Uma Interface de Programação Distribuída para Aplicações em Otimização Combinatória

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s

▼❡str❛❞♦ ❡ ❉♦✉t♦r❛❞♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦

❯♠❛ ■♥t❡r❢❛❝❡ ❞❡ Pr♦❣r❛♠❛çã♦ ❉✐str✐❜✉í❞❛

♣❛r❛ ❆♣❧✐❝❛çõ❡s ❡♠ ❖t✐♠✐③❛çã♦

❈♦♠❜✐♥❛tór✐❛

❆❧❧❜❡rs♦♥ ❇r✉♥♦ ❞❡ ❖❧✐✈❡✐r❛ ❉❛♥t❛s

(2)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s

▼❡str❛❞♦ ❡ ❉♦✉t♦r❛❞♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦

❯♠❛ ■♥t❡r❢❛❝❡ ❞❡ Pr♦❣r❛♠❛çã♦ ❉✐str✐❜✉í❞❛ ♣❛r❛

❆♣❧✐❝❛çõ❡s ❡♠ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛

❆✉t♦r

❆❧❧❜❡rs♦♥ ❇r✉♥♦ ❞❡ ❖❧✐✈❡✐r❛ ❉❛♥t❛s

❖r✐❡♥t❛❞♦r

❘✐❝❛r❞♦ ❈♦r❞❡✐r♦ ❈♦rrê❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❈♦♦r❞❡♥❛çã♦

❞♦

Pr♦❣r❛♠❛

❞❡

Pós✲❣r❛❞✉❛çã♦

❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ❝♦♠♦

♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦

❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛ ❞❛

❈♦♠♣✉t❛çã♦✳

(3)

❘❡s✉♠♦

E

st❡ tr❛❜❛❧❤♦ ❢♦✐ ♠♦t✐✈❛❞♦ ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❛ ❡①♣❧♦r❛çã♦ ❞♦ ♣♦t❡♥❝✐❛❧ ❞♦

♣❛r❛❧❡❧✐s♠♦ ❞✐str✐❜✉í❞♦ ❡♠ ❛♣❧✐❝❛çõ❡s ❡♠ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✳ P❛r❛

t❛♥t♦✱ ♣r♦♣♦♠♦s ✉♠❛ ✐♥t❡r❢❛❝❡ ❞❡ ♣r♦❣r❛♠❛çã♦ ❞✐str✐❜✉í❞❛✱ ♥❛ q✉❛❧ ♣r❡③❛♠♦s ❞♦✐s

r❡q✉✐s✐t♦s ♣r✐♥❝✐♣❛✐s✿ ❡✜❝✐ê♥❝✐❛ ❡ r❡✉s♦✳

❖ ♣r✐♠❡✐r♦ ❛❞✈é♠ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡ ❈❆❉ ✭❈♦♠♣✉t❛çã♦ ❞❡ ❆❧t♦

❉❡s❡♠♣❡♥❤♦✮ ❡①✐❣✐r❡♠ ♠á①✐♠♦ ❞❡s❡♠♣❡♥❤♦ ♣♦ssí✈❡❧✳ ❆ss✐♠ s❡♥❞♦✱ ❡s♣❡❝✐✜❝❛♠♦s

❡st❛ ✐♥t❡r❢❛❝❡ ❝♦♠♦ ✉♠❛ ❡①t❡♥sã♦ ❞❛ ❜✐❜❧✐♦t❡❝❛ ▼P■✱ ❛ q✉❛❧ é ❛ss✉♠✐❞❛ ❝♦♠♦

❡✜❝✐❡♥t❡ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ❞✐str✐❜✉í❞❛s✳ ❖ r❡q✉✐s✐t♦ r❡✉s♦ ❞❡✈❡ t♦r♥❛r ❝♦♠♣❛tí✈❡✐s ❞✉❛s

❝❛r❛❝t❡ríst✐❝❛s ✐♠♣♦rt❛♥t❡s✿ ❛ss✐♥❝r♦♥✐s♠♦ ❡ ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s✳ ❖ ❛ss✐♥❝r♦♥✐s♠♦

❞❡✈❡ ❡st❛r ♣r❡s❡♥t❡ ♥❛ ✐♥t❡r❢❛❝❡✱ ✉♠❛ ✈❡③ q✉❡ ❛s ❛♣❧✐❝❛çõ❡s ❡♠ ❖t✐♠✐③❛çã♦

❈♦♠❜✐♥❛tór✐❛✱ ❡♠ s✉❛ ♠❛✐♦r✐❛✱ ♣♦ss✉❡♠ ✉♠❛ ♥❛t✉r❡③❛ ❛ssí♥❝r♦♥❛✳ ❖♣❡r❛çõ❡s

❝♦❧❡t✐✈❛s sã♦ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s q✉❡ ❞❡✈❡♠ ❡st❛r ❞✐s♣♦♥í✈❡✐s ♥❛ ✐♥t❡r❢❛❝❡✱ ❞❡ ♠♦❞♦

q✉❡ ♣♦ss❛♠ s❡r ✉t✐❧✐③❛❞❛s ♣♦r ❛♣❧✐❝❛çõ❡s ❡♠ s✉❛s ❡①❡❝✉çõ❡s✳

❚❡♥❞♦ ❡♠ ✈✐st❛ ❛t❡♥❞❡r ♦ r❡q✉✐s✐t♦ r❡✉s♦✱ ❜❛s❡❛♠♦s ❡st❛ ✐♥t❡r❢❛❝❡ ♥♦s ▼♦❞❡❧♦s

❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞♦s ♣♦r ❊✈❡♥t♦s ❡ ♣♦r P✉❧s♦s✱ ♣♦✐s ♦s ♠❡s♠♦s sã♦

❛ssí♥❝r♦♥♦s ❡ ♣❡r♠✐t❡♠ ❛ ✐♥❝♦r♣♦r❛çã♦ ❞❡ ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s✳

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❆❜str❛❝t

❚❤✐s ✇♦r❦ ✇❛s ♠♦t✐✈❛t❡❞ ❜② t❤❡ ♥❡❡❞ ♦❢ ❡①♣❧♦✐t✐♥❣ t❤❡ ♣♦t❡♥t✐❛❧ ♦❢ ❞✐str✐❜✉t❡❞

♣❛r❛❧❡❧✐s♠ ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ❛♣♣❧✐❝❛t✐♦♥s✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s ❣♦❛❧✱ ✇❡

♣r♦♣♦s❡ ❛ ❞✐str✐❜✉t❡❞ ♣r♦❣r❛♠♠✐♥❣ ✐♥t❡r❢❛❝❡✱ ✐♥ ✇❤✐❝❤ ✇❡ ❝❤❡r✐s❤ t✇♦ ♠❛✐♥

r❡q✉✐r❡♠❡♥ts✿ ❡✣❝✐❡♥❝② ❛♥❞ r❡✉s❡✳

❚❤❡ ✜rst st❡♠s ❢r♦♠ t❤❡ ♥❡❡❞ ♦❢ ❍P❈ ✭❍✐❣❤ P❡r❢♦r♠❛♥❝❡ ❈♦♠♣✉t✐♥❣✮

❛♣♣❧✐❝❛t✐♦♥s r❡q✉✐r❡ ♠❛①✐♠✉♠ ♣♦ss✐❜❧❡ ♣❡r❢♦r♠❛♥❝❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ s♣❡❝✐❢② ♦✉r

✐♥t❡r❢❛❝❡ ❛s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ▼P■ ❧✐❜r❛r②✱ ✇❤✐❝❤ ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❡✣❝✐❡♥t

❢♦r ❞✐str✐❜✉t❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❚❤❡ ✏r❡✉s❡✑ r❡q✉✐r❡♠❡♥t ♠✉st ♠❛❦❡ ❝♦♠♣❛t✐❜❧❡ t✇♦

✐♠♣♦rt❛♥t ❢❡❛t✉r❡s✿ ❛s②♥❝❤r♦♥✐s♠ ❛♥❞ ❝♦❧❧❡❝t✐✈❡ ♦♣❡r❛t✐♦♥s✳ ❆s②♥❝❤r♦♥✐s♠ ♠✉st ❜❡

♣r❡s❡♥t ❛t ♦✉r ✐♥t❡r❢❛❝❡✱ ♦♥❝❡ ♠♦st ♦❢ ❝♦♠❜✐♥❛t♦r✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ❛♣♣❧✐❝❛t✐♦♥s ❤❛✈❡

❛♥ ❛s②♥❝❤r♦♥♦✉s ♥❛t✉r❡✳ ❈♦❧❧❡❝t✐✈❡ ♦♣❡r❛t✐♦♥s ❛r❡ ❢❡❛t✉r❡s t❤❛t s❤♦✉❧❞ ❜❡ ❛✈❛✐❧❛❜❧❡

✐♥ t❤❡ ✐♥t❡r❢❛❝❡✱ s♦ t❤❛t t❤❡② ❝❛♥ ❜❡ ✉s❡❞ ❜② ❛♣♣❧✐❝❛t✐♦♥s ✐♥ t❤❡✐r ❡①❡❝✉t✐♦♥✳

■♥ ♦r❞❡r r❡❛❝❤ t❤❡ r❡✉s❡ r❡q✉✐r❡♠❡♥t✱ ✇❡ ❜❛s❡❞ t❤✐s ✐♥t❡r❢❛❝❡ ♦♥ t❤❡ ❊✈❡♥t✲ ❛♥❞

P✉❧s❡✲❞r✐✈❡♥ ▼♦❞❡❧s ♦❢ ❉✐str✐❜✉t❡❞ ❈♦♠♣✉t✐♥❣✱ ♦♥❝❡ t❤❡② ❛r❡ ❛s②♥❝❤r♦♥♦✉s ❛♥❞

❛❧❧♦✇ t❤❡ ✐♥❝♦r♣♦r❛t✐♦♥ ♦❢ ❝♦❧❧❡❝t✐✈❡ ♦♣❡r❛t✐♦♥s✳

(5)

❉❡❞✐❝❛tór✐❛

(6)

❆❣r❛❞❡❝✐♠❡♥t♦s

G

♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣❡❧♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ ♣♦r ♠❡

♣r♦♣♦r❝✐♦♥❛r s❛ú❞❡ ❡ ♣❡rs❡✈❡r❛♥ç❛ ♣❛r❛ ❛❧❝❛♥ç❛r t♦❞♦s ♦s ♦❜❥❡t✐✈♦s ❞❛ ♠✐♥❤❛

✈✐❞❛✳

❆♦ ♣r♦❢❡ss♦r ❘✐❝❛r❞♦ ❈♦rrê❛ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡♠ ♠✐♠ ❞❡♣♦s✐t❛❞❛ ❡ ♣❡❧❛

❝♦♠♣❡tê♥❝✐❛ ♥❛ ♦r✐❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦ ♣r♦❢❡ss♦r ❍❡r♦♥ ❈❛r✈❛❧❤♦ ♣❡❧♦s ❝♦♥s❡❧❤♦s✱ ✐♥❝❡♥t✐✈♦s ❡ ❞✐s♣♦s✐çã♦ ♣❛r❛ ❛❥✉❞❛r

♥❛s ❞✐✜❝✉❧❞❛❞❡s ❞❡st❡ tr❛❜❛❧❤♦✳

➚ ❯❋❈✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈♦♠♣✉t❛çã♦✱ ♣❡❧❛

♦♣♦rt✉♥✐❞❛❞❡ ❞❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

➚ ♠✐♥❤❛ ❛✈ó ▲✐❞✉✐♥❛ q✉❡ ♠❡ ❝r✐♦✉ ❝♦♠♦ ✉♠ ✜❧❤♦ ❡ ♠❡ ❡♥s✐♥♦✉ t✉❞♦ ♥❛ ✈✐❞❛ ❡

♠❡ ❢❡③ s❡r t✉❞♦ ♦ q✉❡ s♦✉ ❤♦❥❡✳

➚ ♠✐♥❤❛ ♠ã❡ ❆❧❜❛ q✉❡ ♠❡ ❞❡✉ ♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ é ♣❛r❛ ♠✐♠ ❡①❡♠♣❧♦ ❞❡

♣❡rs❡✈❡r❛♥ç❛✳

❆♦ ♠❡✉ ♣❛✐ ❲✐❧❧❞s♦♥ ✭✐♥ ♠❡♠♦r✐❛♠✮ q✉❡ ❢♦✐ ❡♠ ✈✐❞❛ ❡①❡♠♣❧♦ ❞❡ s✐♠♣❧✐❝✐❞❛❞❡ ❡

❞❡❞✐❝❛çã♦✳

➚s ♠✐♥❤❛s t✐❛s ❋❛❜✐❛♥❡✱ ❑át✐❛ ❡ ❆❧t❛♠✐r❛ q✉❡ ♠✉✐t♦ ❛❥✉❞❛r❛♠ ♥❛ ♠✐♥❤❛ ❝r✐❛çã♦✳

❆♦ ❙❡r♣r♦✱ ❡♠♣r❡s❛ q✉❡ ♠❡ ❛❝♦❧❤❡✉ tã♦ ❜❡♠ ❡ ♣❡r♠✐t✐✉ ❧✐❜❡r❛çã♦ ❞❡ ❤♦r❛s ❞❡

tr❛❜❛❧❤♦ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ ♠❡str❛❞♦✳

(7)

❙✉♠ár✐♦

❆❜str❛❝t

✐✐

✶ ■♥tr♦❞✉çã♦

✶

✶✳✶ ▼♦t✐✈❛çõ❡s ❡ ❖❜❥❡t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✶✳✷ ❖s ▼♦❞❡❧♦s ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✷✳✶ ❈♦♠♣✉t❛çõ❡s ❉✐str✐❜✉í❞❛s ❉✐r✐❣✐❞❛s ♣♦r ❊✈❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✷✳✷ ❈♦♠♣✉t❛çõ❡s ❉✐str✐❜✉í❞❛s ❉✐r✐❣✐❞❛s ♣♦r P✉❧s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✸ ❖♣❡r❛çõ❡s ❈♦❧❡t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✸✳✶ ❘❡❣✐str♦ ❞♦s ❊st❛❞♦s ▲♦❝❛✐s ■♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✸✳✷ ❖r❞❡♥❛çã♦ ❞❡ ▼❡♥s❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✸✳✸ ❉❡t❡❝çã♦ ❞❛ ❚❡r♠✐♥❛çã♦ ●❧♦❜❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✹ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✺ ❖r❣❛♥✐③❛çã♦ ❞♦ ❉♦❝✉♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✷ ❖ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r ❊✈❡♥t♦s

✽

✷✳✶ ❈♦♠♣✉t❛çõ❡s ❉✐str✐❜✉í❞❛s ❝♦♠♦ ❈♦♥❥✉♥t♦ ❞❡ ❊✈❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✷✳✷ ❯♠ ❆❧❣♦r✐t♠♦ ●❡♥ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✷✳✸ ❖♣❡r❛çõ❡s ❈♦❧❡t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✸✳✶ ❘❡❣✐str♦ ❞♦s ❊st❛❞♦s ▲♦❝❛✐s ■♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✸✳✷ ❉❡t❡❝çã♦ ❞❛ ❚❡r♠✐♥❛çã♦ ●❧♦❜❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷✳✸✳✸ ❖r❞❡♥❛çã♦ ❞❡ ▼❡♥s❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸ ❖ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r P✉❧s♦s

✷✶

✸✳✶ ❈♦♠♣✉t❛çõ❡s ❉✐str✐❜✉í❞❛s ❝♦♠♦ ❙❡q✉ê♥❝✐❛s ❞❡ P✉❧s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸✳✷ ❯♠ ❆❧❣♦r✐t♠♦ ●❡♥ér✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✸✳✸ ▼❡❝❛♥✐s♠♦ ❞❡ ●❡r❛çã♦ ❞❡ P✉❧s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸✳✹ ❖♣❡r❛çõ❡s ❈♦❧❡t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✹✳✶ ❘❡❣✐str♦ ❞♦s ❊st❛❞♦s ▲♦❝❛✐s ■♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸✳✹✳✷ ❉❡t❡❝çã♦ ❞❛ ❚❡r♠✐♥❛çã♦ ●❧♦❜❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸✳✹✳✸ ❖r❞❡♥❛çã♦ ❞❡ ▼❡♥s❛❣❡♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✹ ■♥t❡r❢❛❝❡ ❞❡ Pr♦❣r❛♠❛çã♦

✸✹

✹✳✶ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✹✳✷ ❆ ❇✐❜❧✐♦t❡❝❛ ▼P■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹✳✸ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r ❊✈❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

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❙✉♠ár✐♦

✈✐

✹✳✸✳✶ ❘♦t✐♥❛s ❞❡ ❈♦♥✜❣✉r❛çã♦ ❞❡ P❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✹✳✸✳✷ ❘♦t✐♥❛s q✉❡ ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s ❞❡♥tr♦ ❞❡ ❢✉♥çõ❡s ❞❡ ❊✈❡♥t♦

❡ ❊✈❡♥t♦ ❊s♣♦♥tâ♥❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹✳✸✳✸ ❘♦t✐♥❛s q✉❡ ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s ❞❡♥tr♦ ❞❡ ❢✉♥çõ❡s ❞❡ ❊✈❡♥t♦ ✹✵

✹✳✸✳✹ ❘♦t✐♥❛s ❞❡ ❉✐❢✉sã♦ ❡ ❊①❡❝✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✹✳✸✳✺ ❯♠❛ ❆♣❧✐❝❛çã♦ ●❡♥ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✹✳✹ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r P✉❧s♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✹✳✹✳✶ ❘♦t✐♥❛s ❞❡ ❈♦♥✜❣✉r❛çã♦ ❞❡ P❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✹✳✹✳✷ ❘♦t✐♥❛s q✉❡ ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s ❞❡♥tr♦ ❞❡ ❢✉♥çõ❡s ❞❡ ❊✈❡♥t♦ ✹✼

✹✳✹✳✸ ❘♦t✐♥❛s q✉❡ ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s ❞❡♥tr♦ ❞❡ ❢✉♥çõ❡s ❞❡ P✉❧s♦ ✳ ✹✽

✹✳✹✳✹ ❘♦t✐♥❛s ❞❡ ❉✐❢✉sã♦ ❡ ❊①❡❝✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹✳✹✳✺ ❯♠❛ ❆♣❧✐❝❛çã♦ ●❡♥ér✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✺ ❆♣❧✐❝❛çõ❡s

✺✺

✺✳✶ ❆❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲❇♦✉♥❞ ❙❡q✉❡♥❝✐❛❧ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞♦ ❈■▼ ✳ ✳ ✺✺

✺✳✷ ❆❧❣♦r✐t♠♦s ❇r❛♥❝❤✲❛♥❞✲❇♦✉♥❞ ❉✐str✐❜✉í❞♦s ❆❧❡❛tór✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✺✳✷✳✶ ❆s♣❡❝t♦s ●❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✺✳✷✳✷ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r ❊✈❡♥t♦s ✳ ✳ ✳ ✺✽

✺✳✷✳✸ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r P✉❧s♦s ✳ ✳ ✳ ✳ ✻✷

✻ ❊①♣❡r✐♠❡♥t♦s

✻✻

✻✳✶ P❛râ♠❡tr♦s ❞❡ ❆✈❛❧✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✻✳✷ ❉❡s❝r✐çã♦ ❞♦s ❊①♣❡r✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✻✳✸ ❊①♣❡r✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾

✻✳✹ ❆✈❛❧✐❛çã♦ ❞♦s ❊①♣❡r✐♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹

✼ ❈♦♥❝❧✉sõ❡s

✼✻

✼✳✶ Pr✐♥❝✐♣❛✐s ❉✐✜❝✉❧❞❛❞❡s ❊♥❝♦♥tr❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

✼✳✷ ❈♦♥tr✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

✼✳✸ Pr♦❞✉t♦s ●❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

✼✳✹ ❚r❛❜❛❧❤♦s ❢✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❖ t❡r♠♦ ❈♦♠♣✉t❛çã♦ ❞❡ ❆❧t♦ ❉❡s❡♠♣❡♥❤♦✱ ♦✉ ❈❆❉✱ ❡stá ❛ss♦❝✐❛❞♦ ❛♦ ♥✐❝❤♦ ❞❡ ❛♣❧✐❝❛çõ❡s q✉❡ r❡q✉❡r❡♠ ✉♠ ❛❧t♦ ♣♦❞❡r ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ❢r❡q✉❡♥t❡♠❡♥t❡ ❛❧❝❛♥ç❛❞♦ ❛tr❛✈és ❞♦ ✉s♦ ✐♥t❡♥s✐✈♦ ❞❡ ♣❛r❛❧❡❧✐s♠♦✳ ❆ ❈❆❉ é ✉s❛❞❛ ❧❛r❣❛♠❡♥t❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ♠♦❞❡❧♦s ❢ís✐❝♦s ♣❛r❛ ♣r❡✈✐sã♦ ❝❧✐♠át✐❝❛ ❡ ❞❡ ❝❛tástr♦❢❡s ♥❛t✉r❛✐s✱ ♥❛ ❞❡s❝♦❜❡rt❛ ❞❡ ❢ár♠❛❝♦s ❛tr❛✈és ❞❛ ♠♦❞❡❧❛❣❡♠ ♠♦❧❡❝✉❧❛r✱ ♥❛ ❣❡♥ô♠✐❝❛ ❡ ❜✐♦✐♥❢♦r♠át✐❝❛ ❡ ♥♦ s❡t♦r ❞❡ ✜♥❛♥ç❛s ❛tr❛✈és ❞❛ ♣r❡✈✐sã♦ ❞❡ ♠♦✈✐♠❡♥t❛çõ❡s ❞❡ ♠❡r❝❛❞♦ ❬✶✹❪✳ ❚❛✐s ❛♣❧✐❝❛çõ❡s sã♦ ♦r✐✉♥❞❛s✱ s♦❜r❡t✉❞♦✱ ❞❛s ❈✐ê♥❝✐❛s ❈♦♠♣✉t❛❝✐♦♥❛✐s ❡ ❊♥❣❡♥❤❛r✐❛s ❬✷✵❪✳ ◆❡ss❡ ❝♦♥t❡①t♦✱ ✉s❛✲s❡ ♦ t❡r♠♦ ❈❆❉ ♣❛r❛ ❞❡s✐❣♥❛r ❛ ár❡❛ q✉❡ ❡st✉❞❛ ♠♦❞❡❧♦s✱ té❝♥✐❝❛s ❡ ❛❧❣♦r✐t♠♦s q✉❡ ♣❡r♠✐t❡♠ ❡①tr❛✐r ♦ ♠á①✐♠♦ ❞❡ ❞❡s❡♠♣❡♥❤♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡♠ ❝❛❞❛ ❛♣❧✐❝❛çã♦✳ ❆❧é♠ ❞❛s ❛♣❧✐❝❛çõ❡s ❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ r❡ss❛❧t❛♠♦s ❛ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✱ ❛ q✉❛❧✱ ❛tr❛✈és ❞❡ ♣r♦❜❧❡♠❛s s♦❜r❡ ❣r❛❢♦s✱ é ✉♠❛ ❞❛s ♠♦t✐✈❛çõ❡s ❞❡st❡ tr❛❜❛❧❤♦✳

◆♦ ✈❛st♦ ❞♦♠í♥✐♦ ❞❡ ❈❆❉✱ ❞❡❞✐❝❛♠♦s ♥♦ss❛ ❛t❡♥çã♦ ❛♦ ♣❛r❛❧❡❧✐s♠♦ ❞✐str✐❜✉í❞♦✱ ♥♦ q✉❛❧ ♦ ♣♦❞❡r ❝♦♠♣✉t❛❝✐♦♥❛❧ é ♦❜t✐❞♦ ✉t✐❧✐③❛♥❞♦✲s❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥ós ✐♥t❡r❝♦♥❡❝t❛❞♦s ❡♠ ✉♠❛ r❡❞❡ ♣♦r ❝❛♥❛✐s ❞❡ ❝♦♠✉♥✐❝❛çõ❡s ❜✐❞✐r❡❝✐♦♥❛✐s✳ ◆❡ss❛ r❡❞❡✱ ❝❛❞❛ ♥ó ♣♦ss✉✐ s✉❛ ♣ró♣r✐❛ ♠❡♠ór✐❛✱ à q✉❛❧ ❡❧❡ t❡♠ ❛❝❡ss♦ ❡①❝❧✉s✐✈♦ ❡♠ s✉❛s ❝♦♠♣✉t❛çõ❡s ❧♦❝❛✐s✳ ❆s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ♥ós ♦❝♦rr❡ ❛tr❛✈és ❞❡ tr♦❝❛s ❞❡ ♠❡♥s❛❣❡♥s✳ ❚♦❞♦s ♦s ❝❛♥❛✐s ❞❡ ❝♦♠✉♥✐❝❛çã♦ sã♦ ❝♦♥✜á✈❡✐s✱ ♦✉ s❡❥❛✱ s❡ ✉♠❛ ♠❡♥s❛❣❡♠ é ❡♥✈✐❛❞❛ ❛tr❛✈és ❞♦ r❡❢❡r✐❞♦ ❝❛♥❛❧✱ s✉❛ ❝❤❡❣❛❞❛ ❛♦ ♥ó ❞❡ ❞❡st✐♥♦ é ❣❛r❛♥t✐❞❛✱ ♠❡s♠♦ q✉❡ ❝♦♠ ❛❧❣✉♠ ❛tr❛s♦ ✜♥✐t♦✳ ❖s ❝❛♥❛✐s t❛♠❜é♠ ♥ã♦ ♣♦❞❡♠ ❣❛r❛♥t✐r ♦r❞❡♠ ♥❛s ♠❡♥s❛❣❡♥s q✉❡ tr❛❢❡❣❛♠ ♥❡❧❡s✳ ❈❛❞❛ ♥ó é ❝❛♣❛③ ❞❡ r❡❛❧✐③❛r ❝♦♠♣✉t❛çõ❡s s❡q✉❡♥❝✐❛✐s✱ ❛s q✉❛✐s sã♦ ❣✉✐❛❞❛s ♣♦r ✉♠ r❡❧ó❣✐♦ ❧♦❝❛❧✳ ❊♥tr❡t❛♥t♦✱ ♥ã♦ ❡①✐st❡ ✉♠ r❡❧ó❣✐♦ ❣❧♦❜❛❧ q✉❡ ❞✐t❛ ♦ ♣❛ss♦ ❡♠ q✉❡ ❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ s❡ ❞❡s❡♥r♦❧❛✳

✶✳✶ ▼♦t✐✈❛çõ❡s ❡ ❖❜❥❡t✐✈♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❜✉s❝❛♠♦s ❡①♣❧♦r❛r ♦ ♣♦t❡♥❝✐❛❧ ❞♦ ♣❛r❛❧❡❧✐s♠♦ ❞✐str✐❜✉í❞♦ ❡♠ ❛♣❧✐❝❛çõ❡s ❛ssí♥❝r♦♥❛s✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❡♠ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✳ P❛r❛ t❛♥t♦✱ ♣r♦♣♦♠♦s ✉♠❛ ✐♥t❡r❢❛❝❡ ❞❡ ♣r♦❣r❛♠❛çã♦ ❞✐str✐❜✉í❞❛ q✉❡ ♣r❡③❛ ❞♦✐s r❡q✉✐s✐t♦s ♣r✐♥❝✐♣❛✐s✱ ❛♠❜♦s ❡♠ ✉♠ ❝♦♥t❡①t♦ ❞❡ ❝♦♠♣✉t❛çõ❡s ❛ssí♥❝r♦♥❛s✿ ❡✜❝✐ê♥❝✐❛ ❡ r❡✉s♦✳ ❆♠❜♦s ❛❞✈ê♠ ❞❛ ♥❛t✉r❡③❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡ ❈❆❉✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ ♦ ♣r✐♥❝í♣✐♦ ✏❡✜❝✐ê♥❝✐❛✑ ❥á ❡st❛r ♣r❡s❡♥t❡ ♥♦ ♣r♦❥❡t♦ ❞❛ ▼❡ss❛❣❡ P❛ss✐♥❣ ■♥t❡r❢❛❝❡ ✭▼P■✮ ❬✶✽❪✱ ❛ ✐♥t❡r❢❛❝❡ ❛q✉✐ ♣r♦♣♦st❛ é ✉♠❛ ❡①t❡♥sã♦ ❞❛ ▼P■✳

(10)

✶✳✷✳ ❖s ▼♦❞❡❧♦s ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛

✷

✉s♦ ❞❛ ✐♥t❡r❢❛❝❡ ♣♦r ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s✳ P♦ré♠✱ ❡ss❛ ✐❞❡✐❛ tr❛③ ❝♦♥s✐❣♦ ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s ❡s♣❡❝í✜❝❛s às ❝❛r❛❝t❡ríst✐❝❛s ❞❛s ❛♣❧✐❝❛çõ❡s ❡♠ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✳ ❆ ♣r✐♠❡✐r❛ ❞❡❧❛s é ♦ ❛ss✐♥❝r♦♥✐s♠♦✱ ❝❛r❛❝t❡ríst✐❝❛ ♣r❡s❡♥t❡ ♥❛ ♠❛✐♦r✐❛ ❞❛s ❡str❛té❣✐❛s ❞❡ ♣❛r❛❧❡❧✐③❛çã♦ ❞❡ss❡ t✐♣♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❡ q✉❡ ❞❡✈❡ ❡st❛r ♣r❡s❡♥t❡ ♥❛ ✐♥t❡r❢❛❝❡ ❬✻✱ ✼✱ ✶✵❪✳ ❆ s❡❣✉♥❞❛ ❞✐③ r❡s♣❡✐t♦ ❛♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ❛ssí♥❝r♦♥❛s✱ ❢✉♥❝✐♦♥❛❧✐❞❛❞❡s ♣r❡✈✐❛♠❡♥t❡ ✐♥❝♦r♣♦r❛❞❛s à ✐♥t❡r❢❛❝❡✱ q✉❡ ♣♦❞❡♠ s❡r r❡q✉❡r✐❞❛s ♣♦r ❛♣❧✐❝❛çõ❡s ❡♠ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛ ♥♦ ♠♦♠❡♥t♦ ❞❡ s✉❛s ❡①❡❝✉çõ❡s✳ ❆ ❜✐❜❧✐♦t❡❝❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ♣♦r tr♦❝❛ ❞❡ ♠❡♥s❛❣❡♥s ▼P■ ❥á ✐♥❝❧✉✐ ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s✳ ❊♥tr❡t❛♥t♦✱ ❛s ♠❡s♠❛s ♣♦ss✉❡♠ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡ s✐♥❝r♦♥✐s♠♦✱ ❡♠ r❛③ã♦ ❞❛ ❢♦rt❡ ♠♦t✐✈❛çã♦ ❞❡ ❛♣❧✐❝❛çõ❡s ♥✉♠ér✐❝❛s✳ ❊♥tr❡ ♦✉tr♦s ❢❛t♦s✱ é ❝♦♠✉♠ ❡♠ ▼P■ q✉❡ ✉♠❛ ❢✉♥çã♦ ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ ♦♣❡r❛çã♦ ❝♦❧❡t✐✈❛ ❡①✐❥❛✱ ♣❛r❛ ♦ s❡✉ ♣❡r❢❡✐t♦ ❢✉♥❝✐♦♥❛♠❡♥t♦✱ ❛ ✐♥✈♦❝❛çã♦ ❡s♣♦♥tâ♥❡❛ ❡ ❡♠ ✉♠❛ ♦r❞❡♠ ♣ré✲❡st❛❜❡❧❡❝✐❞❛ ❞❛ ♠❡s♠❛ ♣♦r ❞✐✈❡rs♦s ♥ós ❞❛ r❡❞❡✳ ❊ss❡ ❢❛t♦ ♣r❡ss✉♣õ❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡ s✐♥❝r♦♥✐s♠♦ ❡♥tr❡ ♦s ♥ós✳ ❆ ❢♦r♠❛ q✉❡ ❡♥❝♦♥tr❛♠♦s ♣❛r❛ ❝♦♠♣❛t✐❜✐❧✐③❛r ♦ ❛ss✐♥❝r♦♥✐s♠♦ ❡ ❛s ❢✉♥çõ❡s ❞❡ ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ❞❛ ✐♥t❡r❢❛❝❡ ❢♦✐ ❛tr❛✈és ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠♦❞❡❧♦s ❞✐str✐❜✉í❞♦s ❛ssí♥❝r♦♥♦s ❡♠ q✉❡ ❛s ❝♦♠♣✉t❛çõ❡s ❧♦❝❛✐s sã♦ ❣✉✐❛❞❛s ♣❡❧❛s ♠❡♥s❛❣❡♥s r❡❝❡❜✐❞❛s✳ ❆ss✐♠✱ é ♣♦ssí✈❡❧ ❡①♣r❡ss❛r ❢♦r♠❛❧♠❡♥t❡ ❛s ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ❞❡ ❢♦r♠❛ t♦t❛❧♠❡♥t❡ ❞✐str✐❜✉í❞❛ ❡ ❛ssí♥❝r♦♥❛ ❛♣❡♥❛s ♣❡❧❛ tr♦❝❛ ❛❞❡q✉❛❞❛ ❞❡ ♠❡♥s❛❣❡♥s✳ ❆s ú♥✐❝❛s ❢✉♥çõ❡s q✉❡ ♣r❡❝✐s❛♠ s❡r ❡①❡❝✉t❛❞❛s ❝♦❧❡t✐✈❛♠❡♥t❡ sã♦ ❛q✉❡❧❛s q✉❡ ❝♦♥✜❣✉r❛♠ ❡ ❞✐s♣❛r❛♠ ♦s ♠♦❞❡❧♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ❛s ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ✜❝❛♠ ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❛♣❧✐❝❛çõ❡s q✉❡ s❡❥❛♠ ❡①❡❝✉t❛❞❛s ❞❡♥tr♦ ❞❡ ❛❧❣✉♠ ❞❡ss❡s ♠♦❞❡❧♦s✳

✶✳✷ ❖s ▼♦❞❡❧♦s ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛

❖s ♠♦❞❡❧♦s ❞❡ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ ❝♦♥s✐st❡♠ ❡♠ ✉♠ ❢♦r♠❛❧✐s♠♦ ♣❛r❛ ❛ ❞❡s❝r✐çã♦ ♣r❡❝✐s❛ ❞❡ ❝♦♠♣✉t❛çõ❡s ❞✐str✐❜✉í❞❛s ❬✶❪✳ ❈♦♠ ❛ ❛❥✉❞❛ ❞❡ ✉♠ t❛❧ ♠♦❞❡❧♦✱ ✐❞❡♥t✐✜❝❛♠♦s ♣❛❞rõ❡s ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝♦❧❡t✐✈♦ ❡①✐st❡♥t❡s ♥❛s ❛♣❧✐❝❛çõ❡s ❛ssí♥❝r♦♥❛s✳ ◆♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸✱ ❢❛③❡♠♦s ✉♠❛ ❞❡s❝r✐çã♦ ❝♦♠♣❧❡t❛ ❞❡ss❡s ♠♦❞❡❧♦s✳ ◆♦s ❛t❡♠♦s ❛q✉✐ s♦♠❡♥t❡ ❛ ✉♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❛s ❝♦♠♣✉t❛çõ❡s q✉❡ ♣♦❞❡♠ s❡r ❡①♣r❡ss❛s ❛ ♣❛rt✐r ❞❡st❡s ❞♦✐s ♠♦❞❡❧♦s✳

P❛r❛ ♦s ♠♦❞❡❧♦s✱ ❢♦r♥❡❝❡♠♦s ❝♦♠♦ ❡♥tr❛❞❛ ✉♠ ❣r❛❢♦ Γ = (N, L)✱ ♥♦ q✉❛❧N ={1,2, . . . , n} ❡

L❝♦rr❡s♣♦♥❞❡♠ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ós ❞❛ r❡❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❛♥❛✐s ❜✐❞✐r❡❝✐♦♥❛✐s

❡♥tr❡ ❡ss❡s ♥ós✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❘❡❢❡r❡✲s❡ ❛ ✉♠ ❝❛♥❛❧ ❛tr❛✈és ❞❛ ♥♦t❛çã♦ij✱ ❡♠ q✉❡i, j∈N sã♦

♦s ♥ós r❡❢❡r❡♥t❡s às ❞✉❛s ❡①tr❡♠✐❞❛❞❡s ❞♦ ❝❛♥❛❧✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ ❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♥ói∈N

❝♦♠♦ ♦ ❝♦♥❥✉♥t♦N(i)❞❡ t♦❞♦s ♦s ♥ós t❛✐s q✉❡ ❡①✐st❡ ✉♠ ❝❛♥❛❧ ❡♥tr❡ ❝❛❞❛ ✉♠ ❞❡❧❡s ❡ ♦ ♥ói✳ ▼❛✐s

♣r❡❝✐s❛♠❡♥t❡✱N(i) ={j∈N |ij ∈L}✳ ❯♠ ♥ó s♦♠❡♥t❡ ♣♦❞❡ ❡♥✈✐❛r ♠❡♥s❛❣❡♥s ♣❛r❛ ✉♠ ♥ó q✉❡ ♣❡rt❡♥❝❡ à s✉❛ ✈✐③✐♥❤❛♥ç❛✳

✶✳✷✳✶ ❈♦♠♣✉t❛çõ❡s ❉✐str✐❜✉í❞❛s ❉✐r✐❣✐❞❛s ♣♦r ❊✈❡♥t♦s

❆ ♥♦çã♦ ♣r✐♠♦r❞✐❛❧ ❞❡ss❡ t✐♣♦ ❞❡ ❝♦♠♣✉t❛çõ❡s é ❛ ❞❡ ❡✈❡♥t♦✱ ❞❡♥♦♠✐♥❛çã♦ ❞❛❞❛ ❛♦ ❛t♦ ❞❡ r❡❝❡❜❡r ✉♠❛ ♠❡♥s❛❣❡♠ ❡✱ ❡♠ r❡s♣♦st❛ ❛ ❡❧❛✱ r❡❛❧✐③❛r ✉♠❛ ❝♦♠♣✉t❛çã♦ ❞❡ ♠❛♥❡✐r❛ ❛tô♠✐❝❛✱ ♣♦ss✐✈❡❧♠❡♥t❡ ❛❧t❡r❛♥❞♦ ♦ ❡st❛❞♦ ❧♦❝❛❧ ❞♦ ♥ó✳

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✶✳✷✳ ❖s ▼♦❞❡❧♦s ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛

✸

♠❡♥s❛❣❡♠ ❡♥✈✐❛❞❛ ♣♦r ✉♠ ❞❡ s❡✉s ✈✐③✐♥❤♦s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♣♦r ✉♠❛ ❝♦♠♣✉t❛çã♦ ❧♦❝❛❧ ❛ss♦❝✐❛❞❛ à ♠❡♥s❛❣❡♠ r❡❝❡❜✐❞❛✱ ❛ q✉❛❧ ❝❛r❛❝t❡r✐③❛ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✉♠ ❡✈❡♥t♦✳

❖ ❆❧❣♦r✐t♠♦ ✶✳✶ é ✉♠❛ ❞❡s❝r✐çã♦ ❣❡♥ér✐❝❛ ❞❡ ✉♠❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ ❞✐r✐❣✐❞❛ ♣♦r ❡✈❡♥t♦s ❡♠ ✉♠ ♥ó ✐❞❡♥t✐✜❝❛❞♦ ♣♦r i ❞❛ r❡❞❡ Γ✳ ◆♦ ❛❧❣♦r✐t♠♦ ❛❜❛✐①♦✱ ♦ r❡❝❡❜✐♠❡♥t♦ ❞❡ ✉♠❛

♠❡♥s❛❣❡♠ msgi ❞❡ ✉♠ ✈✐③✐♥❤♦ ❞❡i ❛❧t❡r❛ ♦ ❡st❛❞♦ ❧♦❝❛❧ ❞♦ ♥ó i ❛tr❛✈és ❞❛ ❡①❡❝✉çã♦ ❞❛ ❢✉♥çã♦

EV EN Ti q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❝♦♠♣✉t❛çã♦ ❧♦❝❛❧ ♣❛rt✐❝✉❧❛r ❛ss♦❝✐❛❞❛ ❛♦ ♥ói✳ ❆❧é♠ ❞❡ ❛❧t❡r❛r

♦ ❡st❛❞♦ ❧♦❝❛❧ ❞♦ ♥ói✱EV EN Ti ❣❡r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡♥s❛❣❡♥sM SGi✱ ♣♦ss✐✈❡❧♠❡♥t❡ ✈❛③✐♦✱ ❡♠

q✉❡ ❝❛❞❛ ♠❡♥s❛❣❡♠ é ❞❡st✐♥❛❞❛ ❛ ✉♠ ✈✐③✐♥❤♦ ❞❡ i✳ ❆ ♣r✐♠❡✐r❛ ❝❤❛♠❛❞❛ ❞❛ ❢✉♥çã♦ EV EN Ti✱

❝♦♥s✐st❡ ❞♦ ❡✈❡♥t♦ ❡s♣♦♥tâ♥❡♦✱ q✉❡ é ❣❡r❛❞♦ s❡♠ ❛ r❡❝❡♣çã♦ ❞❡ ♠❡♥s❛❣❡♠✳ ❖ ❛❧❣♦r✐t♠♦ t❡r♠✐♥❛ q✉❛♥❞♦ ♦ ♥ó t❡♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ✜♥❛❧ ❞❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ ✭❧✐♥❤❛ ✹✮✳

❆❧❣♦r✐t♠♦ ✶✳✶✿ ❈♦♠♣✉t❛çã♦ ❢❡✐t❛ ♥♦ ♥ói♥♦ ♠♦❞❡❧♦ ❞❡ ❝♦♠♣✉t❛çõ❡s ❞✐r✐❣✐❞❛s ♣♦r ❡✈❡♥t♦s

❊♥tr❛❞❛✿ ❈♦♥❥✉♥t♦N(i)❞❡ ✈✐③✐♥❤♦s ❞❡i❡♠Γ✱ ❡st❛❞♦ ❧♦❝❛❧ ✐♥✐❝✐❛❧ ♣❛r❛i

✶✳ ❘❡❣✐str❡ ♦ ❡st❛❞♦ ❧♦❝❛❧ ✐♥✐❝✐❛❧ ❞♦ ♥ói

✷✳ EV EN Ti(−, M SGi)

✸✳ ❊♥✈✐❡ ❝❛❞❛ ♠❡♥s❛❣❡♠ ❡♠M SGi ♣❛r❛ s❡✉ r❡s♣❡❝t✐✈♦ ✈✐③✐♥❤♦

✹✳ ❡♥q✉❛♥t♦ ❛ t❡r♠✐♥❛çã♦ ❣❧♦❜❛❧ ♥ã♦ ❢♦r ❝♦♥❤❡❝✐❞❛ ♣♦ri❢❛ç❛

✺✳ s❡ ✉♠❛ ♠❡♥s❛❣❡♠msgi ❞❡ ✉♠ ✈✐③✐♥❤♦ ❞❡i❢♦r r❡❝❡❜✐❞❛ ❡♥tã♦

✻✳ EV EN Ti(msgi, M SGi)

✼✳ ❊♥✈✐❡ ❝❛❞❛ ♠❡♥s❛❣❡♠ ❡♠M SGi ♣❛r❛ s❡✉ r❡s♣❡❝t✐✈♦ ✈✐③✐♥❤♦

✶✳✷✳✷ ❈♦♠♣✉t❛çõ❡s ❉✐str✐❜✉í❞❛s ❉✐r✐❣✐❞❛s ♣♦r P✉❧s♦s

❊♥q✉❛♥t♦ ♦ ♠♦❞❡❧♦ ❞❡ ❝♦♠♣✉t❛çõ❡s ❞✐r✐❣✐❞❛s ♣♦r ❡✈❡♥t♦s é ❛❞❡q✉❛❞♦ ♣❛r❛ ❞❡s❝r❡✈❡r s✐t✉❛çõ❡s ❡♠ q✉❡ ❛s ♠✉❞❛♥ç❛s ♥♦s ❡st❛❞♦s ❧♦❝❛✐s ❛❝♦♥t❡❝❡♠ ❡①❝❧✉s✐✈❛♠❡♥t❡ ♣♦r r❡❛çã♦ ❛ ✉♠❛ r❡❝❡♣çã♦ ❞❡ ♠❡♥s❛❣❡♠✱ ❤á ❝♦♠♣✉t❛çõ❡s ❞✐str✐❜✉í❞❛s ❡♠ q✉❡ ❡ss❛s ♠✉❞❛♥ç❛s ❞❡ ❡st❛❞♦ ♣♦❞❡♠ ♦❝♦rr❡r ♠❡s♠♦ q✉❛♥❞♦ ♠❡♥s❛❣❡♥s ♥ã♦ sã♦ r❡❝❡❜✐❞❛s✳

◆❡ss❛ ❝❧❛ss❡ ❞❡ ❝♦♠♣✉t❛çõ❡s✱ ❛ ❡✈♦❧✉çã♦ ❞❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛✱ ❞♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ❛té ♦ ❡st❛❞♦ ✜♥❛❧✱ é ❣✉✐❛❞❛ ♣♦r ✉♠ ♠❡❝❛♥✐s♠♦ q✉❡ ❣❡r❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣✉❧s♦s ✭♠❡❝❛♥✐s♠♦ ❞❡ ❣❡r❛çã♦ ❞❡ ♣✉❧s♦s✮✱ ♦ q✉❛❧ ❣♦✈❡r♥❛ ❛ ❡✈♦❧✉çã♦ ❞♦ ❡st❛❞♦ ❞❡ ❝❛❞❛ ♥ói✳ ❚❛❧ ♠❡❝❛♥✐s♠♦ é ✐♠♣❧❡♠❡♥t❛❞♦ ❛tr❛✈és

❞❛s ❢✉♥çõ❡s ❛❜str❛t❛sgetCurrent❡hasAdvanced❞♦ ❆❧❣♦r✐t♠♦ ✶✳✷ ❛ s❡❣✉✐r✳ ❆ ♣r✐♠❡✐r❛ ❢✉♥çã♦

é ✉s❛❞❛ ♣❛r❛ ♥♦t✐✜❝❛r ♦ ♠❡❝❛♥✐s♠♦ ❞❡ ❣❡r❛çã♦ ❞❡ ♣✉❧s♦s q✉❡ ♦ ♥ói❝♦♠❡ç♦✉ s✉❛ ❝♦♠♣✉t❛çã♦ ❧♦❝❛❧

❛ss♦❝✐❛❞❛ ❛♦ ♣✉❧s♦ ❣❡r❛❞♦ ♠❛✐s r❡❝❡♥t❡✳ ❆ s❡❣✉♥❞❛ é ✉♠❛ ❢✉♥çã♦ ❜♦♦❧❡❛♥❛ ✉t✐❧✐③❛❞❛ ♣❡❧♦ ♠❡❝❛♥✐s♠♦ ❞❡ ❣❡r❛çã♦ ❞❡ ♣✉❧s♦s ♣❛r❛ ❛ s✐♥❛❧✐③❛çã♦ ❞❛ ❣❡r❛çã♦ ❞❡ ✉♠ ♥♦✈♦ ♣✉❧s♦✳ ❙❡ hasAdvanced r❡t♦r♥❛

tr✉❡ ❡♥tã♦ ❡❧❛ ♥ã♦ r❡t♦r♥❛rá tr✉❡ ♥♦✈❛♠❡♥t❡ ❛♥t❡s ❞❡getCurrents❡r ❝❤❛♠❛❞❛✳ EV EN Tit❡♠ ❛

❢✉♥çã♦ s♦♠❡♥t❡ ❞❡ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s r❡❧❡✈❛♥t❡s ❞❛ ♠❡♥s❛❣❡♠ r❡❝❡❜✐❞❛✱ ♥ã♦ ❣❡r❛♥❞♦ ♠❡♥s❛❣❡♥s✳ ❆ tr❛♥s✐çã♦ ❞♦ ❡st❛❞♦ ❞♦ ♥ó é ❡①❡❝✉t❛❞❛ ♣❡❧❛ ❢✉♥çã♦P U LSEi✱ ❛ q✉❛❧ r❡❝❡❜❡ ❝♦♠♦ ♣❛rt❡ ❞❡ s✉❛

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✶✳✸✳ ❖♣❡r❛çõ❡s ❈♦❧❡t✐✈❛s

✹

❆❧❣♦r✐t♠♦ ✶✳✷✿ ❈♦♠♣✉t❛çã♦ ❢❡✐t❛ ♥♦ ♥ói♥♦ ♠♦❞❡❧♦ ❞❡ ❝♦♠♣✉t❛çõ❡s ❞✐r✐❣✐❞❛s ♣♦r ♣✉❧s♦s

❊♥tr❛❞❛✿ ❈♦♥❥✉♥t♦N(i)❞❡ ✈✐③✐♥❤♦s ❞❡i❡♠Γ✱ ❡st❛❞♦ ❧♦❝❛❧ ✐♥✐❝✐❛❧ ♣❛r❛i

✶✳ ℓi←getCurrent()

✷✳ P U LSEi(ℓi, M SGi)

✸✳ ❊♥✈✐❡ ❝❛❞❛ ♠❡♥s❛❣❡♠ ❡♠M SGi ♣❛r❛ s❡✉ r❡s♣❡❝t✐✈♦ ✈✐③✐♥❤♦

✹✳ ❡♥q✉❛♥t♦ ❛ t❡r♠✐♥❛çã♦ ❣❧♦❜❛❧ ♥ã♦ ❢♦r ❝♦♥❤❡❝✐❞❛ ♣♦ri❢❛ç❛

✺✳ s❡hasAdvanced()❡♥tã♦

✻✳ ℓi←getCurrent()

✼✳ P U LSEi(ℓi, M SGi)

✽✳ ❊♥✈✐❡ ❝❛❞❛ ♠❡♥s❛❣❡♠ ❡♠M SGi ♣❛r❛ s❡✉ r❡s♣❡❝t✐✈♦ ✈✐③✐♥❤♦

✾✳ s❡♥ã♦ s❡ ✉♠❛ ♠❡♥s❛❣❡♠msgi ❞❡ ✉♠ ✈✐③✐♥❤♦ ❞❡i❢♦r r❡❝❡❜✐❞❛ ❡♥tã♦

✶✵✳ EV EN Ti(msgi)

✶✳✸ ❖♣❡r❛çõ❡s ❈♦❧❡t✐✈❛s

❖♣❡r❛çõ❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❝♦❧❡t✐✈❛s ♣♦r ❡♥✈♦❧✈❡r❡♠ ❞✐✈❡rs❛s ❝♦♠♣✉t❛çõ❡s ❧♦❝❛✐s ❡♠ ♠❛✐s ❞❡ ✉♠ ♥ó ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❡♠ s✉❛ r❡❛❧✐③❛çã♦✳ ❖ ♣♦ssí✈❡❧ ❛ss✐♥❝r♦♥✐s♠♦ ♣r❡s❡♥t❡ ♥❡ss❛s ♦♣❡r❛çõ❡s ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛♦ ❢❛t♦ ❞❡ ❛s ❝♦♠♣✉t❛çõ❡s ❧♦❝❛✐s ❡♥✈♦❧✈✐❞❛s ♥ã♦ ❡st❛r❡♠ r❡str✐t❛s ❛ s❡❣✉✐r ✉♠❛ ú♥✐❝❛ ♦r❞❡♠ ❞❡ ❡①❡❝✉çã♦✳ ❊♠ ♥♦ss♦ ❡st✉❞♦✱ ❡♥✉♠❡r❛♠♦s ❞✐✈❡rs❛s ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ❛ssí♥❝r♦♥❛s ❡✱ ❞❡♥tr❡ ❡❧❛s✱ ❡s❝♦❧❤❡♠♦s ❛❧❣✉♠❛s✱ ❛s q✉❛✐s ❝♦♥s✐❞❡r❛♠♦s ❛❜r❛♥❣❡r ✉♠❛ ❣❛♠❛ ♠❛✐♦r ❞❡ ❛♣❧✐❝❛çõ❡s✳ ❆s ♠❡s♠❛s sã♦ ❞❡✜♥✐❞❛s ❛ s❡❣✉✐r✳

✶✳✸✳✶ ❘❡❣✐str♦ ❞♦s ❊st❛❞♦s ▲♦❝❛✐s ■♥✐❝✐❛✐s

◆♦s r❡❢❡r✐❞♦s ♠♦❞❡❧♦s✱ s✉♣♦♠♦s q✉❡ ♦ ❣r❛❢♦Γq✉❡ ❞❡s❝r❡✈❡ ❛ r❡❞❡ ❡ ♦s ❡st❛❞♦s ✐♥✐❝✐❛✐s ❞♦s ♥ós

sã♦ ❝♦♥❤❡❝✐❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♣❡❧♦ ♥ó0✳ ❆ss✐♠✱ ❞❡✈❡♠♦s ❞✐❢✉♥❞✐r ❡ss❛s ✐♥❢♦r♠❛çõ❡s ♣❛r❛ ♦s ❞❡♠❛✐s

♥ós ❞❛ r❡❞❡✳ ❆ ❞✐❢✉sã♦ ❞❡ss❛s ✐♥❢♦r♠❛çõ❡s ♣❛r❛ t♦❞♦s ♦s ♦✉tr♦s ♥ós ❞❛ r❡❞❡ ❝♦♥st✐t✉✐ ♦ ♣r♦❜❧❡♠❛ ❞♦ r❡❣✐str♦ ❞♦s ❡st❛❞♦s ❧♦❝❛✐s ✐♥✐❝✐❛✐s✳

❯♠❛ ♣r♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ♠❡♥s❛❣❡♠ ❝♦♥s✐st❡ ❡♠ ✉♠ ♥ó i✐♥✐❝✐❛❧♠❡♥t❡ ❡♥✈✐á✲❧❛ ♣❛r❛ t♦❞♦s ♦s

s❡✉s ✈✐③✐♥❤♦s ❡ ❝❛❞❛ ✈✐③✐♥❤♦j✱ r❡❝✉rs✐✈❛♠❡♥t❡✱ ❡♥✈✐á✲❧❛ ❛ s❡✉s ✈✐③✐♥❤♦s✱ ❝♦♠ ❡①❝❡çã♦ ❞❡i✳ ❖❜s❡r✈❡

q✉❡ ✉♠❛ ♣r♦♣❛❣❛çã♦ ❞❡ ✉♠❛ ♠❡♥s❛❣❡♠ ❣❡r❛ ✉♠❛ ár✈♦r❡ ❡♠ q✉❡ ❝❛❞❛ ♥ó é ✜❧❤♦ ❞♦ ♥ó q✉❡ ♣r✐♠❡✐r♦ ❧❤❡ ❡♥✈✐♦✉ ❛ ♠❡♥s❛❣❡♠✳

❯♠❛ ♠❡♥s❛❣❡♠ ❞❡ r❡❛❧✐♠❡♥t❛çã♦ ❞❡j ♣❛r❛ié ✉♠❛ ♠❡♥s❛❣❡♠ ❡♥✈✐❛❞❛ ❝♦♠♦ r❡s♣♦st❛ ❛ ✉♠❛

♠❡♥s❛❣❡♠ ❛♥t❡r✐♦r q✉❡ ❢♦✐ ❡♥✈✐❛❞❛ ❞❡i♣❛r❛j✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❞❡s❡♥✈♦❧✈❡♠♦s✱ ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸✱ ❛❧❣♦r✐t♠♦s q✉❡ ✉t✐❧✐③❛♠ ✉♠ ♠❡❝❛♥✐s♠♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❝♦♠ r❡❛❧✐♠❡♥t❛çã♦ ♥♦s ❞♦✐s ♠♦❞❡❧♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦ r❡❣✐str♦ ❞♦s ❡st❛❞♦s ❧♦❝❛✐s ✐♥✐❝✐❛✐s✳

✶✳✸✳✷ ❖r❞❡♥❛çã♦ ❞❡ ▼❡♥s❛❣❡♥s

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✶✳✹✳ ❆♣❧✐❝❛çõ❡s

✺

❞❡ ♣r♦t♦❝♦❧♦s ❞❡ ♦r❞❡♥❛çã♦ ❞❡ ♠❡♥s❛❣❡♥s ♣❛r❛ ♦ ❝♦♥tr♦❧❡ ❞♦ ♥ã♦✲❞❡t❡r♠✐♥✐s♠♦ r❡✈❡❧❛ ❣❛♥❤♦ s✐❣♥✐✜❝❛t✐✈♦ ❞❡ ❞❡s❡♠♣❡♥❤♦ ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❬✶✻❪✳

❉❡♥tr❡ ♦s ♠❡❝❛♥✐s♠♦s ❞❡ ♦r❞❡♥❛çã♦ ❞❡ ♠❡♥s❛❣❡♥s ♠❛✐s ✉t✐❧✐③❛❞♦s✱ ❡s❝♦❧❤❡♠♦s ✐♥❝♦r♣♦r❛♠♦s à ♥♦ss❛ ✐♥t❡r❢❛❝❡ ❛s ♦r❞❡♥s ❋■❋❖ ✭❞♦ ✐♥❣❧ês ❋✐rst ■♥✱ ❋✐rst ❖✉t✮ ❡ ❈❛✉s❛❧✳ ❇✉s❝❛♠♦s ♥❛ ❧✐t❡r❛t✉r❛ ❛❧❣♦r✐t♠♦s ♣❛r❛ ❛ ❡ss❛s ♦r❞❡♥s ❬✹✱ ✾❪✳ ◆❛ ✐♥t❡r❢❛❝❡✱ ♦ ✉s✉ár✐♦✱ ♥♦ ♠♦♠❡♥t♦ ❞❛ ❡①❡❝✉çã♦ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ✉t✐❧✐③❛ ✉♠ ❞♦s ♠♦❞❡❧♦s✱ ♣♦❞❡ ❡s❝♦❧❤❡r✱ ❛tr❛✈és ❞❛ ❛tr✐❜✉✐çã♦ ❞❡ ✉♠ ♣❛râ♠❡tr♦✱ s❡ ✉t✐❧✐③❛rá ❛❧❣✉♠ ♠❡❝❛♥✐s♠♦ ❞❡ ♦r❞❡♥❛çã♦ ❡✱ ❝❛s♦ s✐♠✱ q✉❛❧ ♦r❞❡♠ ✉t✐❧✐③❛rá✳ ❆ ❞❡s❝r✐çã♦ ❞❡ss❛s ♦r❞❡♥s✱ ❜❡♠ ❝♦♠♦ ♦s ❛❧❣♦r✐t♠♦s ❛❞♦t❛❞♦s s❡rã♦ ❞❡t❛❧❤❛❞♦s ♣♦st❡r✐♦r♠❡♥t❡✳

✶✳✸✳✸ ❉❡t❡❝çã♦ ❞❛ ❚❡r♠✐♥❛çã♦ ●❧♦❜❛❧

❖ ❡st❛❞♦ ❣❧♦❜❛❧ ❡♠ ✉♠❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ é ❞❡✜♥✐❞♦ ♣❡❧❛ ✉♥✐ã♦ ❞♦s ❡st❛❞♦s ❧♦❝❛✐s ❞♦s ♥ós ♠❛✐s ♦s ❡st❛❞♦s ❞♦s ❝❛♥❛✐s ❞❡ ❝♦♠✉♥✐❝❛çã♦✳ ◆♦s ♠♦❞❡❧♦s s✉♣♦♠♦s q✉❡ ♥❡♥❤✉♠ ♥ó ♣♦ss✉✐ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♠♣❧❡t♦ ❞♦ ❡st❛❞♦ ❣❧♦❜❛❧ ❡ ♥ã♦ s✉♣♦♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ r❡❧ó❣✐♦ ❣❧♦❜❛❧✳ ▲♦❣♦✱ ❞❡t❡r♠✐♥❛r s❡ ❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ t❡r♠✐♥♦✉ s❡ t♦r♥❛ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ r❡q✉❡r ✉♠ tr❛t❛♠❡♥t♦ ❝✉✐❞❛❞♦s♦ ❬✶✸❪✳ ❉❡ss❛ ❢♦r♠❛✱ ♣r♦♣♦♠♦s t❛♠❜é♠ ♥❡st❡ tr❛❜❛❧❤♦ ❛❧❣♦r✐t♠♦s ❜❛s❡❛❞♦s ♥♦ ❛❧❣♦r✐t♠♦ ❉✐❥❦str❛✲❙❝❤♦❧t❡♥ ❬✶✼❪ ♥♦s ❞♦✐s ♠♦❞❡❧♦s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞❡t❡❝çã♦ ❞❛ t❡r♠✐♥❛çã♦ ❣❧♦❜❛❧✱ ❞❡t❛❧❤❛❞♦s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸✳ ◆❛ ✐♥t❡r❢❛❝❡✱ ♣❡r♠✐t✐♠♦s q✉❡ ♦ ✉s✉ár✐♦ ❡s❝♦❧❤❛✱ ❛tr❛✈és ❞❡ ✉♠ ♣❛râ♠❡tr♦ ♣❛ss❛❞♦ ❛♦ ♠♦❞❡❧♦✱ s❡ ♦ ♠♦❞❡❧♦ ❞❡✈❡rá ❞❡t❡❝t❛r ❛ t❡r♠✐♥❛çã♦ ❣❧♦❜❛❧ ♦✉ s❡ ❛ ♣ró♣r✐❛ ❛♣❧✐❝❛çã♦ ✐♥❞✐❝❛rá ❛♦ ♠♦❞❡❧♦ q✉❡ ❛ t❡r♠✐♥❛çã♦ ❢♦✐ ❞❡t❡❝t❛❞❛✳

✶✳✹ ❆♣❧✐❝❛çõ❡s

❯♠ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠❜✐♥❛tór✐❛✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❣❡♥ér✐❝❛✱ t❡♠ ❝♦♠♦ ❡♥tr❛❞❛ ✉♠❛ ❞❡s❝r✐çã♦ ❝♦♠♣❛❝t❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧S❞❡ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❡ ✉♠❛ ❢✉♥çã♦f q✉❡ ❛tr✐❜✉✐

✉♠ ✈❛❧♦r f(S) ❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ S ❞❡ S✳ ❖ ♣r♦❜❧❡♠❛ ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r S ∈ S q✉❡ s❡❥❛ ✏♠❡❧❤♦r✑ q✉❡ t♦❞♦s ♦s ❞❡♠❛✐s✳ ❊ss❛ ♥♦çã♦ ❞❡ ✏♠❡❧❤♦r✑ ♣♦❞❡ s❡r ❞❡ ❞♦✐s t✐♣♦s✳ ❊♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛çã♦✱ ❞❡s❡❥❛✲s❡ ❡♥❝♦♥tr❛rS ∈ S t❛❧ q✉❡∀S′ _{∈ S ⇒}_f₍_S₎_≤_f₍_S′₎_{✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉♠}

♣r♦❜❧❡♠❛ ❞❡ ♠❛①✐♠✐③❛çã♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ ❛ ♣r♦❝✉r❛ ❞❡S ∈ S t❛❧ q✉❡∀S′ _{∈ S ⇒}_f₍_S₎_≥_f₍_S′₎_✳

❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❛ s♦❧✉çã♦S é ✉♠❛ s♦❧✉çã♦ ót✐♠❛✳

❆ tít✉❧♦ ❞❡ ✐❧✉str❛çã♦✱ ❝♦♥s✐❞❡r❡ G = (V, E) ✉♠ ❣r❛❢♦✳ ❯♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡

S é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ V ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ♠✉t✉❛♠❡♥t❡ ♥ã♦ ❛❞❥❛❝❡♥t❡s ❡♠ G✳ ❯♠

♣r♦❜❧❡♠❛ ❞❡ ♠❛①✐♠✐③❛çã♦ ❛ss♦❝✐❛❞♦ ❛ ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❞❡♥♦♠✐♥❛❞♦ ♣r♦❜❧❡♠❛ ❞♦ ❈♦♥❥✉♥t♦ ■♥❞❡♣❡♥❞❡♥t❡ ▼á①✐♠♦ ✭❈■▼✮✱ é ❞❡✜♥✐❞♦ t♦♠❛♥❞♦ S ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s

✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ G ❡ ❛ ❞❡✜♥✐♥❞♦ ❛ ❢✉♥çã♦ f(S) = |S|✱ ♣❛r❛ t♦❞♦ S ∈ S✳ ❆ ❢♦r♠❛ ❝♦♠♣❛❝t❛ ✉s✉❛❧♠❡♥t❡ ❛❞♦t❛❞❛ ♣❛r❛ ❞❡s❝r❡✈❡rS ♥❡st❡ ❝❛s♦ é ♦ ♣ró♣r✐♦ ❣r❛❢♦G✳ ❆ ❞✐✜❝✉❧❞❛❞❡ ❞❡ r❡s♦❧✉çã♦

❞❡ss❡ ♣r♦❜❧❡♠❛ ♣❛r❛ ❣r❛❢♦s ❛r❜✐trár✐♦s r❡s✐❞❡ ♥♦ ❢❛t♦ ❞❡ q✉❡|S|é✱ ♣❛r❛ ♠✉✐t♦s ❣r❛❢♦s✱ ❡①♣♦♥❡♥❝✐❛❧

❡♠ |V|✱ ✐♥✈✐❛❜✐❧✐③❛♥❞♦ ❛ ❡♥✉♠❡r❛çã♦ ❡①❛✉st✐✈❛ ❞❡ S ❬✷✶✱ ✷✷❪✳ ❯t✐❧✐③❛♠♦s ❛ ♥♦t❛çã♦ α(G) ♣❛r❛

r❡♣r❡s❡♥t❛r ♦ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♠á①✐♠♦ ♣❛r❛G✳

❖ Pr♦❜❧❡♠❛ ❈■▼ é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ❞✐❢í❝✐❧✱ ♣♦✐s ♣❡rt❡♥❝❡ à ❝❧❛ss❡ ◆P✲❉✐❢í❝✐❧ ❬✽❪✱ ❢❛t♦ ❡ss❡ q✉❡ ❡stá ❧✐❣❛❞♦ à ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ ✉♠ ❣r❛❢♦ ✭❡ss❡ ♥ú♠❡r♦ ♣♦❞❡ ❝❤❡❣❛r ❛3n3✮ ❬✽❪✳

❆ ❞✐✜❝✉❧❞❛❞❡ ❞❡ r❡s♦❧✉çã♦ ❡①✐❜✐❞❛ ♣❡❧♦ ❈■▼✱ ❧✐❣❛❞❛ ❛♦ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❡♠ S✱ é

(14)

✶✳✺✳ ❖r❣❛♥✐③❛çã♦ ❞♦ ❉♦❝✉♠❡♥t♦

✻

♠ét♦❞♦s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠❜✐♥❛tór✐❛ ❞✐❢í❝❡✐s✱ ❛ q✉❛❧ ❡stá ❧✐❣❛❞❛ ❛♦ ❢❛t♦ ❞❡ ❞✐✈❡rs❛s s♦❧✉çõ❡s s❡r❡♠ ❡♥✉♠❡r❛❞❛s ✭♠❡s♠♦ q✉❡ ❡♠ ♥ú♠❡r♦ r❡❞✉③✐❞♦ ❝♦♠ r❡❧❛çã♦ ❛|S|✱ ❛

q✉❛♥t✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❡♥✉♠❡r❛❞❛s t❡♥❞❡ ❛ s❡r ❣r❛♥❞❡✮✳ ❯♠❛ ✈❡③ q✉❡ ✈ár✐♦s ♥ós ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❡stã♦ ❞✐s♣♦♥í✈❡✐s✱ ♣♦❞❡✲s❡ ❛❝❡❧❡r❛r ❛ ❡♥✉♠❡r❛çã♦ ✈✐s✐t❛♥t♦ ❞✐✈❡rs❛s s♦❧✉çõ❡s ❡♠ ♣❛r❛❧❡❧♦✳ ❙❡❣✉✐♥❞♦ ❡ss❡ ♣r✐♥❝í♣✐♦✱ ❛ ❡str❛té❣✐❛ ♥❛t✉r❛❧ ❞❡ ♣❛r❛❧❡❧✐③❛çã♦ ❝♦♥s✐st❡ ❡♠ r❡♣❛rt✐r ❛s s♦❧✉çõ❡s ❛ ❡♥✉♠❡r❛r ❡♥tr❡ ♦s ♥ós ❞❡ ♣r♦❝❡ss❛♠❡♥t♦✳ ❈❛s♦ ❡ss❛ r❡♣❛rt✐çã♦ ♣✉❞❡ss❡ s❡r ❢❡✐t❛ ❞❡ ❢♦r♠❛ ❡q✉✐tá✈❡❧ ❛ ♣r✐♦r✐✱ ♦s ♥ós ♣♦❞❡r✐❛♠ tr❛❜❛❧❤❛r ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ✭❡ ❞❡ ❢♦r♠❛ t♦t❛❧♠❡♥t❡ ❛ssí♥❝r♦♥❛✮ ❡♠ s✉❛s r❡s♣❡❝t✐✈❛s ❡♥✉♠❡r❛çõ❡s✳ ❆ ❞✐✜❝✉❧❞❛❞❡ r❡s✐❞❡ ♥♦ ❢❛t♦ ❞❡ ♥ã♦ s❡ ❝♦♥❤❡❝❡r✱ ❞❡ ❛♥t❡♠ã♦✱ ❛s s♦❧✉çõ❡s ❛ s❡r❡♠ ❡♥✉♠❡r❛❞❛s✱ ✈✐st♦ q✉❡ ❡❧❛s sã♦ ❡s❝♦❧❤✐❞❛s ❞✉r❛♥t❡ ❛ ♣ró♣r✐❛ ❡♥✉♠❡r❛çã♦✱ s❡❣✉♥❞♦ ❛ ❤❡✉ríst✐❝❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❣✉✐á✲❧❛ ❬✸✱✺✕✼✱✶✵❪✳

❆♣❡s❛r ❞❡ ♦ ❛ss✐♥❝r♦♥✐s♠♦ s❡r ✉♠ ❡❧❡♠❡♥t♦ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ❛❧❣♦r✐t♠♦s ♣❛r❛❧❡❧♦s ❡✜❝✐❡♥t❡s ❜❛s❡❛❞♦s ❡♠ ♠ét♦❞♦s ❡♥✉♠❡r❛t✐✈♦s ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠❜✐♥❛tór✐❛✱ ✐♥t❡r❛çõ❡s ❡♥tr❡ ❞✐❢❡r❡♥t❡s ♥ós ❞❡✈❡♠ ♦❝♦rr❡r ❞✉r❛♥t❡ ✉♠❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ss❛s ✐♥t❡r❛çõ❡s é ♠❛♥t❡r ✉♠ ❜♦♠ ❜❛❧❛♥❝❡❛♠❡♥t♦ ❞❛s ❡♥✉♠❡r❛çõ❡s r❡❛❧✐③❛❞❛s ♣♦r ❞✐❢❡r❡♥t❡s ♥ós ❛ ✜♠ ❞❡ ♠✐♥✐♠✐③❛r ♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ❡♥✉♠❡r❛❞❛s ♥❛ ✈❡rsã♦ ♣❛r❛❧❡❧❛ q✉❡ ♥ã♦ ♦ sã♦ ♥❛ ✈❡rsã♦ s❡q✉❡♥❝✐❛❧✳ ❖s ♠♦❞❡❧♦s ❜❛s❡❛❞♦s ❡♠ ❡✈❡♥t♦s ♦✉ ♣✉❧s♦s sã♦ ❛❞❡q✉❛❞♦s ❛ ❡ss❡ ❝❡♥ár✐♦✳

❖ ❜r❛♥❝❤✲❛♥❞✲❜♦✉♥❞ é ✉♠ ♠ét♦❞♦ ❣❡♥ér✐❝♦ ♣❛r❛ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ót✐♠❛s ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✳ ❈♦♥s✐st❡ ❞❡ ✉♠❛ ❡♥✉♠❡r❛çã♦ s✐st❡♠át✐❝❛ ❞❡ t♦❞❛s ❛s s♦❧✉çõ❡s ❝❛♥❞✐❞❛t❛s✱ ♦♥❞❡ ❣r❛♥❞❡s s✉❜❝♦♥❥✉♥t♦s ❞❡ s♦❧✉çõ❡s ❝❛♥❞✐❞❛t❛s ✐♥❢r✉tí❢❡r❛s sã♦ ❞❡s❝❛rt❛❞♦s ❡♠ ♠❛ss❛✱ ✉t✐❧✐③❛♥❞♦ ❧✐♠✐t❡s s✉♣❡r✐♦r❡s ❡ ✐♥❢❡r✐♦r❡s ❡st✐♠❛❞♦s ♣❛r❛ ❛ q✉❛♥t✐❞❛❞❡ ❛ s❡r ♦t✐♠✐③❛❞❛ ❬✶✺❪✳ ❖ ❜r❛♥❝❤✲❛♥❞✲❜♦✉♥❞ ♣❛r❛ ♦ ❈■▼ é ❞❡s❝r✐t♦ ❛ s❡❣✉✐r✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❞❡t❡r♠✐♥❛✲s❡ ✉♠❛ ❡st✐♠❛t✐✈❛✱ ♥❛ ❢♦r♠❛ ❞❡ ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r✱ ♣❛r❛α(G)✳ ❊♠ s❡❣✉✐❞❛✱ ❡s❝♦❧❤❡✲s❡ ✉♠ ✈ért✐❝❡v∈V ❡ ❞✐✈✐❞❡✲s❡ ❛ ❡♥✉♠❡r❛çã♦ ❡♠ ❞✉❛s ♣❛rt❡s✿ ❡♠ ✉♠❛✱ sã♦ ❝♦♥s✐❞❡r❛❞♦s ❛♣❡♥❛s ♦s ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s q✉❡ ❝♦♥tê♠ v❀ ♥❛ ♦✉tr❛✱ s♦♠❡♥t❡ ❛q✉❡❧❡s q✉❡ ♥ã♦ ❝♦♥tê♠ v✳ ❈❛❞❛ ✉♠❛ ❞❡ss❛s ♣❛rt❡s r❡❝❡❜❡

♦ ♥♦♠❡ ❞❡ s✉❜♣r♦❜❧❡♠❛✳ ❆ ❡♥✉♠❡r❛çã♦ ♣r♦ss❡❣✉❡ ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❞♦s s✉❜♣r♦❜❧❡♠❛s ✭❛q✉❡❧❡ ❞♦s ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♥t❡♥❞♦ v✮✱ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ❛ ❡♥✉♠❡r❛çã♦ ❞♦s ❝♦♥❥✉♥t♦s

✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ s✉❜❣r❛❢♦ ❞❡G❞❡✜♥✐❞♦ ♣♦r t♦❞♦s ♦s ✈ért✐❝❡s q✉❡ ♥ã♦ sã♦ ❛❞❥❛❝❡♥t❡s ❛v✳ ❉❡ss❛

❢♦r♠❛✱ ❛♣❧✐❝❛✲s❡ r❡❝✉rs✐✈✐❞❛❞❡ ✭❝❛❧❝✉❧❛♥❞♦ ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦ s✉❜♣r♦❜❧❡♠❛ ❡ ❞✐✈✐❞✐♥❞♦ ❛ s✉❛ ❡♥✉♠❡r❛çã♦✮ ❛té q✉❡ s❡ ❞❡t❡r♠✐♥❡ ♦ ♠❛✐♦r ❞♦s ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♥t❡♥❞♦v✳ P❛r❛ tr❛t❛r

❞♦s ❝♦♥❥✉♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s q✉❡ ♥ã♦ ❝♦♥tê♠ v✱ ❝♦♠♣❛r❛✲s❡ ♦ ♠❡❧❤♦r ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❥á

♦❜t✐❞♦ ❝♦♠ ✉♠❛ ❡st✐♠❛t✐✈❛ ❞♦ ♠❛✐♦r q✉❡ s❡ ♣♦❞❡ ♦❜t❡r s❡♠ ✐♥❝❧✉✐rv✳ ❈❛s♦ ❛ ❡st✐♠❛t✐✈❛ ✐♥❞✐q✉❡

❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♠❡❧❤♦r✱ ♣r♦❝❡❞❡✲s❡ à ❡♥✉♠❡r❛çã♦ ❞❛ s❡❣✉♥❞❛ ♣❛rt❡✱ t❛♠❜é♠ r❡❝✉rs✐✈❛♠❡♥t❡✳ ❖s ❞❡t❛❧❤❡s ❞❡ss❡ ❛❧❣♦r✐t♠♦ ✭✐♥❝❧✉✐♥❞♦ ♦ ❝á❧❝✉❧♦ ❞❛ ❡st✐♠❛t✐✈❛ ❞❡ ❝❛❞❛ s✉❜♣r♦❜❧❡♠❛ ❡ ♦ ❝r✐tér✐♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡s❝♦❧❤❡r ♦ ✈ért✐❝❡ ✉s❛❞♦ ♣❛r❛ ❣❡r❛r s✉❜♣r♦❜❧❡♠❛s✮ ❡stã♦ ❡♠ ❬✷✸❪✱ ❡ ❛❧❣✉♥s ❞❡❧❡s sã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✺✳

✶✳✺ ❖r❣❛♥✐③❛çã♦ ❞♦ ❉♦❝✉♠❡♥t♦

(15)

✶✳✺✳ ❖r❣❛♥✐③❛çã♦ ❞♦ ❉♦❝✉♠❡♥t♦

✼

(16)

❈❛♣ít✉❧♦ ✷

❖ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦

❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r ❊✈❡♥t♦s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡s❝r❡✈❡♠♦s ♦ ▼♦❞❡❧♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❉✐str✐❜✉í❞❛ ❉✐r✐❣✐❞❛ ♣♦r ❊✈❡♥t♦s✳ ❊st❡ ♠♦❞❡❧♦ ❛♣r❡s❡♥t❛ ✉♠ ❢♦r♠❛❧✐s♠♦ ♣❛r❛ ❛ ♣❡r❢❡✐t❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❝♦♠♣✉t❛çõ❡s ❞❡s❝r✐t❛s ♣❡❧♦ ❆❧❣♦r✐t♠♦ ✶✳✶ ❞♦ ❈❛♣ít✉❧♦ ✶✳ ❆tr❛✈és ❞❡st❡ ❢♦r♠❛❧✐s♠♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♣r❡❝✐s❛♠❡♥t❡ ♦ s✐❣♥✐✜❝❛❞♦✱ ❞❡♥tr♦ ❞❡st❡ ♠♦❞❡❧♦✱ ❞❛s ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ♠❡♥❝✐♦♥❛❞❛s ♥❛ ❙❡çã♦ ✶✳✸✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ♦s ❛❧❣♦r✐t♠♦s q✉❡ r❡❛❧✐③❛♠ t❛✐s ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s✳

✷✳✶ ❈♦♠♣✉t❛çõ❡s ❉✐str✐❜✉í❞❛s ❝♦♠♦ ❈♦♥❥✉♥t♦ ❞❡ ❊✈❡♥t♦s

❆ ♣❛rt✐r ❞♦ ❆❧❣♦r✐t♠♦ ✶✳✶ ❞♦ ❈❛♣ít✉❧♦ ✶✱ ♣♦❞❡♠♦s ❝❛r❛❝t❡r✐③❛r ✉♠ ❡✈❡♥t♦ξ♣♦r ✉♠❛6✲t✉♣❧❛

❞❡✜♥✐❞❛ ❝♦♠♦✿

ξ ❂hi, r, msg, σ, σ′_{, M SG}_i

❡♠ q✉❡✿

◮

_i é ♦ ♥ó ♦♥❞❡ ♦ ❡✈❡♥t♦ ♦❝♦rr❡❀

◮

_r é ♦ t❡♠♣♦ ♥♦ r❡❧ó❣✐♦ ❧♦❝❛❧ ❞❡_i❡♠ q✉❡ ♦ ❡✈❡♥t♦ ♦❝♦rr❡✉❀

◮

_msg é ❛ ♠❡♥s❛❣❡♠ q✉❡ ❞✐s♣❛r❛ ♦ ❡✈❡♥t♦✱ s❡ ❡①✐st❡ ❛❧❣✉♠❛❀

◮

_σ é ♦ ❡st❛❞♦ ❞♦ ♥ó_i❛♥t❡s ❞❛ ♦❝♦rrê♥❝✐❛ ❞♦ ❡✈❡♥t♦❀

◮

_σ✬ é ♦ ❡st❛❞♦ ❞♦ ♥ó_i❛♣ós ❞❛ ♦❝♦rrê♥❝✐❛ ❞♦ ❡✈❡♥t♦❀

◮

_{M SG} é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡♥s❛❣❡♥s ❣❡r❛❞❛s ❝♦♠♦ r❡s✉❧t❛❞♦ ❞❛ ♦❝♦rrê♥❝✐❛ ❞♦ ❡✈❡♥t♦✳ ❊st❡

❝♦♥❥✉♥t♦ ♣♦❞❡ s❡r ✈❛③✐♦✳

❊ss❛ ❞❡✜♥✐çã♦ ❞❡ ❡✈❡♥t♦ é ❜❛s❡❛❞❛ ♥❛ ♣r❡♠✐ss❛ ❞❡ q✉❡ ❝❛❞❛ ♥ói∈N♦♣❡r❛ ❝♦♠♦ ✉♠❛ ♠áq✉✐♥❛

❞❡ ❡st❛❞♦s ❝✉❥❛s tr❛♥s✐çõ❡s sã♦ ♦s ❡✈❡♥t♦s ♦❝♦rr❡♥❞♦ ❡♠ i ❞✉r❛♥t❡ ❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛✳

❯♠❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ ♣♦❞❡ s❡r ✈✐st❛ s✐♠♣❧❡s♠❡♥t❡ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❝♦❧❡çã♦ ❞❡ ❡✈❡♥t♦s

(17)

✷✳✷✳ ❯♠ ❆❧❣♦r✐t♠♦ ●❡♥ér✐❝♦

✾

❆s ❞❡♣❡♥❞ê♥❝✐❛s ❝❛✉s❛✐s ❞♦s ❡✈❡♥t♦s ❡♠ ✉♠❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ sã♦ ❞❡s❝r✐t❛s ❢♦r♠❛❧♠❡♥t❡ ❛tr❛✈és ❞❛s ♦r❞❡♥s ♣❛r❝✐❛✐s→❡ s♦❜r❡ Ξq✉❡ ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r✳ ❙❡❥❛ξ✉♠ ❡✈❡♥t♦✳ ❆s ❢✉♥çõ❡s node(ξ)❡timei(ξ)✱ i∈N✱ r❡t♦r♥❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ♥ó ❡♠ q✉❡ ξ ♦❝♦rr❡ ❡ ✉♠ ✈❛❧♦rri s❡ξ é

♦ri✲és✐♠♦ ❡✈❡♥t♦ q✉❡ ♦❝♦rr❡ ♥♦ ♥ói✳ ▼❛✐s ❛✐♥❞❛✱msgi(ξ)r❡♣r❡s❡♥t❛ ❛ ♠❡♥s❛❣❡♠ r❡❝❡❜✐❞❛ ♥♦ ♥ó

i❡ q✉❡ ❞✐s♣❛r❛ ♦ ❡✈❡♥t♦ ξ′_{✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱} _{M SG}

i(ξ)r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♠❡♥s❛❣❡♥s

r❡s✉❧t❛♥t❡s ❞❛ ♦❝♦rrê♥❝✐❛ ❞♦ ❡✈❡♥t♦ξ✳

❆ r❡❧❛çã♦→é t❛❧ q✉❡ξ→ξ′_{✱ ♣❛r❛} _{ξ, ξ}′ _∈_Ξ_{✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✈❛❧❡ ✉♠❛ ❞❛s ❞✉❛s ❝♦♥❞✐çõ❡s ❛}

s❡❣✉✐r✿

◮

_i₌_node₍_ξ_{) =}_node₍_ξ′_{) =}_j ❡_time_i₍_ξ_{) =}_time_i₍_ξ′₎₋₁❀

◮

_i₌_node₍_ξ₎₆₌_node₍_ξ′_{) =}_j ❡_msg_j₍_ξ′₎_∈_{M SG}_i₍_ξ₎✳

❊ss❛s ❝♦♥❞✐çõ❡s ✐♥❝❧✉❡♠ ♥❛ r❡❧❛çã♦→ ♣❛r❡s ❞❡ ❡✈❡♥t♦s q✉❡ sã♦ ❝♦♥s❡❝✉t✐✈♦s ❡♠ ✉♠ ♠❡s♠♦

♥ó ♦✉ sã♦ ❛ ♦r✐❣❡♠ ❡ ♦ ❞❡st✐♥♦ ❞❡ ✉♠❛ ♠❡s♠❛ ♠❡♥s❛❣❡♠✳ ❖❜s❡r✈❡ q✉❡ ❡ss❛s sã♦ ❛s s✐t✉❛çõ❡s q✉❡ r❡✢❡t❡♠ ❛ ✐♥t✉✐çã♦ ♥❛t✉r❛❧ s♦❜r❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝❛✉s❛❧ ❞✐r❡t❛ ❡♥tr❡ ❡✈❡♥t♦s✳

❆ r❡❧❛çã♦ r❡✢❡t❡ ❛ s✐t✉❛çã♦ ❡♠ q✉❡ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ é ❞❡s❡♥❝❛❞❡❛❞❛ ♣❡❧❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡✈❡♥t♦s r❡❧❛❝✐♦♥❛❞♦s ♣♦r →✳ P♦st♦ ❞❡ ♦✉tr❛ ❢♦r♠❛✱ é ♦ ❢❡❝❤♦ tr❛♥s✐t✐✈♦

❞❡ →✳ P♦rt❛♥t♦✱ s❡ ξ1, . . . , ξt ∈ Ξ✱ ♣❛r❛ ❛❧❣✉♠ t ≥ 1✱ s❡ ξ1 ξt✱ ❡♥tã♦ ξs → ξs+1✱ ♣❛r❛ ❝❛❞❛

s∈1, ..., t−1✳

◆❛ ✜❣✉r❛ ❛❞✐❛♥t❡✱ ✐❧✉str❛♠♦s ✉♠❛ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ ❞✐r✐❣✐❞❛ ♣♦r ❡✈❡♥t♦s✳ ❆s ❧✐♥❤❛s ❤♦r✐③♦♥t❛✐s r❡♣r❡s❡♥t❛♠ ❛s ❧✐♥❤❛s ❞❛ ❡✈♦❧✉çã♦ ❞♦ t❡♠♣♦ ❞❡ três ♥ósi, j, k∈N✳ ❯♠ ❝ír❝✉❧♦ ♣r❡t♦

✐♥❞✐❝❛ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ ♥♦ r❡s♣❡❝t✐✈♦ ♥ó✳ ❯♠❛ s❡t❛ ❡♥tr❡ ❞♦✐s ❡✈❡♥t♦s ✐♥❞✐❝❛ q✉❡ ❡❧❡s s❡ r❡❧❛❝✐♦♥❛♠ ❛tr❛✈és ❞❡→✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ♣❛r ❞❛ r❡❧❛çã♦ é r❡♣r❡s❡♥t❛❞❛ tr❛✈és ❞❡ ✉♠❛ s❡t❛

♦♥❞✉❧❛❞❛✳

j

k

i

ξ

4

ξ

1

ξ

2

ξ

3

ξ

5

❋✐❣✉r❛ ✷✳✶✿ ❊①❡♠♣❧♦ ❞❡ ❡✈❡♥t♦s ♥♦ ♠♦❞❡❧♦ ❞❡ ❝♦♠♣✉t❛çã♦ ❞✐str✐❜✉í❞❛ ❞✐r✐❣✐❞❛ ♣♦r ❡✈❡♥t♦s

✷✳✷ ❯♠ ❆❧❣♦r✐t♠♦ ●❡♥ér✐❝♦

◆❡st❛ s❡çã♦✱ ❞❡s❝r❡✈❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ❣❡♥ér✐❝♦✱ ❞♦t❛❞♦ ❞❛s ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ❞❡s❝r✐t❛s ♥♦ ❈❛♣ít✉❧♦ ✶✱ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❝♦♠♣✉t❛çõ❡s ❞✐r✐❣✐❞❛s ♣♦r ❡✈❡♥t♦s✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ♦s ❝♦♥tr♦❧❡s r❡❧❛t✐✈♦s às ♦r❞❡♥❛çõ❡s ❞❡ ♠❡♥s❛❣❡♥s ♥ã♦ ❡stã♦ ♣r❡s❡♥t❡s ♥❡ss❡ ❛❧❣♦r✐t♠♦✳

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✷✳✷✳ ❯♠ ❆❧❣♦r✐t♠♦ ●❡♥ér✐❝♦

✶✵

❝♦❧❡t✐✈❛s✳ ❆ ❛♣r❡s❡♥t❛çã♦ ❢❡✐t❛ ♥❡st❡ ❝❛♣ít✉❧♦ ♠♦str❛ ❡♠ ❞❡t❛❧❤❡s ❛♣❡♥❛s ♠❡♥s❛❣❡♥s ❞❛s ♦♣❡r❛çõ❡s ❝♦❧❡t✐✈❛s ❡ ♦s r❡s♣❡❝t✐✈♦s ❡✈❡♥t♦s✳

◆❛ ❧✐♥❤❛ ✶ é ❢❡✐t❛ ✐♥✐❝✐❛❧✐③❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ❞❡ ❝♦♥tr♦❧❡ ❧♦❝❛✐s ❞♦ ❛❧❣♦r✐t♠♦✳ ❙♦♠❡♥t❡ ♦ ♥ó ✵ r❡❝❡❜❡ ❝♦♠♦ ❡♥tr❛❞❛ ♦ ❣r❛❢♦Γq✉❡ ❞❡s❝r❡✈❡ ❛ r❡❞❡ ❡ ♦s ❡st❛❞♦s ❧♦❝❛✐s ✐♥✐❝✐❛✐s ❞♦s ♥ós✳ ◆❛ ❧✐♥❤❛s ✷ ❡

✸✱ ♦ ♥ó ✵ ✐♥✐❝✐❛ ❛ ♣r♦♣❛❣❛çã♦ ❞❡st❛s ✐♥❢♦r♠❛çõ❡s s❡❣✉♥❞♦ ❛ ♦♣❡r❛çã♦ ❝♦❧❡t✐✈❛ ❞❡ r❡❣✐str♦ ❞♦s ❡st❛❞♦s ❧♦❝❛✐s ✐♥✐❝✐❛✐s✳ ❊♥q✉❛♥t♦ ♦ ♥ó ♥ã♦ t✐✈❡r r❡❝❡❜✐❞♦ ❡ss❛s ✐♥❢♦r♠❛çõ❡s✱ ❛s ♠❡♥s❛❣❡♥s ❞♦ s✉❜str❛t♦ sã♦ ❛❝✉♠✉❧❛❞❛s ❡♠Φi❡ ❛s ♠❡♥s❛❣❡♥s ❞❡ ❝♦♥tr♦❧❡ ❞♦ r❡❣✐str♦ ❞♦s ❧♦❝❛✐s ✐♥✐❝✐❛✐s sã♦ ♣r♦❝❡ss❛❞❛s ✭❧✐♥❤❛s

✺✲✶✵✮✳

P♦r ❝♦♥✈❡♥✐ê♥❝✐❛✱ ❝r✐❛♠♦s ❛ ❢✉♥çã♦ SP ON T AN EOU S❴EV EN Ti ♣❛r❛ tr❛t❛r ♦s ❡✈❡♥t♦s

❡s♣♦♥tâ♥❡♦s✱ q✉❡ sã♦ ❡✈❡♥t♦s ❣❡r❛❞♦s s❡♠ ❛ r❡❝❡♣çã♦ ❞❡ ♠❡♥s❛❣❡♥s✳ ❖ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❛ ❞❡t❡❝çã♦ ❞❛ t❡r♠✐♥❛çã♦ ❣❧♦❜❛❧ ❡①✐❣❡ q✉❡ s♦♠❡♥t❡ ♦ ♥ó ✵ ❡❢❡t✉❡ ❡✈❡♥t♦ ❡s♣♦♥tâ♥❡♦✳ ◆❛s ❧✐♥❤❛s ✶✶ ❛ ✶✸✱ ♦ ♥ó ✵ ♣r♦❝❡ss❛ ♦ ❡✈❡♥t♦ ❡s♣♦♥tâ♥❡♦✳ ❙❡ ♦ r❡t♦r♥♦ ❞❛s ❢✉♥çõ❡sSP ON T AN EOU S❴EV EN Ti❡EV EN Ti

é ✵✱ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ♥ó s❡ ♠❛♥t❡✈❡ ❛t✐✈♦ ❛♣ós ♦ ú❧t✐♠♦ ❡✈❡♥t♦ ❣❡r❛❞♦✳ ◗✉❛♥❞♦ ✐ss♦ ♦❝♦rr❡✱ ❝♦♠♣✉t❛çõ❡s r❡❢❡r❡♥t❡s à ❞❡t❡❝çã♦ ❞❛ t❡r♠✐♥❛çã♦ ❣❧♦❜❛❧ sã♦ ❡❢❡t✉❛❞❛s✳ ❯♠❛ ♦❜s❡r✈❛çã♦ ✐♠♣♦rt❛♥t❡ é q✉❡ s♦❧✐❝✐t❛çõ❡s ❛♦s ♠♦❞❡❧♦s ❢❡✐t❛s ✐♥t❡r♥❛♠❡♥t❡ às ❢✉♥çõ❡s SP ON T AN EOU S❴EV EN Ti ❡

EV EN Ti ♣♦❞❡♠ ❞❡s❡♥❝❛❞❡❛r ❝♦♠♣✉t❛çõ❡s ❧♦❝❛✐s ❞♦ ♠♦❞❡❧♦ ❡ ❡♥✈✐♦ ❞❡ ♠❡♥s❛❣❡♥s ❞❡ ❝♦♥tr♦❧❡

✭❧✐♥❤❛s ✶✶✲✶✸✱ ✶✾✲✷✶ ❡ ✷✼✲✷✾✮✳ ❚r❛t❛r❡♠♦s ❡st❡ ❢❛t♦ ❝♦♠ ♠❛✐s ❞❡t❛❧❤❡s ♥♦ ❈❛♣ít✉❧♦ ✹✳

P❛r❛ ♦s ❞❡♠❛✐s ♥ós✱ ❛s ♠❡♥s❛❣❡♥s ❡♠Φi sã♦ ♣r♦❝❡ss❛❞❛s ✭❧✐♥❤❛s ✶✹✲✷✶✮✳ ❙❡ ❛ ♠❡♥s❛❣❡♠ ❢♦r

❞♦ s✉❜str❛❞♦✱ ✉♠ ❡✈❡♥t♦ é ♣r♦❝❡ss❛❞♦✳ ❙❡ ❛ ♠❡♥s❛❣❡♠ ❢♦r ❞❡ ❝♦♥tr♦❧❡✱ ❛ ❝♦♠♣✉t❛çã♦ ❛ss♦❝✐❛❞❛ é r❡❛❧✐③❛❞❛✳ ❆ ❝♦♠♣✉t❛çã♦ ❡♥tã♦ ♣r♦ss❡❣✉❡ ❛té q✉❡ ❛ t❡r♠✐♥❛çã♦ ❣❧♦❜❛❧ s❡❥❛ ❞❡t❡❝t❛❞❛ ✭❧✐♥❤❛s ✷✷✲✷✾✮✳ ❉❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡ às ♠❡♥s❛❣❡♥s ❝♦♥t✐❞❛s ❡♠ Φi✱ ✉♠❛ ❝♦♠♣✉t❛çã♦ é ❢❡✐t❛ ♣❛r❛ ❝❛❞❛