ ❙t❡✐♥✳ ❆❧❣♦r✐t♠♦s✿ ❚❡♦r✐❛ ❡ Prát✐❝❛✳ ❊❧s❡✈✐❡r✱ ✷✵✶✷✳

✲

◆❡st❛ ❛✉❧❛✿ ❈❛♣ít✉❧♦ ✶✻✳

❖✉tr♦s ♣r♦❜❧❡♠❛s✿

❤tt♣s✿✴✴✇✇✇✳❣❡❡❦s❢♦r❣❡❡❦s✳♦r❣✴❣r❡❡❞②✲❛❧❣♦r✐t❤♠s✴

❤tt♣s✿✴✴✇✇✇✳✐♠❡✳✉s♣✳❜r✴⑦♣❢✴❛♥❛❧✐s❡❴❞❡❴❛❧❣♦r✐t♠♦s✴❛✉❧❛s✴ ❣✉❧♦s♦✳❤t♠❧