GRASP com path-relinking para agrupamento de dados biológicos

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❘❆❋❆❊▲ ❉❊ ▼❆●❆▲❍➹❊❙ ❉■❆❙ ❋❘■◆❍❆◆■

●❘❆❙P ❈❖▼ P❆❚❍✲❘❊▲■◆❑■◆● P❆❘❆

❆●❘❯P❆▼❊◆❚❖ ❉❊ ❉❆❉❖❙ ❇■❖▲Ó●■❈❖❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳

❖r✐❡♥t❛❞♦r✿ ❘✐❝❛r❞♦ ▼❛rt✐♥s ❆❜r❡✉ ❙✐❧✈❛

❈♦♦r✐❡♥t❛❞♦r✿ ●❡r❛❧❞♦ ❘♦❜s♦♥ ▼❛t❡✉s

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❝

✷✵✶✶✱ ❘❛❢❛❡❧ ❞❡ ▼❛❣❛❧❤ã❡s ❉✐❛s ❋r✐♥❤❛♥✐✳ ❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳

❋r✐♥❤❛♥✐✱ ❘❛❢❛❡❧ ❞❡ ▼❛❣❛❧❤ã❡s ❉✐❛s

❋✾✶✹❣ ●❘❆❙P ❝♦♠ P❛t❤✲❘❡❧✐♥❦✐♥❣ ♣❛r❛ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s ✴ ❘❛❢❛❡❧ ❞❡ ▼❛❣❛❧❤ã❡s ❉✐❛s ❋r✐♥❤❛♥✐✳ ✖ ❇❡❧♦ ❍♦r✐③♦♥t❡✱ ✷✵✶✶

①✈✐✱ ✽✺ ❢✳ ✿ ✐❧✳ ❀ ✷✾❝♠

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✖ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✖ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛ ❞❛

❈♦♠♣✉t❛çã♦✳

❖r✐❡♥t❛❞♦r✿ ❘✐❝❛r❞♦ ▼❛rt✐♥s ❆❜r❡✉ ❙✐❧✈❛ ❈♦♦r✐❡♥t❛❞♦r✿ ●❡r❛❧❞♦ ❘♦❜s♦♥ ▼❛t❡✉s

✶✳ ❆❣r✉♣❛♠❡♥t♦ ✲ ❚❡s❡s✳ ✷✳ ❇✐♦❧♦❣✐❛ ❈♦♠♣✉t❛❝✐♦♥❛❧ ✲ ❚❡s❡s✳ ✸✳ ▼❡t❛✲❤❡✉ríst✐❝❛ ✲ ❚❡s❡s✳ ■✳❖r✐❡♥t❛❞♦r✳

■■✳❈♦♦r✐❡♥t❛❞♦r✳ ■■■✳❚ít✉❧♦✳

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UNIVERSIDADE FEDERAL DE MINAS GERAIS

INSTITUTO DE CIÊNCIAS EXATAS

PROGRAMA DE PÓS-GRADUAÇÃO EM CIÊNCIA DA COMPUTAÇÃOzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

FOLHA DE APROVAÇÃO

GRASP com path relinking para agrupamento de dados biológicos

RAFAEL DE MAGALHÃES DIAS FRINHANI

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s✱ ♣♦r ✐❧✉♠✐♥❛r ♠❡✉s ♣❡♥s❛♠❡♥t♦s ❡ ♠❡ ❛❥✉❞❛r ♥❛s ❡s❝♦❧❤❛s ❝♦rr❡t❛s✳ ❆ ♠❡✉s ♣❛✐s✱ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ ❢♦rç❛ ❡ ❞❡t❡r♠✐♥❛çã♦✳

❆ ♠✐♥❤❛ ❡s♣♦s❛ ❱❛♥❡ss❛✱ ♣❡❧♦ ❝❛r✐♥❤♦✱ ❞❡❞✐❝❛çã♦ ❡ ♣♦r ❡♥❝❤❡r ♠❡✉ ❝♦r❛çã♦ ❞❡ ♦r❣✉❧❤♦ ❡ ❢❡❧✐❝✐❞❛❞❡✳

❆♦ ♠❡✉s ♦r✐❡♥t❛❞♦r❡s ❘✐❝❛r❞♦ ❡ ❘♦❜s♦♥✱ ♣♦r ♠♦str❛r❡♠ ❝❛♠✐♥❤♦s ♥✉♥❝❛ ❛♥t❡s ✐♠❛❣✐♥❛❞♦s✳

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❘❡s✉♠♦

❖ ❛❣r✉♣❛♠❡♥t♦ é ✉♠ ♠ét♦❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❞❛❞♦s ❡♠ ❣r✉♣♦s ✭❝❧✉st❡rs✮✳ ❊♠ ♣r♦❜❧❡♠❛s ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ ♥ã♦ s❡ s❛❜❡ ♣r❡✈✐❛♠❡♥t❡ q✉❛♥t❛s ❡ q✉❛✐s ❛s ❝❧❛ss❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❞❡s❝r❡✈❡r ❝♦❡r❡♥t❡♠❡♥t❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✳

◆❛ ❜✐♦❧♦❣✐❛ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ♦ ❛❣r✉♣❛♠❡♥t♦ ♠♦str♦✉✲s❡ ❢❡rr❛♠❡♥t❛ út✐❧ ❡♠ ♣r♦✲ ❜❧❡♠❛s ❞❡ ❞❡s❝♦❜❡rt❛ ❞❡ ♣❛❞rõ❡s ❡♠ ❞❛❞♦s ❝♦♠♦ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ♣r♦t❡í♥❛s✱ ♣r❡❞✐çã♦ ❞❛ ❧♦❝❛❧✐③❛çã♦ ❞❡ ♣r♦t❡í♥❛s ❡♠ ✉♥✐❞❛❞❡s ❝❡❧✉❧❛r❡s ❡ ❞✐❛❣♥óst✐❝♦ ❞❡ ❝â♥❝❡r✳ ❘❡❝❡♥t❡♠❡♥t❡✱ té❝♥✐❝❛s ❞❡ ♦t✐♠✐③❛çã♦✱ ❝♦♠♦ ❛s ♠❡t❛❤❡✉ríst✐❝❛s✱ tê♠ s✐❞♦ ✉t✐❧✐③❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ ♠ét♦❞♦ ❛❧t❡r♥❛t✐✈♦ ♦✉ ❛✉①✐❧✐❛r ♣❛r❛ ❛✉♠❡♥t❛r ❛ ❡✜❝✐ê♥❝✐❛ ❡ ❡✜❝á❝✐❛ ❞❛s ❢❡rr❛♠❡♥t❛s ❝❧áss✐❝❛s ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ ❣❡r❛❧♠❡♥t❡ ❜❛s❡❛❞❛s ❡♠ ♠ét♦❞♦s ❡st❛tíst✐❝♦s ❡ ♠❛t❡♠át✐❝♦s✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ♣r♦♣♦r ❛❧❣♦r✐t♠♦s ❤í❜r✐❞♦s ❢✉♥❞❛♠❡♥t❛❞♦s ♥❛s ♠❡t❛❤❡✉ríst✐❝❛s ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t❛t✐✈❡ ❙❡❛r❝❤ Pr♦❝❡❞✉r❡ ✭●❘❆❙P✮ ❡ P❛t❤✲ ❘❡❧✐♥❦✐♥❣ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s✱ ❝♦♠ ✐♥t✉✐t♦ ❞❡ ♦❜t❡r ♠❡❧❤♦r❡s s♦❧✉çõ❡s q✉❛♥❞♦ ❝♦♥s✐❞❡r❛❞♦ ✉♥✐❝❛♠❡♥t❡ ♦ ●❘❆❙P ♥❛ s✉❛ ❢♦r♠❛ ♣❛❞rã♦✳ P♦rt❛♥t♦✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛✲s❡ ❛ ❤✐♣ót❡s❡ q✉❡ ♦ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❝♦♠♦ ❡str❛té❣✐❛ ❞❡ ✐♥t❡♥s✐✜❝❛çã♦ ❞♦ ●❘❆❙P✱ ♠❡❧❤♦r❛ ♦ ❞❡s❡♠♣❡♥❤♦ ❡ q✉❛❧✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s✳

❈♦♥s✐❞❡r❛✲s❡ ❛ ❤✐❜r✐❞✐③❛çã♦ ❞♦ ●❘❆❙P ♣r♦♣♦st♦ ♣♦r ◆❛s❝✐♠❡♥t♦ ❡t ❛❧✳ ❬✷✵✶✵❜❪✱ ❝♦♠ q✉❛tr♦ ✈❛r✐❛çõ❡s ❞♦ P❛t❤✲❘❡❧✐♥❦✐♥❣✿ ❋♦r✇❛r❞✱ ❇❛❝❦✇❛r❞✱ ▼✐①❡❞✱ ❡ ●r❡❡❞② ❘❛♥✲ ❞♦♠✐③❡❞ ❆❞❛♣t❛t✐✈❡✳ ❆ ✈❛❧✐❞❛çã♦ ❞❛s s♦❧✉çõ❡s ♦❜t✐❞❛s ❞❛✲s❡ ♣♦r ♠❡✐♦ ❞❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦s ❛❧❣♦r✐t♠♦s ❝❧áss✐❝♦s ❦✲♠❡❛♥s✱ ❦✲♠❡❞✐❛♥s✱ P❆▼ ❜❡♠ ❝♦♠♦ ♦ ●❘❆❙P ♣r♦♣♦st♦ ♣♦r ◆❛s❝✐♠❡♥t♦ ❡t ❛❧✳ ❬✷✵✶✵❜❪✳

❖s ❡①♣❡r✐♠❡♥t♦s ❢♦r❛♠ r❡❛❧✐③❛❞♦s ❝♦♠ ❞❛❞♦s r❡❛✐s ❞❡ ✶✵ ✐♥stâ♥❝✐❛s ❜✐♦❧ó❣✐❝❛s✱ ❡ ♦s r❡s✉❧t❛❞♦s ♠♦str❛r❛♠ q✉❡ ♦ ♠♦❞❡❧♦ ❤í❜r✐❞♦ ♠❡❧❤♦r♦✉ ❛ t❛r❡❢❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦✳ ❖❜t❡✈❡✲ s❡ ♠❛✐♦r ❡①♣❧♦r❛çã♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❜✉s❝❛✱ ♠❛✐♦r ❝♦❡sã♦ ❞❡ ❛❣r✉♣❛♠❡♥t♦s✱ ❛✉♠❡♥t♦ ❞❛ r♦❜✉st❡③ ❡ r❡❞✉çã♦ ❞♦ t❡♠♣♦ ❝♦♠♣✉t❛❝✐♦♥❛❧✳ ❆s ✈❛r✐❛♥t❡s ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t❛t✐✈❡ ❡ ▼✐①❡❞ ❛♣r❡s❡♥t❛r❛♠ ♦s ♠❡❧❤♦r❡s r❡s✉❧t❛❞♦s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❆❣r✉♣❛♠❡♥t♦✱ ❇✐♦❧♦❣✐❛ ❈♦♠♣✉t❛❝✐♦♥❛❧✱ ▼❡t❛❤❡✉ríst✐❝❛s✱ ●❘❆❙P✱ P❛t❤✲❘❡❧✐♥❦✐♥❣✳

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❆❜str❛❝t

❈❧✉st❡r✐♥❣ ✐s ❛♥ ✉♥s✉♣❡r✈✐s❡❞ ♠❡t❤♦❞ ♦❢ ❝❧❛ss✐❢②✐♥❣ ❞❛t❛ ✐♥t♦ ❝❧✉st❡rs✳ ■♥ ❝❧✉st❡r✐♥❣ ♣r♦❜❧❡♠s✱ ✐t ✐s ♥♦t ♣r❡✈✐♦✉s❧② ❦♥♦✇♥ ❤♦✇ ♠❛♥② ❛♥❞ ✇❤✐❝❤ ❝❧❛ss❡s ❛r❡ ♥❡❡❞❡❞ t♦ ❞❡s❝r✐❜❡ ❝♦❤❡r❡♥t❧② ❛ s❡t ♦❢ ❞❛t❛✳

■♥ ❝♦♠♣✉t❛t✐♦♥❛❧ ❜✐♦❧♦❣②✱ ❞❛t❛ ❝❧✉st❡r✐♥❣ ♣r♦✈❡❞ t♦ ❜❡ ✉s❡❢✉❧ ✐♥ ♣r♦❜❧❡♠s ♦❢ ♣❛t✲ t❡r♥s ❞✐s❝♦✈❡r② ✐♥ ❞❛t❛ s✉❝❤ ❛s ♣r♦t❡✐♥ ❝❧❛ss✐✜❝❛t✐♦♥✱ ♣r❡❞✐❝t✐♦♥ ♦❢ ♣r♦t❡✐♥ ❧♦❝❛❧✐③❛t✐♦♥ ✐♥ ❝❡❧❧ ✉♥✐ts ❛♥❞ ❝❛♥❝❡r ❞✐❛❣♥♦s✐s✳

❘❡❝❡♥t❧②✱ ♦♣t✐♠✐③❛t✐♦♥ t❡❝❤♥✐q✉❡s ❧✐❦❡ ♠❡t❛❤❡✉r✐st✐❝s✱ ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❢r❡q✉❡♥t❧② ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛s ❛♥ ❛❧t❡r♥❛t✐✈❡ ♦r ❛❞❥✉♥❝t t♦ ✐♥❝r❡❛s❡ t❤❡ ❡✣❝✐❡♥❝② ❛♥❞ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ ❝❧❛ss✐❝ t♦♦❧s ♦❢ ❝❧✉st❡r✐♥❣✱ ✉s✉❛❧❧② ❜❛s❡❞ ♦♥ st❛t✐st✐❝❛❧ ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ♠❡t❤♦❞s✳ ■♥ t❤✐s st✉❞② ✇❡ ♣r♦♣♦s❡❞ ❤②❜r✐❞ ❛❧❣♦r✐t❤♠s ❜❛s❡❞ ♦♥ ♠❡t❛❤❡✉r✐st✐❝s ●❘❆❙P ❛♥❞ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❢♦r t❤❡ ❝❧✉st❡r✐♥❣ ♣r♦❜❧❡♠ ♦❢ ❜✐♦❧♦❣✐❝❛❧ ❞❛t❛✱ ❛✐♠✐♥❣ t♦ ♦❜t❛✐♥ ❜❡tt❡r s♦❧✉t✐♦♥s ✇❤❡♥ ❝♦♠♣❛r❡❞ t♦ ✉s✐♥❣ ♦♥❧② t❤❡ ●❘❆❙P ✐♥ ✐ts st❛♥❞❛r❞ ❢♦r♠✳ ❚❤❡r❡❢♦r❡✱ ✐♥ t❤✐s ✇♦r❦✱ ✇❡ ❝♦♥s✐❞❡r❡❞ t❤❡ ❤②♣♦t❤❡s✐s t❤❛t ✉s✐♥❣ t❤❡ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❛s ❛ str❛t❡❣② ❢♦r ✐♥t❡♥s✐❢②✐♥❣ ●❘❆❙P✱ ✐♠♣r♦✈❡s ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ q✉❛❧✐t② ♦❢ s♦❧✉t✐♦♥s✳

❲❡ ❝♦♥s✐❞❡r ❤②❜r✐❞✐③❛t✐♦♥ ♦❢ t❤❡ ●❘❆❙P ♣r♦♣♦s❡❞ ❜② ◆❛s❝✐♠❡♥t♦ ❡t ❛❧✳ ❬✷✵✶✵❜❪✱ ✇✐t❤ ❢♦✉r ✈❛r✐❛♥ts ♦❢ P❛t❤✲❘❡❧✐♥❦✐♥❣✿ ❢♦r✇❛r❞✱ ❜❛❝❦✇❛r❞✱ ♠✐①❡❞✱ ❛♥❞ ❣r❡❡❞② r❛♥❞♦♠✐✲ ③❡❞ ❛❞❛♣t❛t✐✈❡✳ ❚❤❡ ✈❛❧✐❞❛t✐♦♥ ♦❢ s♦❧✉t✐♦♥s ✐s ❣✐✈❡♥ ❜② ❝♦♠♣❛r✐♥❣ ✇✐t❤ t❤❡ ❝❧❛ss✐❝ ❛❧✲ ❣♦r✐t❤♠s ❦✲♠❡❛♥s✱ ❦✲♠❡❞✐❛♥s✱ P❆▼ ✭P❛rt✐t✐♦♥✐♥❣ ❆r♦✉♥❞ ▼❡❞♦✐❞s✮ ❛s ✇❡❧❧ ❛s ●❘❆❙P ♣r♦♣♦s❡❞ ❜② ◆❛s❝✐♠❡♥t♦ ❡t ❛❧✳ ✭✷✵✶✵✮✳

❚❤❡ ❡①♣❡r✐♠❡♥ts ✇❡r❡ ♣❡r❢♦r♠❡❞ ✇✐t❤ r❡❛❧ ❞❛t❛ ❢r♦♠ ✶✵ ❜✐♦❧♦❣✐❝❛❧ ✐♥st❛♥❝❡s✱ ❛♥❞ r❡s✉❧ts s❤♦✇❡❞ t❤❛t t❤❡ ❤②❜r✐❞ ♠♦❞❡❧ ✐♠♣r♦✈❡❞ t❤❡ ❝❧✉st❡r✐♥❣ t❛s❦✳ ❲❡ ♦❜t❛✐♥❡❞ ❢✉rt❤❡r ❡①♣❧♦r❛t✐♦♥ ♦❢ t❤❡ s❡❛r❝❤ s♣❛❝❡✱ ♠♦r❡ ❝♦❤❡s✐✈❡ ❝❧✉st❡rs✱ ✐♥❝r❡❛s❡ ♦❢ r♦❜✉st♥❡ss ❛♥❞ r❡❞✉❝t✐♦♥ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡✳ ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t✐✈❡ ❛♥❞ ▼✐①❡❞ s❤♦✇❡❞ t❤❡ ❜❡st r❡s✉❧ts✳

❑❡②✇♦r❞s✿ ❈❧✉st❡r✐♥❣✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ❇✐♦❧♦❣②✱ ▼❡t❛❤❡✉r✐st✐❝s✱ ●❘❆❙P✱ P❛t❤✲ ❘❡❧✐♥❦✐♥❣✳

(12)

(13)

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❚❛①♦♥♦♠✐❛ ❞❡ té❝♥✐❝❛s ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❊①❡♠♣❧♦ ❞❡ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❛♣❧✐❝❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s✳ ✷✶

✸✳✶ ❋❧✉①♦❣r❛♠❛ ❞❛s ❛t✐✈✐❞❛❞❡s r❡❛❧✐③❛❞❛s ❞✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♣r♦❥❡t♦✳ ✷✺ ✸✳✷ ❋❧✉①♦❣r❛♠❛ ❞❛s ❡t❛♣❛s r❡❛❧✐③❛❞❛s ❞✉r❛♥t❡ ♦ ●❘❆❙P✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✹✳✶ ◗✉❛♥t✐❞❛❞❡ ❞❡ ♠❡❧❤♦r❡s ❈❘❛♥❞s ♣❛r❛ ❝❛❞❛ ♠étr✐❝❛ ❡ ❛❧❣♦r✐t♠♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✷ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❇r❡❛st✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✸ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❇r❡❛st✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✹ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❇r❡❛st❆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✺ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❇r❡❛st❇✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✻ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❇r❡❛st❇✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✼ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❉▲❇❈▲❆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✽ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❉▲❇❈▲❇✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✾ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ■r✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✶✵ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ▼✉❧t✐❆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✶✶ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ◆♦✈❛rt✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✶✷ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ ❨❡❛st✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✶✸ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ Pr♦t❡✐♥✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✶✹ ●rá✜❝♦ ❚✐♠❡✲t♦✲❚❛r❣❡t ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ Pr♦t❡✐♥✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

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(15)

❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✈✐✐

❘❡s✉♠♦ ✐①

❆❜str❛❝t ①✐

▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐✐✐

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❊st❛❞♦ ❞❛ ❆rt❡ ✺

✷✳✶ ❆❣r✉♣❛♠❡♥t♦ ❞❡ ❉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✶✳✶ ▼ét♦❞♦s ❞❡ ❆❣r✉♣❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✶✳✷ ❆♥á❧✐s❡ ❞❡ ❆❣r✉♣❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ ▼❡t❛❤❡✉ríst✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷✳✶ ●❘❆❙P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷✳✷ P❛t❤✲❘❡❧✐♥❦✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷✳✸ ●❘❆❙P ❝♦♠ P❛t❤✲❘❡❧✐♥❦✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✸ ▼❡t♦❞♦❧♦❣✐❛ ✷✺

✸✳✶ ■♥stâ♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✶✳✶ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ❉♦❜r❛s ✭❋♦❧❞ ❈❧❛ss✐✜❝❛t✐♦♥✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶✳✷ Pr❡❞✐çã♦ ❞❛ ▲♦❝❛❧✐③❛çã♦ ❞❡ Pr♦t❡í♥❛s ✭Pr❡❞✐❝t✐♦♥ ♦❢ Pr♦t❡✐♥ ▲♦✲

❝❛❧✐③❛t✐♦♥ ❙✐t❡s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶✳✸ ❉✐❛❣♥óst✐❝♦ ❞❡ ❈â♥❝❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶✳✹ Pr♦❜❧❡♠❛s ❚❛①♦♥ô♠✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ Pré✲Pr♦❝❡ss❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸ ■♠♣❧❡♠❡♥t❛çã♦ ●❘❆❙P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸✳✶ ❋❛s❡ ❈♦♥str✉t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

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✸✳✸✳✷ ❋❛s❡ ❞❡ ❇✉s❝❛ ▲♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✹ ■♠♣❧❡♠❡♥t❛çã♦ ●❘❆❙P ❝♦♠ P❛t❤✲❘❡❧✐♥❦✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹✳✶ ❋♦r✇❛r❞ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❡ ❇❛❝❦✇❛r❞ P❛t❤✲❘❡❧✐♥❦✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✹✳✷ ▼✐①❡❞ P❛t❤✲❘❡❧✐♥❦✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✹✳✸ ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t❛t✐✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✺ ❆♠❜✐❡♥t❡ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✺✳✶ ❆♠❜✐❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✺✳✷ P❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✹ ❘❡s✉❧t❛❞♦s ❡ ❉✐s❝✉ssã♦ ✹✸

✹✳✶ ❈♦♠♣❛r❛çã♦ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❞❛ ❧✐t❡r❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✷ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ❛❧❣♦r✐t♠♦s ♣r♦♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✺ ❈♦♥❝❧✉sã♦ ❡ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ✼✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✼

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❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❖ ❝r❡s❝✐♠❡♥t♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞❛❞♦s ❡♠ ❢♦r♠❛t♦ ❞✐❣✐t❛❧ ❡ ❞❛s t❡❝♥♦❧♦❣✐❛s ❞❡ ❛r♠❛✲ ③❡♥❛♠❡♥t♦ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❡ ❣r❛♥❞❡s ❜❛s❡s ❞❡ ❞❛❞♦s✳ ❙❡❣✉♥❞♦ ❇✉❝❡♥❡ ❡t ❛❧✳ ❬✷✵✵✷❪✱ ❝❛❞❛ ✈❡③ ♠❛✐s✱ ♦ ✈♦❧✉♠❡ ❞❡ ✐♥❢♦r♠❛çõ❡s ❡①❝❡❞❡ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛♥á❧✐s❡ ♣❡❧♦s ♠ét♦❞♦s tr❛❞✐❝✐♦♥❛✐s✱ ✐♥❝❛♣❛③❡s ❞❡ ❛♥❛❧✐s❛r ♦s ❞❛❞♦s s♦❜ ♦ ❡♥❢♦q✉❡ ❞♦ ❝♦♥❤❡❝✐✲ ♠❡♥t♦✳ ❈♦♠♦ r❡s✉❧t❛❞♦ ❞❡ss❡ ❛✉♠❡♥t♦ ❡❢❡t✐✈♦✱ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ss❛s ✐♥❢♦r♠❛çõ❡s t♦r♥♦✉✲s❡ ❝♦♠♣❧❡①♦ ❡ ❞✐❢í❝✐❧✱ ❝✉❧♠✐♥❛♥❞♦ ❡♠ ❞❛❞♦s ❛r♠❛③❡♥❛❞♦s✱ ♠❛s s❡♠ q✉❡ s❡❥❛♠ ✉t✐❧✐③❛❞♦s ❞❡ ✉♠❛ ❢♦r♠❛ r❡❛❧♠❡♥t❡ ❡✜❝✐❡♥t❡ ❬❍❛❧♠❡♥s❝❤❧❛❣❡r✱ ✷✵✵✵❪✳

❑♥♦✇❧❡❞❣❡ ❉✐s❝♦✈❡r② ✐♥ ❉❛t❛❜❛s❡ ✭❑❉❉✮ ❬❋r❛✇❧❡② ❡t ❛❧✳✱ ✶✾✾✷❀ P✐❛t❡ts❦②✲❙❤❛♣✐r♦ ✫ ❋r❛✇❧❡②✱ ✶✾✾✶❪✱ ❉❛t❛ ▼✐♥✐♥❣ ❬❲✐tt❡♥ ✫ ❋r❛♥❦✱ ✷✵✵✺❀ ❋❛②②❛❞ ❡t ❛❧✳✱ ✶✾✾✻❀ ❍❛♥ ✫ ❑❛♠❜❡r✱ ✷✵✵✻❪✱ P❛tt❡r♥ ❘❡❝♦❣♥✐t✐♦♥ ❬❲❡❜❜✱ ✶✾✾✾❀ ❘✐♣❧❡②✱ ✷✵✵✽❪✱ P❛tt❡r♥ ❉✐s❝♦✈❡r② ❬❲❛♥❣ ❡t ❛❧✳✱ ✶✾✾✾❀ ❘✐❣♦✉ts♦s ❡t ❛❧✳✱ ✷✵✵✵❪ ❡ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ❬❇✐s❤♦♣✱ ✷✵✵✻❀ ▼✐❝❤❛❧s❦✐ ❡t ❛❧✳✱ ✶✾✾✹❪ sã♦ s❡❣♠❡♥t♦s ❞❛ ❝♦♠♣✉t❛çã♦ q✉❡ tr❛t❛♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ✉t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ♣❛r❛ ✐❞❡♥t✐✜❝❛r ❡♠ ❞❛❞♦s✱ ♣❛❞rõ❡s ✈á❧✐❞♦s✱ ♣r❡✈✐❛♠❡♥t❡ ❞❡s❝♦♥❤❡❝✐❞♦s✱ ♣♦t❡♥❝✐❛❧♠❡♥t❡ út❡✐s ❡ ❝♦♠♣r❡❡♥sí✈❡✐s✱ q✉❡ ✈✐s❛♠ ♠❡❧❤♦r❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♦✉ ♣r♦❝❡ss♦ ❞❡ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦✳

❉✐❢❡r❡♥t❡s té❝♥✐❝❛s ❛♥❛❧ít✐❝❛s ♣♦❞❡♠ s❡r ❡♠♣r❡❣❛❞❛s ♥❛ ❞❡s❝♦❜❡rt❛ ❞❡ ♣❛❞rõ❡s✱ ✈❛r✐❛♥❞♦ ❡♥tr❡ ❛❜♦r❞❛❣❡♥s ❞❡s❝r✐t✐✈❛s ❡ ❣rá✜❝❛s ♠❛✐s ❜ás✐❝❛s✱ té❝♥✐❝❛s ♠✉❧t✐✈❛r✐❛❞❛s ✭❡①✳ ❛♥á❧✐s❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦s ♦✉ r❡❣r❡ssã♦ ♠ú❧t✐♣❧❛ ♦✉ ❧♦❣íst✐❝❛✮ ❡ ♠♦❞❡❧♦s ❞❡ ❛♣r❡♥❞✐✲ ③❛❞♦ ✭❝♦♠♦ r❡❞❡s ♥❡✉r❛✐s ❡ ❛❧❣♦r✐t♠♦s ❣❡♥ét✐❝♦s✮ ❬❍❛✐r ❏r ❡t ❛❧✳✱ ✷✵✵✻❪✳ ❊ss❛s té❝♥✐❝❛s ♣♦❞❡♠ s❡r ♦r❣❛♥✐③❛❞❛s ❝♦♠♦ ❆♥á❧✐s❡ ❞❡ ❘❡❣r❛s ❞❡ ❆ss♦❝✐❛çã♦ ❩❤❛♥❣ ✫ ❩❤❛♥❣ ❬✷✵✵✷❪✱ ❈❧❛ss✐✜❝❛çã♦ ❡ Pr❡❞✐çã♦ ❬P❤②✉✱ ✷✵✵✾❪✱ ❆♥á❧✐s❡ ❞❡ P❛❞rõ❡s ❙❡q✉ê♥❝✐❛✐s ❬❩❤❛♦ ✫ ❇❤♦✇✲ ♠✐❝❦✱ ✷✵✵✸❪✱ ❆♥á❧✐s❡ ❞❡ ♦✉t❧✐❡rs ❬❍♦❞❣❡ ✫ ❆✉st✐♥✱ ✷✵✵✹❪ ❡ ❆♥á❧✐s❡ ❞❡ ❆❣r✉♣❛♠❡♥t♦s ❬❇❡r❦❤✐♥✱ ✷✵✵✻❪✳

❆ ❛♥á❧✐s❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦s ✭❝❧✉st❡r✐♥❣✮ ❝♦♥s✐st❡ ❡♠ ✐❞❡♥t✐✜❝❛r ❝❧❛ss❡s ❞❡ ✐t❡♥s ❡♠ ✉♠❛ ❜❛s❡ ❞❡ ❞❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛❧❣✉♠❛ ♠❡❞✐❞❛ ❞❡ s✐♠✐❧❛r✐❞❛❞❡✳ ❈❛❞❛ ❣r✉♣♦

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✷ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦

❝♦♥s✐st❡ ❞❡ ♦❜❥❡t♦s q✉❡ sã♦ s✐♠✐❧❛r❡s ❡♥tr❡ s✐ ❡ ❞✐❢❡r❡♥t❡s ❞♦s ♦❜❥❡t♦s ❞♦s ♦✉tr♦s ❣r✉♣♦s✳ ❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ♦❜❥❡t♦s ❡ ❡♥tr❡ ❣r✉♣♦s é ❞❛❞❛ ♣♦r ✉♠ ✈❛❧♦r✱ ❝❛❧❝✉❧❛❞♦ ♣♦r ♠❡❞✐❞❛s ❞❡ ❞✐stâ♥❝✐❛ ♦✉ s✐♠✐❧❛r✐❞❛❞❡✳ ❖s ❛❧❣♦r✐t♠♦s ❝❧áss✐❝♦s ❞❡ ❛♥á❧✐s❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦s sã♦ ❜❛s❡❛❞♦s ❡♠ té❝♥✐❝❛s ♠❛t❡♠át✐❝❛s✱ ❡st❛tíst✐❝❛s ❡ ❞❡ ❛♥á❧✐s❡ ♥✉♠ér✐❝❛ ❡✱ ❞✐❢❡r❡♥t❡ ❞❡ ♦✉tr❛s t❛r❡❢❛s ❝♦♠♦ ❛ ❝❧❛ss✐✜❝❛çã♦ ❡ ❛ ♣r❡❞✐çã♦✱ tr❛❜❛❧❤❛♠ s♦❜r❡ ❞❛❞♦s ♥♦s q✉❛✐s ❛s ❝❧❛ss❡s ♥ã♦ ❡stã♦ ♣r❡✈✐❛♠❡♥t❡ ❞❡✜♥✐❞❛s✳ ❆♣❡s❛r ❞♦ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❛❧❣♦r✐t♠♦s ♣❛r❛ ❛♥á❧✐s❡s ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ ❝❛❞❛ ❛❧❣♦r✐t♠♦ t❡♠ s✉❛s ✈❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s✱ s❡♥❞♦ ♠❛✐s ❛❞❡q✉❛❞♦ ♣❛r❛ t✐♣♦s ❡s♣❡❝í✜❝♦s ❞❡ ❞❛❞♦s✳

❆ ❜✐♦❧♦❣✐❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ tr❛t❛ ❞❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❡ ❛♥❛✲ ❧ít✐❝♦s ♣❛r❛ ♣r♦❜❧❡♠❛s ❜✐♦❧ó❣✐❝♦s✱ ❡ é ✉♠❛ ár❡❛ ❡♠ rá♣✐❞❛ ❡✈♦❧✉çã♦ ❝✐❡♥tí✜❝❛✳ ❇❛s❡s ❞❡ ❞❛❞♦s ❡ ❧✐t❡r❛t✉r❛ ❜✐♦❧ó❣✐❝❛s ❡stã♦ ❝r❡s❝❡♥❞♦ ❞❡ ❢♦r♠❛ ❛❝❡❧❡r❛❞❛ ❬●✐❜❛s ✫ ❏❛♠❜❡❝❦✱ ✷✵✵✶❪✳ ❙❡❣✉♥❞♦ ❑✐t❛♥♦ ❬✷✵✵✷❪✱ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s ❝♦♠♣❧❡①♦s r❡q✉❡r ❛ ✐♥t❡❣r❛çã♦ ❞❡ ♣❡sq✉✐s❛ ❡①♣❡r✐♠❡♥t❛❧ ❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ ✈✐s❛ ❛❧❝❛♥ç❛r ✐♥♦✈❛çõ❡s ♣rá✲ t✐❝❛s ♥❛ ♠❡❞✐❝✐♥❛✳ ❉✐✈❡rs♦s ❡①♣❡r✐♠❡♥t♦s ✉t✐❧✐③❛♥❞♦ t❛r❡❢❛s ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s ♣❛r❛ ❜❛s❡s ❜✐♦❧ó❣✐❝❛s tê♠ s✐❞♦ ♣✉❜❧✐❝❛❞♦s ❬❍❛♥❞ ✫ ❍❡❛r❞✱ ✷✵✵✺❀ ❆♥ ✫ ❈❤❡♥✱ ✷✵✵✾❀ ❆❧♦♥ ❡t ❛❧✳✱ ✶✾✾✾❀ ▼♦♥t✐ ❡t ❛❧✳✱ ✷✵✵✸❀ ▼❛ ❡t ❛❧✳✱ ✷✵✵✻❪✳

❆ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❞❛❞♦s ❡ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❣r✉♣♦s ❞❡s❡✲ ❥❛❞❛✱ t♦r♥❛♠ ❛ t❛r❡❢❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ✉♠ ❝♦♠♣❧✐❝❛❞♦ ♣r♦❜❧❡♠❛ ❝♦♠❜✐♥❛tór✐♦✳ ❆❧❣♦r✐t✲ ♠♦s q✉❡ r❡❛❧✐③❛♠ ❡st❛ t❛r❡❢❛✱ ❡♠ ❣❡r❛❧✱ ❛♣r❡s❡♥t❛♠ ❛❧t❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡✴♦✉ ❜❛✐①❛ ♣r❡❝✐sã♦✳ ◆❡st❡ ❝♦♥t❡①t♦✱ té❝♥✐❝❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠♦ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r✱ ♣r♦❣r❛♠❛çã♦ ✐♥t❡✐r❛ ❡ ♠❡t❛❤❡✉ríst✐❝❛s✱ tê♠ s✐❞♦ ✉t✐❧✐③❛❞❛s ❝♦♠♣❧❡♠❡♥t❛r♠❡♥t❡ ❛♦s ❛❧✲ ❣♦r✐t♠♦s ❝❧áss✐❝♦s✱ ❞❡ ❢♦r♠❛ ❛ ♠❡❧❤♦r❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❡✴♦✉ ♣r❡❝✐sã♦ ❞❛s t❛r❡❢❛s ❞❡ ❛❣r✉♣❛♠❡♥t♦✳

❘❡❝❡♥t❡♠❡♥t❡✱ ♦ ✐♥t❡r❡ss❡ ♥♦ ✉s♦ ❞❡ ♠❡t❛❤❡✉ríst✐❝❛s ♣❛r❛ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❛✉♠❡♥t♦✉ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❬▼❛ ❡t ❛❧✳✱ ✷✵✵✻❀ ❇❛♥❞②♦♣❛❞❤②❛② ✫ ▼❛✉❧✐❦✱ ✷✵✵✷❀ P❛✲ ❝❤❡❝♦✱ ✷✵✵✺❀ ❆❧✲❙✉❧t❛♥✱ ✶✾✾✺❪✳ ◆❛s❝✐♠❡♥t♦ ❡t ❛❧✳ ❬✷✵✶✵❜❪ ❢♦r❛♠ ♣✐♦♥❡✐r♦s ♥❛ ✉t✐❧✐③❛çã♦ ❞❛ ♠❡t❛❤❡✉ríst✐❝❛ ●❘❆❙P ✲ ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t❛t✐✈❡ ❙❡❛r❝❤ Pr♦❝❡❞✉r❡ ✭Pr♦❝❡✲ ❞✐♠❡♥t♦ ❛❞❛♣t❛❞♦ ❞❡ ❜✉s❝❛ ❣✉❧♦s❛ ❛❧❡❛tór✐❛✮ ♣❛r❛ ♦ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s✱ ♦❜t❡♥❞♦ s✉❝❡ss♦ ❡♠ ❝♦♠♣❛r❛çã♦ ❛ ❛❧❣♦r✐t♠♦s ❝❧áss✐❝♦s ❝♦♠♦ ❦✲♠❡❛♥s✱ ❦✲♠❡❞✐❛♥s ❡ P❆▼ ✭P❛rt✐t✐♦♥✐♥❣ ❆r♦✉♥❞ ▼❡❞♦✐s✮✳

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✸

♥❛♠❡♥t♦ ❞♦ ●❘❆❙P tr❛❞✐❝✐♦♥❛❧ ♥ã♦ s❡ ❜❛s❡✐❛ ❡♠ ❛♣r❡♥❞✐③❛❣❡♠ s♦❜r❡ ❛ s✉❛ ❡①❡❝✉çã♦✳ ❖ ✉s♦ ❞❡ s♦❧✉çõ❡s ♣r❡✈✐❛♠❡♥t❡ ❡♥❝♦♥tr❛❞❛s ♣❛r❛ ✐♥✢✉❡♥❝✐❛r ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♥❛ ✐t❡r❛çã♦ ❝♦rr❡♥t❡ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞❡ ♠❡❝❛♥✐s♠♦s ❞❡ ♠❡♠ór✐❛✳ ❚❛✐s ♠❡❝❛♥✐s♠♦s ❝♦♥s✲ tr♦❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s ❡❧✐t❡✱ ❡✈✐t❛♥❞♦ ❛ ❡①♣❧♦r❛çã♦ ❞❡ s♦❧✉çõ❡s ♠✉✐t♦ r✉✐♥s ❡✴♦✉ r❡❞✉♥❞❛♥t❡s✳

❯♠❛ ❞❛s ❢♦r♠❛s ❞❡ ✐♥❝♦r♣♦r❛r ♠❡♠ór✐❛ ♥♦ ●❘❆❙P é ❝♦♠ ❛ ❡str❛té❣✐❛ P❛t❤✲ ❘❡❧✐♥❦✐♥❣ ❬●❧♦✈❡r✱ ✶✾✾✻✱ ✷✵✵✵❀ ●❧♦✈❡r ✫ ▼❛rt✐✱ ✷✵✵✻❀ ●❧♦✈❡r ❡t ❛❧✳✱ ✷✵✵✵❪✳ ❈♦♠ P❛t❤✲ ❘❡❧✐♥❦✐♥❣✱ ❛♣ós ✉♠❛ s♦❧✉çã♦ t❡r s✐❞♦ ❣❡r❛❞❛ ♣❡❧♦ ●❘❆❙P✱ ❡❧❛ é ❝♦♠❜✐♥❛❞❛ ❝♦♠ ✉♠❛ s♦❧✉çã♦ ❛❧❡❛t♦r✐❛♠❡♥t❡ s❡❧❡❝✐♦♥❛❞❛ ❡♠ ✉♠ ♣♦♦❧ ❞❡ s♦❧✉çõ❡s ❡❧✐t❡✳ ❖ ♦❜❥❡t✐✈♦ ❞♦ ●❘❆❙P ❝♦♠ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❬▲❛❣✉♥❛ ✫ ▼❛rt✐✱ ✶✾✾✾❪ é ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ✐♥t❡r♠❡❞✐ár✐❛s ❞❡ ♠❡❧❤♦r q✉❛❧✐❞❛❞❡✱ q✉❡ ❡st❡❥❛♠ ❡♥tr❡ ❞✉❛s ❜♦❛s s♦❧✉çõ❡s✳ ●❘❆❙P ❝♦♠ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❢♦✐ ❛♣❧✐❝❛❞♦ ❝♦♠ s✉❝❡ss♦ ❡♠ ❬❋❡st❛ ❡t ❛❧✳✱ ✷✵✵✼❀ ▼❛t❡✉s ❡t ❛❧✳✱ ✷✵✶✵❀ ❇❛st♦s ❡t ❛❧✳✱ ✷✵✵✺❪✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ tr❛❜❛❧❤♦ ❢♦✐ ♣r♦♣♦r ❛❧❣♦r✐t♠♦s ❤í❜r✐❞♦s ❢✉♥❞❛♠❡♥t❛❞♦s ♥❛s ❤❡✉ríst✐❝❛s ●❘❆❙P ❡ P❛t❤✲❘❡❧✐♥❦✐♥❣ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s ❜✐✲ ♦❧ó❣✐❝♦s✱ ❝♦♠ ✐♥t✉✐t♦ ❞❡ ♦❜t❡r ♠❡❧❤♦r❡s s♦❧✉çõ❡s q✉❛♥❞♦ ❝♦♥s✐❞❡r❛❞♦ ✉♥✐❝❛♠❡♥t❡ ♦ ●❘❆❙P ♥❛ s✉❛ ❢♦r♠❛ ♣❛❞rã♦✳ P♦rt❛♥t♦✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r♦✉✲s❡ ❛ ❤✐♣ót❡s❡ q✉❡ ♦ P❛t❤✲❘❡❧✐♥❦✐♥❣ ❝♦♠♦ ❡str❛té❣✐❛ ❞❡ ✐♥t❡♥s✐✜❝❛çã♦ ❞♦ ●❘❆❙P ♠❡❧❤♦r❛ ♦ ❞❡s❡♠♣❡✲ ♥❤♦ ❡ q✉❛❧✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s✳ P❛r❛ t❡st❛r ❡ss❛ ❤✐♣ót❡s❡✱ ❢♦✐ r❡❛❧✐③❛❞❛ ❛ ❤✐❜r✐❞✐③❛çã♦ ❞♦ ●❘❆❙P ♣r♦♣♦st♦ ♣♦r ◆❛s❝✐♠❡♥t♦ ❡t ❛❧✳ ❬✷✵✶✵❜❪✱ ❝♦♠ q✉❛tr♦ ✈❛r✐❛çõ❡s ❞♦ P❛t❤✲ ❘❡❧✐♥❦✐♥❣✿ ❢♦r✇❛r❞✱ ❜❛❝❦✇❛r❞✱ ♠✐①❡❞✱ ❡ ❣r❡❡❞② r❛♥❞♦♠✐③❡❞ ❛❞❛♣t❛t✐✈❡✳ ❆ ✈❛❧✐❞❛çã♦ ❞❛s s♦❧✉çõ❡s ♦❜t✐❞❛s ❞❡✉✲s❡ ♣♦r ♠❡✐♦ ❞❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ♦s ❛❧❣♦r✐t♠♦s ❝❧áss✐❝♦s ❦✲ ♠❡❛♥s✱ ❦✲♠❡❞✐❛♥s ❡ P❆▼ ❡ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ♣r♦♣♦st♦ ♣♦r ◆❛s❝✐♠❡♥t♦ ❡t ❛❧✳ ❬✷✵✶✵❜❪✱ q✉❡ ✉t✐❧✐③❛ ♦ ●❘❆❙P✳ P❛r❛ r❡❛❧✐③❛çã♦ ❞♦s ❡①♣❡r✐♠❡♥t♦s✱ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ✶✵ ✐♥stâ♥❝✐❛s ❜✐♦❧ó❣✐❝❛s ❡ ♦s ❛❧❣♦r✐t♠♦s ❢♦r❛♠ ❛♥❛❧✐s❛❞♦s q✉❛♥t♦ ❛ s✉❛ r♦❜✉st❡③✱ ❡✜❝á❝✐❛ ❡ ❝♦❡sã♦ ❞♦s ❛❣r✉♣❛♠❡♥t♦s ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❛s ❝❧❛ss✐✜❝❛çõ❡s r❡❛✐s ❢♦r♥❡❝✐❞❛s ❝♦♠ ❛s ✐♥stâ♥❝✐❛s s✉♣r❛❝✐t❛❞❛s✳

(20)

(21)

❈❛♣ít✉❧♦ ✷

❊st❛❞♦ ❞❛ ❆rt❡

✷✳✶ ❆❣r✉♣❛♠❡♥t♦ ❞❡ ❉❛❞♦s

❆ ❝❧❛ss✐✜❝❛çã♦ é ✉♠❛ ❢✉♥çã♦ ❞❡ ❛♣r❡♥❞✐③❛❞♦ q✉❡ ♠❛♣❡✐❛ ❞❛❞♦s ❞❡ ❡♥tr❛❞❛ ❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❝❛t❡❣♦r✐❛s✳ ❚❛❧ ❢✉♥çã♦ ✉t✐❧✐③❛ ✉♠ ❝♦♥❥✉♥t♦ ♣ré✲❞❡✜♥✐❞♦ ❞❡ ❝❧❛ss❡s✱ q✉❡ s❡r✈❡♠ ❞❡ ❡①❡♠♣❧♦ ♣❛r❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ♦❜❥❡t♦s ♣❡rt❡♥❝❡♥t❡s ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ é ❡♥❝♦♥tr❛r ❛❧❣✉♠ r❡❧❛❝✐♦♥❛♠❡♥t♦ ❡♥tr❡ ♦s ❛tr✐❜✉t♦s ❞❡ ✉♠ ♦❜❥❡t♦ ❡ ✉♠❛ ❝❧❛ss❡✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♣♦ss❛ ✉s❛r ❡ss❡ r❡❧❛❝✐♦♥❛♠❡♥t♦ ♣❛r❛ ♣r❡❞✐③❡r ❛ ❝❧❛ss❡ ❞❡ ✉♠ ❡①❡♠♣❧♦ ♥♦✈♦ ❡ ❞❡s❝♦♥❤❡❝✐❞♦✳ ❆ ❝❧❛ss✐✜❝❛çã♦ é ❝❤❛♠❛❞❛ ❞❡ ♠ét♦❞♦ ♣r❡❞✐t✐✈♦ ❬❘❡③❡♥❞❡✱ ✷✵✵✸❪✳

❖ ❛❣r✉♣❛♠❡♥t♦ é ✉♠ ♠ét♦❞♦ ♥ã♦ s✉♣❡r✈✐s✐♦♥❛❞♦ ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ♣❛❞rõ❡s ❡♠ ❣r✉♣♦s✱ ♦✉ s❡❥❛✱ ♥❡ss❡ ♣r♦❜❧❡♠❛ ♥ã♦ s❡ s❛❜❡ q✉❛✐s sã♦ ❛s ❝❧❛ss❡s✱ ♥❡♠ ♠❡s♠♦ q✉❛♥t❛s sã♦✳ ❇❛s✐❝❛♠❡♥t❡✱ ❛ t❛r❡❢❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❝♦♥s✐st❡ ❡♠ ❞✐✈✐❞✐r ♦s ❞❛❞♦s ❡♠ ❣r✉♣♦s ❞❡ ♦❜❥❡t♦s s✐♠✐❧❛r❡s✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s é ✉♠❛ t❛r❡❢❛ ❞❡s❝r✐t✐✈❛✱ q✉❡ ♣r♦❝✉r❛ ✐❞❡♥t✐✜❝❛r ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ❣r✉♣♦s✱ ❛ ♣❛rt✐r ❞♦s ❞❛❞♦s✳ ■ss♦ é ❢❡✐t♦✱ ❣❡r❛❧♠❡♥t❡✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦❜❥❡t♦s ❝♦♠ ✈❛❧♦r❡s ❞❡ ❛tr✐❜✉t♦s s✐♠✐❧❛r❡s s❡❥❛♠ r❡✉♥✐❞♦s ❡♠ ✉♠ ♠❡s♠♦ ❣r✉♣♦✳ ▼❡❞✐❞❛s ❞❡ ❞✐stâ♥❝✐❛ sã♦ ❣❡r❛❧♠❡♥t❡ ✉t✐❧✐③❛❞❛s ❝♦♠♦ ❜❛s❡ ♣❛r❛ ❛❣r✉♣❛r ♦s ♦❜❥❡t♦s✳ ❖ q✉❡ s❡ ❡s♣❡r❛ é q✉❡ ♦s ❣r✉♣♦s ♦❜t✐❞♦s s❡❥❛♠ ♦s ♠❛✐s ❤♦♠♦❣ê♥❡♦s ✭❝♦❡sã♦ ✐♥t❡r♥❛✮ ❡ ❜❡♠ s❡♣❛r❛❞♦s ♣♦ssí✈❡✐s ✭✐s♦❧❛♠❡♥t♦ ❡①t❡r♥♦✮ ❬❆♥❞❡r❜❡r❣✱ ✶✾✼✸❀ ❇r❛❣❛✱ ✷✵✵✺❀ ❇❡r❦❤✐♥✱ ✷✵✵✻❀ ❏❛✐♥ ❡t ❛❧✳✱ ✶✾✾✾❪✳

❋♦r♠❛❧♠❡♥t❡✱ ♣♦❞❡✲s❡ ❞❡✜♥✐r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ❬❇❡r❦❤✐♥✱ ✷✵✵✻❀ ●r❛❜♠❡✐❡r ✫ ❘✉❞♦❧♣❤✱ ✷✵✵✷❀ ❘♦❞r✐❣✉❡s✱ ✷✵✵✾❪✿ ❙❡❥❛ X = {x1, x2,

✳✳✳✱ xn}✱ ❝♦♥t❡♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ n ♦❜❥❡t♦s ♣❛r❛ ❛♥á❧✐s❡✳ ❈❛❞❛ ♦❜❥❡t♦ xi é ✉♠ ✈❡t♦r ❝♦♠ d ❞✐♠❡♥sõ❡s✱ ❝✉❥♦s ❝♦♠♣♦♥❡♥t❡s sã♦ ❡s❝❛❧❛r❡s ❝❤❛♠❛❞♦s ❞❡ ❛tr✐❜✉t♦s✳ ❖ ♦❜❥❡t✐✈♦

❞♦ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s é ❡♥❝♦♥tr❛r ❣r✉♣♦sC ={C1, C2,✳✳✳✱CK}✱ ♦♥❞❡K é ♦ ♥ú♠❡r♦ ❞❡ ❣r✉♣♦s✳ ❉✐③✲s❡ q✉❡ C é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❣r✉♣♦ ♦❜t✐❞♦ ♣❛r❛ X s❡ três ❝❛r❛❝t❡ríst✐❝❛s

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✻ ❈❛♣ít✉❧♦ ✷✳ ❊st❛❞♦ ❞❛ ❆rt❡

♣♦❞❡♠ s❡r ♦❜s❡r✈❛❞❛s ♥❡ss❡ ❝♦♥❥✉♥t♦✿

✶✳ C1∪C2 ✳✳✳ ∪ Ck =X❀

✷✳ Ci 6=∅❀∀ i❀

✸✳ Ci∩Cj =∅❀∀ i6=j❀

❖ ♣r♦❜❧❡♠❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ ❡♠ t❡♦r✐❛✱ ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦ ❡♥✉♠❡r❛♥❞♦ t♦❞❛s ❛s ♣♦ssí✈❡✐s ❢♦r♠❛s ❞❡ ❛❣r✉♣❛r ♦s ♦❜❥❡t♦s ❡♠ ❣r✉♣♦s ❡ ❛✈❛❧✐❛r ❝❛❞❛ s♦❧✉çã♦ ❞❡ ❢♦r♠❛ ❡①❛✉st✐✈❛ ♦✉ ♣♦r ♠ét♦❞♦s ❡①❛t♦s ❞❡ ❢♦r♠❛ ✐♠♣❧í❝✐t❛ ❬❇❡rts✐♠❛s ✫ ❙❤✐♦❞❛✱ ✷✵✵✼❪✳ ❆ ❛❜♦r❞❛❣❡♠ ❡①❛✉st✐✈❛ é ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♥✈✐á✈❡❧ ♣❛r❛ ♣r♦❜❧❡♠❛s ❣r❛♥❞❡s✱ ❝♦♠♦ r❡s✉❧t❛❞♦✱ ♠✉✐t❛s té❝♥✐❝❛s ❞❡ ❛❣r✉♣❛♠❡♥t♦ sã♦ ❜❛s❡❛❞❛s ♥❛ t❡♥t❛t✐✈❛ ❞❡ ♠❛①✐♠✐③❛r ♦✉ ♠✐♥✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ q✉❡ r❡♣r❡s❡♥t❛ ❛s s✐♠✐❧❛r✐❞❛❞❡s ❡♥tr❡ ♦s ♦❜❥❡t♦s ❛tr❛✈és ❞❡ ♠ét♦❞♦s ❤❡✉ríst✐❝♦s✳

❲❛r❞ ❏r✳ ❬✶✾✻✸❪ ❡①♣❧✐❝❛ q✉❡✱ ❡♠ ❣❡r❛❧✱ ✉♠❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♣♦❞❡ s❡r q✉❛❧q✉❡r r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ q✉❡ ♦ ✐♥✈❡st✐❣❛❞♦r s❡❧❡❝✐♦♥❛ ♣❛r❛ r❡✢❡t✐r ❛ ♥❡❝❡ss✐❞❛❞❡ r❡❧❛t✐✈❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞♦s ♦❜❥❡t♦s✳ P❛r❛ ❈❤❛✈❡s ❬✷✵✵✾❪✱ ♦s ♠ét♦❞♦s ❞❡ ♦t✐♠✐③❛çã♦ ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♣r♦❝✉r❛♠ ❛❝r❡s❝❡♥t❛r ✉♠ ♣♦✉❝♦ ❞❡ ✐♥t❡❧✐❣ê♥❝✐❛ ❛♦ ♠ét♦❞♦ ❡①❛t♦✱ r❡❞✉③✐♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ♣❛r❛ s❡r❡♠ ❛♥❛❧✐s❛❞❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❜✉s❝❛ ❡ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛ r❡s♦❧✉✲ çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❞✐♠❡♥sõ❡s ♠❛✐s ❡❧❡✈❛❞❛s✳ ❉❡t❛❧❤❡s s♦❜r❡ ♠ét♦❞♦s ❞❡ ❛❣r✉♣❛♠❡♥t♦ ✐♥❝❧✉✐♥❞♦ ♦ ❡st✉❞♦ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡s ❞❡ t❡♠♣♦✱ ✈❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s ♣♦❞❡♠ s❡r ♦❜s❡r✈❛❞♦s ❡♠ ❬❏❛✐♥ ❡t ❛❧✳✱ ✶✾✾✾❪✳

✷✳✶✳✶ ▼ét♦❞♦s ❞❡ ❆❣r✉♣❛♠❡♥t♦

❖s ♠ét♦❞♦s ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♣♦❞❡♠ s❡r ❞✐✈✐❞✐❞♦s ❡♠ ❍✐❡rárq✉✐❝♦s ❡ P❛rt✐❝✐♦♥❛✐s ✭❋✐✲ ❣✉r❛ ✷✳✶✮✳ ▼ét♦❞♦s ❤✐❡rárq✉✐❝♦s ❝♦♥str♦❡♠ ✉♠❛ ár✈♦r❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦s✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞❡♥❞♦❣r❛♠❛✳ ❆ ♠❛✐♦r✐❛ ❞♦s ❛❧❣♦r✐t♠♦s ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❤✐❡rárq✉✐❝♦s sã♦ ✈❛r✐❛✲ çõ❡s ❞♦s ❛❧❣♦r✐t♠♦s s✐♥❣❧❡✲❧✐♥❦ ❬❙♥❡❛t❤✱ ✷✵✵✺❪ ❡ ❝♦♠♣❧❡t❡✲❧✐♥❦ ❬❑✐♥❣✱ ✶✾✻✼❪✳

❆❧❣♦r✐t♠♦s ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♣❛rt✐❝✐♦♥❛✐s ♣r♦❝✉r❛♠ ❡♥❝♦♥tr❛r ❛ ♠❡❧❤♦r ♣❛rt✐çã♦ ❞♦s n ♦❜❥❡t♦s ❞❡ ❡♥tr❛❞❛ ❡♠ k ❣r✉♣♦s ❞❡ s❛í❞❛✱ ♦♥❞❡ k é ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❡♥tr❛❞❛ ❞♦

❛❧❣♦r✐t♠♦✳ P❛r❛ ❣r❛♥❞❡s ❜❛s❡s ❞❡ ❞❛❞♦s✱ ♦s ❛❧❣♦r✐t♠♦s ♣❛rt✐❝✐♦♥❛✐s ❛♣r❡s❡♥t❛♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦✱ q✉❛♥❞♦ ❝♦♠♣❛r❛❞♦ ❛♦s ❤✐❡rárq✉✐❝♦s✱ ♣♦✐s✱ ♥❡ss❡ ❝❛s♦✱ ❛ ❝♦♥str✉çã♦ ❞♦ ❞❡♥❞♦❣r❛♠❛ ❡❧❡✈❛✱ ❡♠ ❞❡♠❛s✐❛✱ ♦ ❝✉st♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❛ t❛r❡❢❛ ❞❡ ❛❣r✉♣❛♠❡♥t♦✳

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✷✳✶✳ ❆❣r✉♣❛♠❡♥t♦ ❞❡ ❉❛❞♦s ✼

❋✐❣✉r❛ ✷✳✶✳ ❚❛①♦♥♦♠✐❛ ❞❡ té❝♥✐❝❛s ❞❡ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ❞❛❞♦s✳

✷✳✶✳✶✳✶ ▼ét♦❞♦s P❛rt✐❝✐♦♥❛✐s

❆s té❝♥✐❝❛s ♣❛rt✐❝✐♦♥❛✐s ♥♦r♠❛❧♠❡♥t❡ ♣r♦❞✉③❡♠ ❣r✉♣♦s ♣♦r ♠❡✐♦ ❞❛ ♦t✐♠✐③❛çã♦ ❞❡ ❢✉♥çõ❡s ♦❜❥❡t✐✈♦✳ P♦r ❡①❡♠♣❧♦✱ ♦ ❛❧❣♦r✐t♠♦ ♠❛✐s ♣♦♣✉❧❛r ❞♦ ♠ét♦❞♦ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♣❛rt✐❝✐♦♥❛❧✱ ♦ ❦✲♠❡❛♥s ❬▼❛❝◗✉❡❡♥✱ ✶✾✻✼❪✱ t❡♥t❛ ♠✐♥✐♠✐③❛r ❛ s♦♠❛ ❞♦ ❡rr♦ q✉❛❞rát✐❝♦ ❡♥tr❡ ♦s ♦❜❥❡t♦s ❡ ♦ ❝❡♥tr♦ ❞♦ ❣r✉♣♦✳ ❏á ♦ ❦✲♠❡❞✐❛♥s✱ ✉♠❛ ✈❛r✐❛çã♦ ❞♦ ❦✲♠❡❛♥s✱ t❡♥t❛ ♠✐♥✐♠✐③❛r ❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞♦s ♦❜❥❡t♦s à s✉❛ ♠❡❞✐❛♥❛ ♠❛✐s ♣ró①✐♠❛✳ ❊ss❛ t❛♠❜é♠ é ❛ ✐♥t❡♥çã♦ ❞♦ ♠ét♦❞♦ P❆▼✱ ❜❛s❡❛❞♦ ♥♦ ❦✲♠❡❞✐❛♥s✳ ❖s ❞❡t❛❧❤❡s ❞❡ss❡s três ❛❧❣♦r✐t♠♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ❛❜❛✐①♦✳

❛✮ ❦✲♠❡❛♥s

❆ ✐❞é✐❛ ❝❡♥tr❛❧ ❞♦ ❦✲♠❡❛♥s é ❡♥❝♦♥tr❛r k ❞✐❢❡r❡♥t❡s ❝❡♥tró✐❞❡s ❡♥tr❡ s❡✉s ♦❜❥❡t♦s

❡ ❛❣r✉♣❛r ♦s ❞❡♠❛✐s ♦❜❥❡t♦s ❝♦♠ ❜❛s❡ ♥♦ q✉❛❞r❛❞♦ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦ ♦❜❥❡t♦ ❡ ♦ ❝❡♥tró✐❞❡ ♠❛✐s ♣ró①✐♠♦ ❛ ❡❧❡✱ ♦♥❞❡ k é ♦ ♥ú♠❡r♦ ❞❡ ❣r✉♣♦s✱ ♣❛ss❛❞♦ ❝♦♠♦ ♣❛râ♠❡tr♦

♣❡❧♦ ✉s✉ár✐♦✳

▲✐st❛❞♦ ♣♦r ❲✉ ✫ ❑✉♠❛r ❬✷✵✵✾❪ ❡♥tr❡ ♦s ❞❡③ ♠❛✐s ✐♥✢✉❡♥t❡s ❛❧❣♦r✐t♠♦s ❞❛ ♠✐♥❡✲ r❛çã♦ ❞❡ ❞❛❞♦s✱ ♦ ❦✲♠❡❛♥s ❞❡✈❡ s✉❛ ❣r❛♥❞❡ ♣♦♣✉❧❛r✐❞❛❞❡ ♣♦r s✉❛ ❡✜❝✐ê♥❝✐❛ ❡♠ r❡❛❧✐③❛r ❛❣r✉♣❛♠❡♥t♦s ❡♠ ❜❛s❡s ❞❡ ❞❛❞♦s ♥✉♠❡r♦s❛s✱ ❢❛❝✐❧✐❞❛❞❡ ❞❡ ✐♠♣❧❡♠❡♥t❛çã♦ ❡ ❝♦♠♣❧❡✲ ①✐❞❛❞❡ O(k)✱ ♦♥❞❡ k é ♦ ♥ú♠❡r♦ ❞❡ ❣r✉♣♦s✳ ❖✉tr❛ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ❦✲♠❡❛♥s q✉❡

❝♦♥tr✐❜✉✐ ♣❛r❛ s✉❛ ♣♦♣✉❧❛r✐❞❛❞❡ é ♦ ❢❛t♦ ❞❡❧❡ s❡r ✉s❛❞♦ ♣❛r❛ ✐♥✐❝✐❛❧✐③❛r ❛❧❣♦r✐t♠♦s ❞❡ ❛❣r✉♣❛♠❡♥t♦s ♠❛✐s ❝♦♠♣❧❡①♦s ❡ ❝❛r♦s ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✳

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✽ ❈❛♣ít✉❧♦ ✷✳ ❊st❛❞♦ ❞❛ ❆rt❡

❆❧❣♦r✐t♠♦ ✶✿ ❦✲♠❡❛♥s✭✮

✶ ❋❛ç❛ ❛❧❡❛t♦r✐❛♠❡♥t❡ ✉♠❛ ♣❛rt✐çã♦ ✐♥✐❝✐❛❧ ♥♦s ❞❛❞♦s ❡♠ k ❣r✉♣♦s{C1,✳✳✳✱ Ck}❀ ✷ ❈❛❧❝✉❧❡ ♦s ❝❡♥tró✐❞❡swi= _k1_iPk

i

j=1 wij❀ ✐❂✶✱ ✳✳✳✱k❀ ✸ r❡♣❡❛t

✹ ❆tr✐❜✉❛ ♦s ♦❜❥❡t♦s wij ❛♦ ❣r✉♣♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ❝❡♥tró✐❞❡ ♠❛✐s ♣ró①✐♠♦❀ ✺ ❘❡❝❛❧❝✉❧❡ ♦s ❝❡♥tró✐❞❡s ❞♦s ❣r✉♣♦s✱ ❛ ♣❛rt✐r ❞♦s ♦❜❥❡t♦s r❡❛❧♦❝❛❞♦s❀

✻ ✉♥t✐❧ ♥ã♦ ♦❝♦rr❛ ❛❧t❡r❛çã♦ s✐❣♥✐✜❝❛t✐✈❛ ♥♦s ❝❡♥tró✐❞❡s❀

❜✮ ❦✲♠❡❞✐❛♥s

❖ ❛❧❣♦r✐t♠♦ ❦✲♠❡❞✐❛♥s é ✉♠❛ ❡✈♦❧✉çã♦ ❞♦ ❦✲♠❡❛♥s✱ ♦♥❞❡ ♦s ❝❡♥tró✐❞❡s sã♦ s✉❜s✲ t✐t✉í❞♦s ♣♦r ♠❡❞ó✐❞❡s✱ ❧♦❝❛❧✐③❛❞♦s ♣ró①✐♠♦s ❛♦ ❝❡♥tr♦ ❞♦ ❣r✉♣♦ ✭❛❣♦r❛ ✉♠❛ ♠❡❞✐❛♥❛✮ ❛♦ ✐♥✈és ❞❡ ✉♠ ❝❡♥tr♦ ♠é❞✐♦ ✭♠é❞✐❛✮✳ ❖ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣r✉♣♦s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❝❛❞❛ ❣r✉♣♦ t❡♥❤❛ ✉♠ ♣♦♥t♦ ❝❡♥tr❛❧ ♠❛✐s r❡♣r❡s❡♥t❛t✐✈♦ ❡♠ r❡❧❛çã♦ ❛ ❛❧❣✉♠❛ ♠❡❞✐❞❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❞✐stâ♥❝✐❛✳ ❊st❡ ♠ét♦❞♦ é ♠❛✐s r♦❜✉st♦ ♣❛r❛ ♦✉t❧✐❡rs✶ _{♣♦r ✉s❛r ♠❡❞✐❛♥❛s ❡ ♣♦r ♠✐♥✐♠✐③❛r ✉♠❛ s♦♠❛ ❞❡ ❞✐stâ♥❝✐❛s ✭❞✐ss✐♠✐❧❛r✐❞❛✲}

❞❡s✮ ❛♦ ✐♥✈és ❞♦ ❞❡s✈✐♦ q✉❛❞rát✐❝♦ ♠é❞✐♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉t✐❧✐③❛ ✉♠ t❡♠♣♦ ♠❛✐♦r ❞❡ ♣r♦❝❡ss❛♠❡♥t♦ ❬❘♦❞r✐❣✉❡s✱ ✷✵✵✾❀ ❑❛✉❢♠❛♥ ✫ ❘♦✉ss❡❡✉✇✱ ✶✾✽✼❀ ❙♣❛❡t❤✱ ✶✾✽✵❪✳

❖ ♣s❡✉❞♦❝ó❞✐❣♦ ❞♦ ❦✲♠❡❞✐❛♥s ♣♦❞❡ s❡r ✈✐st♦ ♥♦ ❆❧❣♦r✐t♠♦ ✷✿

❆❧❣♦r✐t♠♦ ✷✿ ❦✲♠❡❞✐❛♥s✭✮

✶ r❡♣❡❛t

✷ ❙❡❧❡❝✐♦♥❛r✱ ❞❡♥tr❡ ♦sn♦❜❥❡t♦s ❞✐s♣♦♥í✈❡✐s ❞❡X✱ ♦sk♦❜❥❡t♦s q✉❡ ❞❡✜♥❡♠ ✉♠

❝♦♥❥✉♥t♦M ❞❡ ♠❡❞✐❛♥❛s❀

✸ ❆ss♦❝✐❛r ♦s ✭n✲k✮ ♦❜❥❡t♦s r❡st❛♥t❡s à s✉❛ ♠❡❞✐❛♥❛ ♠❛✐s ♣ró①✐♠❛❀

✹ ❙✉❜st✐t✉✐r ❛sk♠❡❞✐❛♥❛s ❞❡ ❢♦r♠❛ ❛ ♠✐♥✐♠✐③❛r ❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❞♦s ✭n✲k✮ ♦❜❥❡t♦s

à s✉❛ ♠❡❞✐❛♥❛ ♠❛✐s ♣ró①✐♠❛✳

✺ ✉♥t✐❧ ♥ã♦ ♦❝♦rr❛ ❛❧t❡r❛çã♦ s✐❣♥✐✜❝❛t✐✈❛ ♥♦s ♠❡❞ó✐❞s❀

❝✮ P❛rt✐t✐♦♥✐♥❣ ❆r♦✉♥❞ ▼❡❞♦✐❞s ✭P❆▼✮

❖✉tr♦ ❛❧❣♦r✐t♠♦ ❝❧áss✐❝♦ é ♦ ♣❛rt✐❝✐♦♥❛♠❡♥t♦ ❡♠ t♦r♥♦ ❞❡ ♠❡❞✐❛♥❛s ✭P❆▼✱ ❞♦ ✐♥❣❧ês P❛rt✐t✐♦♥✐♥❣ ❆r♦✉♥❞ ▼❡❞♦✐❞s✮ ❬❑❛✉❢♠❛♥ ✫ ❘♦✉ss❡❡✉✇✱ ✶✾✽✼❪✱ ❜❛s❡❛❞♦ ♥♦ ❦✲ ♠❡❞✐❛♥s✳ ❆ s❡❧❡çã♦ ❞❡ ❦ ♠❡❞♦✐❞❡s é ❡①❡❝✉t❛❞❛ ❡♠ ❞✉❛s ❢❛s❡s✳ ◆❛ ♣r✐♠❡✐r❛ ❢❛s❡✱ ✉♠ ❛❣r✉♣❛♠❡♥t♦ ✐♥✐❝✐❛❧ é ♦❜t✐❞♦ ♣❡❧❛ s✉❝❡ss✐✈❛ s❡❧❡çã♦ ❞❡ ♦❜❥❡t♦s ❛té q✉❡ k ♦❜❥❡t♦s✱ q✉❡

♠❡❧❤♦r r❡♣r❡s❡♥t❡♠ ♦s ❝❡♥tr♦s ❞❡ k ❣r✉♣♦s✱ t❡♥❤❛♠ s✐❞♦ ❡♥❝♦♥tr❛❞♦s✳ ❖ ♣r✐♠❡✐r♦

♦❜❥❡t♦ é ❛q✉❡❧❡ ♣❛r❛ ♦ q✉❛❧ ❛ s♦♠❛ ❞❛s ❞✐ss✐♠✐❧❛r✐❞❛❞❡s ❡♠ r❡❧❛çã♦ ❛ t♦❞♦s ♦s ♦✉tr♦s

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✷✳✶✳ ❆❣r✉♣❛♠❡♥t♦ ❞❡ ❉❛❞♦s ✾

♦❜❥❡t♦s é ❛ ♠❡♥♦r ♣♦ssí✈❡❧✳ ❙✉❜s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❝❛❞❛ ♣❛ss♦✱ P❆▼ s❡❧❡❝✐♦♥❛ ♦ ♦❜❥❡t♦ q✉❡ ❞✐♠✐♥✉✐ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ✭s♦♠❛ ❞❡ ❞✐ss✐♠✐❧❛r✐❞❛❞❡s✮ t❛♥t♦ q✉❛♥t♦ ♣♦ssí✈❡❧✳

◆❛ s❡❣✉♥❞❛ ❢❛s❡✱ t♦❞♦s ♦s ♦❜❥❡t♦s sã♦ ❛♥❛❧✐s❛❞♦s ❡ é ❡s❝♦❧❤✐❞♦ ❝♦♠♦ ♥♦✈♦ ❝❡♥tr♦ ❞❡ ❝❛❞❛ ❣r✉♣♦ ❛q✉❡❧❡ ♦❜❥❡t♦ q✉❡ ♠✐♥✐♠✐③❛ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ t❛♥t♦ q✉❛♥t♦ ♣♦ssí✈❡❧✳ ❊ss❛ ❢✉♥çã♦ é s♦♠❛ ❞❡ t♦❞❛s ❛s ❞✐ss✐♠✐❧❛r✐❞❛❞❡s ❡♥tr❡ ♦ ♦❜❥❡t♦ Oi ❡ ♦s ❞❡♠❛✐s ♦❜❥❡t♦s

Oj ❞♦ ❣r✉♣♦ ❛ q✉❡ ❡❧❡ ♣❡rt❡♥❝❡✳

❖ ❛❧❣♦r✐t♠♦ P❆▼ ❢✉♥❝✐♦♥❛ ❡❢❡t✐✈❛♠❡♥t❡ ❝♦♠ ♣❡q✉❡♥❛s ❜❛s❡s ❞❡ ❞❛❞♦s ❬❘♦❞r✐❣✉❡s✱ ✷✵✵✾❀ ❑❛✉❢♠❛♥ ✫ ❘♦✉ss❡❡✉✇✱ ✷✵✵✺❪✳

✷✳✶✳✷ ❆♥á❧✐s❡ ❞❡ ❆❣r✉♣❛♠❡♥t♦

❘♦♠❡s❜✉r❣ ❬✷✵✵✹❪ ❛♣r❡s❡♥t❛ ♦s ♣❛ss♦s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ q✉❡ ♣♦❞❡♠ s❡r r❡s✉♠✐❞♦s ❝♦♠♦✿

• ❖❜t❡r ❛ ♠❛tr✐③ ❞❡ ❞❛❞♦s❀

• ❈♦♠♣✉t❛r ❛ ♠❛tr✐③ ❞❡ s✐♠✐❧❛r✐❞❛❞❡❀

• ❊①❡❝✉t❛r ♦ ♠ét♦❞♦ ❞❡ ❛❣r✉♣❛♠❡♥t♦❀

• ❈♦♠♣✉t❛r ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦❀

❆ ♠❛tr✐③ ❞❡ ❞❛❞♦s é ❢♦r♠❛❞❛ ♣♦r ♦❜❥❡t♦s ❡ s❡✉s ❛tr✐❜✉t♦s✳ ❆ ♠❛tr✐③ ❞❡ ❞✐ss✐♠✐✲ ❧❛r✐❞❛❞❡ ✭t❛♠❜é♠ ❝❤❛♠❛❞❛ ♠❛tr✐③ ❞❡ ❞✐stâ♥❝✐❛s✮ ❝♦♥té♠ ♦ ❣r❛✉ ❞❡ s✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ❝❛❞❛ ♣❛r ❞❡ ♦❜❥❡t♦s✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ♦❜❥❡t♦s✱ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ s✐♠✐❧❛✲ r✐❞❛❞❡ é ❝❛❧❝✉❧❛❞♦ ❝♦♠ ♦ ✉s♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❞❡ s✐♠✐❧❛r✐❞❛❞❡✱ ❛ ♣❛rt✐r ❞♦s ✈❛❧♦r❡s ❞❡ s❡✉s ❛tr✐❜✉t♦s ❛r♠❛③❡♥❛❞♦s ♥❛ ♠❛tr✐③ ❞❡ ❞❛❞♦s✳ ◆❛ ♠❛tr✐③ ❞❡ s✐♠✐❧❛r✐❞❛❞❡M S✱ ❧✐♥❤❛s

❡ ❝♦❧✉♥❛s r❡♣r❡s❡♥t❛♠ ♦❜❥❡t♦s ❡ ♦ ✈❛❧♦r ❞❡ M S[ij] é ❛ ❞✐ss✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ♦s ♦❜❥❡t♦s

i ❡j✳ ❊①✐st❡♠ ❞✐✈❡rs❛s ❢♦r♠❛s ♣❛r❛ s❡ ❝❛❧❝✉❧❛r ♦ q✉ã♦ s✐♠✐❧❛r é ✉♠ ♣❛r ❞❡ ♦❜❥❡t♦s✳ ◆❛

❙❡çã♦ ✷✳✶✳✷✳✶ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛s ♠étr✐❝❛s ❞❡ s✐♠✐❧❛r✐❞❛❞❡ ✉t✐❧✐③❛❞❛s ♥❡st❡ tr❛❜❛❧❤♦✳ ❆♦ ❡①❡❝✉t❛r ♦ ♠ét♦❞♦ ❞❡ ❛❣r✉♣❛♠❡♥t♦✱ ❝❛❞❛ ♦❜❥❡t♦ ❞❛ ♠❛tr✐③ ❞❡ ❞❛❞♦s é ❛ss♦❝✐✲ ❛❞♦ ❛ ✉♠ ú♥✐❝♦ ❣r✉♣♦✳ P♦st❡r✐♦r♠❡♥t❡✱ ♥❛ ❢❛s❡ ❞❡ ❝♦♠♣✉t❛r ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦✱ ✉s❛✲s❡ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♣❛r❛ ❛✈❛❧✐❛r ♦ ❛❣r✉♣❛♠❡♥t♦ ❣❡r❛❞♦✳

✷✳✶✳✷✳✶ ▼❡❞✐❞❛s ❞❡ ❙✐♠✐❧❛r✐❞❛❞❡

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✶✵ ❈❛♣ít✉❧♦ ✷✳ ❊st❛❞♦ ❞❛ ❆rt❡

❞ij = L X

k=1

|aik−ajk|r !1r

✭✷✳✶✮

❖♥❞❡✱

dij é ❛ s✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ♦s ♦❜❥❡t♦s i ❡ j❀ ▲ é ❛ ❞✐♠❡♥sã♦ ❞♦s ❞❛❞♦s❀

aik ❡ajk sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ❦✲és✐♠♦ ❛tr✐❜✉t♦ ❞♦si✲és✐♠♦ ❡j✲és✐♠♦ ♦❜❥❡t♦sai ❡aj❀ r é ✉♠ ❢❛t♦r q✉❡ ❞❡t❡r♠✐♥❛ ❛ ♠étr✐❝❛ ❞♦ ❡s♣❛ç♦❀

❆s ❞✐stâ♥❝✐❛s ❡✉❝❧✐❞✐❛♥❛ ❡ ❈✐t② ❇❧♦❝❦ ✭t❛♠❜é♠ ❝❤❛♠❛❞❛ ▼❛♥❤❛tt❛♥✮ sã♦ ❝❛s♦s ❡s♣❡❝✐❛✐s ❞❛ ❢♦r♠✉❧❛çã♦ ❛❝✐♠❛ ♠❡♥s✐♦♥❛❞❛✳ ◗✉❛♥❞♦ r ❂ ✶✱ t❡♠✲s❡ ❛ ❞✐stâ♥❝✐❛ ❈✐t② ❇❧♦❝❦ ✭❡q✉❛çã♦ ✷✳✷✮✳ ◗✉❛♥❞♦ r ❂ ✷✱ t❡♠✲s❡ ❛ ❞✐stâ♥❝✐❛ ❡✉❝❧✐❞✐❛♥❛ ✭❡q✉❛çã♦ ✷✳✸✮✱ ❛ ♠❛✐s ❝♦♠✉♠ ♠❡❞✐❞❛ ❞❡ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✳

❞ij = L X

k=1

|aik−ajk| ✭✷✳✷✮

❞ij = v u u t L X k=1

|aik−ajk|2 ✭✷✳✸✮

❖✉tr❛ ♠étr✐❝❛ ❝♦♠✉♠❡♥t❡ ✉t✐❧✐③❛❞❛ é ♦ ✉s♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❝♦rr❡❧❛çã♦ ❝♦♠♦ ♠❡❞✐❞❛ ❞❡ s✐♠✐❧❛r✐❞❛❞❡✳ ❖s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❝♦rr❡❧❛çã♦ ♠❡❞❡♠ ❛ ❢♦rç❛ ❞♦ r❡❧❛❝✐♦♥❛♠❡♥t♦ ❡♥tr❡ ❞✉❛s ✈❛r✐á✈❡✐s✳ ❖ ❈♦s✐♥❡ é ✉♠ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ q✉❡ ❝❛❧❝✉❧❛ ❛ ❞✐stâ♥❝✐❛ ✈❡t♦r✐❛❧ ❡✉❝❧✐❞✐❛♥❛ ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s✱ ✉t✐❧✐③❛♥❞♦ ✈❛❧♦r❡s ❝♦♥t✐❞♦s ♥♦ ✐♥t❡r✈❛❧♦ ❬✲✶✱ ✶❪✳

❆ s✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s ✉t✐❧✐③❛♥❞♦ ❛ ❢✉♥çã♦ ❈♦s✐♥❡ é ❞❛❞❛ ♣❡❧❛ ❋ór♠✉❧❛ ✷✳✹ ❬❆♥❞❡r❜❡r❣✱ ✶✾✼✸❪✳

❉ij =

PL

k=1aikajk

q PL

k=1a 2 ik

PL k=1a

2 jk

✭✷✳✹✮

❖♥❞❡✱

Dij é ❛ s✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ♦s ♦❜❥❡t♦s i ❡j❀ ▲ é ❛ ❞✐♠❡♥sã♦ ❞♦s ❞❛❞♦s❀

aik ❡ajk sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ❦✲és✐♠♦ ❛tr✐❜✉t♦ ❞♦si✲és✐♠♦ ❡j✲és✐♠♦ ♦❜❥❡t♦sai ❡aj❀

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✷✳✷✳ ▼❡t❛❤❡✉ríst✐❝❛s ✶✶

♦ ♦❜❥❡t♦ é ❞❡ ✵♦_{✱ ❡♥q✉❛♥t♦} _D

ij❂✲✶ s✐❣♥✐✜❝❛ q✉❡ ♦s ✈❡t♦r❡s ❢♦r♠❛♠ ✉♠ â♥❣✉❧♦ ❞❡ ✾✵♦✳ ❈♦s✐♥❡ é ✉♠❛ ♠étr✐❝❛ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞❛ ♥❛ r❡❝✉♣❡r❛çã♦ ❞❡ ✐♥❢♦r♠❛çã♦ t❡①t✉❛❧✳

❖✉tr♦ ♠ét♦❞♦ ✉s✉❛❧♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ♣❛r❛ ♠❡❞✐r ❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ✈❛r✐á✈❡✐s é ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❝♦rr❡❧❛çã♦ ❧✐♥❡❛r ❞❡ P❡❛rs♦♥ ❬▲✐r❛ ✫ ◆❡t♦✱ ✷✵✵✽❪✱ ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ❡♠ ❛♥á❧✐s❡ ❞❡ ❛❣r✉♣❛♠❡♥t♦✳ ❙❡❣✉♥❞♦ ❆♥❞❡r❜❡r❣ ❬✶✾✼✸❪✱ ❡ss❡ ❝♦❡✜❝✐❡♥t❡ ♠♦str❛ ❛ q✉❛♥t✐✲ ❞❛❞❡ ❞❡ ❞✐st♦rçã♦ ❝❛✉s❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❛❣r✉♣❛♠❡♥t♦ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ s✐♠✐❧❛r✐✲ ❞❛❞❡ ❡♥tr❡ ♣❛r❡s ❞❡ ♦❜❥❡t♦s✳ ❖ ❝♦❡✜❝✐❡♥t❡ ❞❡ P❡❛rs♦♥ é ❝❛❧❝✉❧❛❞♦ ✉t✐❧✐③❛♥❞♦ ❋ór♠✉❧❛ ✷✳✺ ❞❡s❝r✐t❛ ❛ s❡❣✉✐r✿

Dij =

LPL

k=1aikajk

−PL k=1aik

PL k=1ajk

r

LPL k=1a

2 ik

−PL k=1aik

2r

LPL k=1a

2 jk

−PL k=1ajk

2

✭✷✳✺✮

❖♥❞❡✱

Dij é ❛ s✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ♦s ♦❜❥❡t♦s i❡ j❀ ▲ é ❛ ❞✐♠❡♥sã♦ ❞♦s ❞❛❞♦s❀

aik ❡ajk sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ❦✲és✐♠♦ ❛tr✐❜✉t♦ ❞♦si✲és✐♠♦ ❡j✲és✐♠♦ ♦❜❥❡t♦sai ❡aj❀

❆ ❝♦rr❡❧❛çã♦ ❞❡ P❡❛rs♦♥ ♠❡❞❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s ❡ ♣r♦❞✉③ ✉♠ ✈❛❧♦r ♥♦ ✐♥t❡r✈❛❧♦ ❬✲✶✱ ✶❪✳ ◆❡ss❛ ♠❡❞✐❞❛✱ ♦ ✈❛❧♦r ✶ s✐❣♥✐✜❝❛ ✉♠❛ ❛ss♦❝✐❛çã♦ ♣❡r❢❡✐t❛ ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s✱ ❡♥q✉❛♥t♦ ♦ ✈❛❧♦r ✲✶ s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ✉♠❛ r❡❧❛çã♦ ❧✐♥❡❛r ♥❡❣❛t✐✈❛ ♣❡r❢❡✐t❛ ❡♥tr❡ ❞♦✐s ♦❜❥❡t♦s✳ ❆ss✐♠ ❝♦♠♦ ♥❛ ❝♦rr❡❧❛çã♦ ❈♦s✐♥❡✱ ❝❛❧❝✉❧❛✲s❡ ❛ ❞✐ss✐♠✐❧❛r✐❞❛❞❡ ❡♥tr❡ ♦s ♦❜❥❡t♦s i ❡ j ♣♦rdij = 1− |Dij|✳

✷✳✷ ▼❡t❛❤❡✉ríst✐❝❛s

❆ ♣❛❧❛✈r❛ ❤❡✉ríst✐❝❛ t❡♠ s✉❛ ♦r✐❣❡♠ ♥❛ ●ré❝✐❛ ❛♥t✐❣❛ ❡ ❞❡r✐✈❛ ❞❛ ♣❛❧❛✈r❛ ❤❡✉r✐s❦❡✐♥✱ q✉❡ s✐❣♥✐✜❝❛ ✏❛ ❛rt❡ ❞❡ ❞❡s❝♦❜r✐r ♥♦✈❛s ❡str❛té❣✐❛s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s✑✳ ❖ t❡r♠♦ ✏♠❡t❛❤❡✉ríst✐❝❛✑ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ●❧♦✈❡r ❬✶✾✽✻❪✳ ❖ s✉✜①♦ ♠❡t❛✱ t❛♠❜é♠ é ✉♠❛ ♣❛❧❛✈r❛ ❣r❡❣❛✱ q✉❡ s✐❣♥✐✜❝❛ ✏♠❡t♦❞♦❧♦❣✐❛ ❞❡ ♥í✈❡❧ s✉♣❡r✐♦r✑✳ ❆s ♠❡t❛❤❡✉ríst✐❝❛s ❧✐❞❛♠ ❝♦♠ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠❜✐♥❛tór✐❛✱ ♦♥❞❡ ❤❡✉ríst✐❝❛s ❝❧áss✐❝❛s ♥ã♦ sã♦ ❡✜❝✐❡♥t❡s ❡ ❡✜❝❛③❡s✷_{❬❚❛❧❜✐✱ ✷✵✵✾❀ ❖s♠❛♥ ✫ ❑❡❧❧②✱ ✶✾✾✻❪✳}

❆s ♠❡t❛❤❡✉ríst✐❝❛s ❣❛♥❤❛r❛♠ ♠✉✐t❛ ♣♦♣✉❧❛r✐❞❛❞❡ ♥❛s ú❧t✐♠❛s ❞✉❛s ❞é❝❛❞❛s ❡ r❡♣r❡s❡♥t❛♠ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ té❝♥✐❝❛s ❞❡ ♦t✐♠✐③❛çã♦ q✉❡ ♦❜té♠ ❜♦❛s s♦❧✉çõ❡s✳ ◆❡ss❡s

✷_{❯♠ ❛❧❣♦r✐t♠♦ é ❡✜❝❛③ q✉❛♥❞♦ r❡s♦❧✈❡ ♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ♦ q✉❛❧ ❢♦✐ ♣r♦♣♦st♦✳ ❯♠ ❛❧❣♦r✐t♠♦ é ❡✜❝✐✲}

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✶✷ ❈❛♣ít✉❧♦ ✷✳ ❊st❛❞♦ ❞❛ ❆rt❡

♠ét♦❞♦s ❛ ❣❛r❛♥t✐❛ ❞❡ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ót✐♠❛s é s❛❝r✐✜❝❛❞❛ ❡♠ ♣r♦❧ ❞❛ ♦❜t❡♥çã♦ ❞❡ ❜♦❛s s♦❧✉çõ❡s ❡♠ ♠❡♥♦r t❡♠♣♦ ❬❇❧✉♠ ✫ ❘♦❧✐✱ ✷✵✵✸❀ ❚❛❧❜✐✱ ✷✵✵✾❪✳

◆❛ ❞❡✜♥✐çã♦ ❞❡ ❖s♠❛♥ ✫ ❑❡❧❧② ❬✶✾✾✻❪✱ ♠❡t❛❤❡✉ríst✐❝❛ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ q✉❡ ❣✉✐❛ ✉♠❛ ❤❡✉ríst✐❝❛ s✉❜♦r❞✐♥❛❞❛✱ ❝♦♠❜✐♥❛♥❞♦ ✐♥t❡❧✐❣❡♥t❡♠❡♥t❡ ❞✐❢❡r❡♥t❡s ❝♦♥❝❡✐t♦s ❞❡ ❡①♣❧♦r❛çã♦ ❡ ❛♣r♦✈❡✐t❛♠❡♥t♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❜✉s❝❛✱ ❛ ✜♠ ❞❡ ❡♥❝♦♥tr❛r ❡✜❝✐❡♥t❡♠❡♥t❡ s♦❧✉çõ❡s ♣ró①✐♠❛s ❞❛ ✐❞❡❛❧✳

P❛r❛ ●❧♦✈❡r ✫ ❑♦❝❤❡♥❜❡r❣❡r ❬✷✵✵✸❪✱ ♠❡t❛❤❡✉ríst✐❝❛s sã♦ ♠ét♦❞♦s q✉❡ ♦rq✉❡str❛♠ ✉♠❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♠❡❧❤♦r✐❛ ❞❡ ♣r♦❝❡❞✐♠❡♥t♦s ❞❡ ❜✉s❝❛ ❧♦❝❛❧ ❡ ❡str❛té❣✐❛s ❞❡ ❛❧t♦ ♥í✈❡❧ ♣❛r❛ ❝r✐❛r ♣r♦❝❡ss♦s ❝❛♣❛③❡s ❞❡ ❡s❝❛♣❛r ❞♦s ót✐♠♦s ❧♦❝❛✐s ❡ ❞❡s❡♠♣❡♥❤❛r ✉♠❛ ❜✉s❝❛ ♠❛✐s r♦❜✉st❛ ♥♦ ❡s♣❛ç♦ ❞❡ s♦❧✉çõ❡s✳

❙❡❣✉♥❞♦ ❇❧✉♠ ✫ ❘♦❧✐ ❬✷✵✵✸❪✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♠❡t❛❤❡✉ríst✐❝❛s sã♦✿

• ▼❡t❛❤❡✉ríst✐❝❛s sã♦ ❡str❛té❣✐❛s q✉❡ ❞✐r❡❝✐♦♥❛♠ ♦ ♣r♦❝❡ss♦ ❞❡ ❜✉s❝❛❀

• ❖ ♦❜❥❡t✐✈♦ é ❡①♣❧♦r❛r ❞❡ ❢♦r♠❛ ❡✜❝✐❡♥t❡ ♦ ❡s♣❛ç♦ ❞❡ ❜✉s❝❛ ♣❛r❛ ❡♥❝♦♥tr❛r ♦✉ s❡ ❛♣r♦①✐♠❛r ❞❛s ♠❡❧❤♦r❡s s♦❧✉çõ❡s❀

• ❆s té❝♥✐❝❛s q✉❡ ❝♦♥st✐t✉❡♠ ♦s ❛❧❣♦r✐t♠♦s ❞❡ ♠❡t❛❤❡✉ríst✐❝❛ ✈❛r✐❛♠ ❞❡ s✐♠♣❧❡s ♣r♦❝❡❞✐♠❡♥t♦s ❞❡ ❜✉s❝❛ ❧♦❝❛❧ ❛té ❝♦♠♣❧❡①♦s ♣r♦❝❡ss♦s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠❀

• ❖s ❛❧❣♦r✐t♠♦s ❞❡ ♠❡t❛❤❡✉ríst✐❝❛ ❜✉s❝❛♠ ❜♦❛s s♦❧✉çõ❡s ❡✱ ❣❡r❛❧♠❡♥t❡✱ sã♦ ♥ã♦✲ ❞❡t❡r♠✐♥íst✐❝♦s✸_❀

• ❖s ❛❧❣♦r✐t♠♦s ❞❡ ♠❡t❛❤❡✉ríst✐❝❛ ♣♦❞❡♠ ✐♥❝♦r♣♦r❛r ♠❡❝❛♥✐s♠♦s q✉❡ ❡✈✐t❡♠ q✉❡ ♦s ♠❡s♠♦s ✜q✉❡♠ ♣r❡s♦s ❡♠ ár❡❛s r❡str✐t❛s ❞♦ ❡s♣❛ç♦ ❞❡ ❜✉s❝❛❀

• ▼❡t❛❤❡✉ríst✐❝❛s ♣♦❞❡♠ ❢❛③❡r ✉s♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❡s♣❡❝í✜❝♦s✱ ♥❛ ❢♦r♠❛ ❞❡ ❤❡✉✲ ríst✐❝❛s q✉❡ sã♦ ❝♦♥tr♦❧❛❞❛s ♣❡❧❛ ❡str❛té❣✐❛ ❞❡ ♥í✈❡❧ s✉♣❡r✐♦r✳

◆❛ ♣rát✐❝❛✱ ❛s ♠❡t❛❤❡✉ríst✐❝❛s ❢♦r♥❡❝❡♠ ♠♦❞❡❧♦s ❣❡♥ér✐❝♦s q✉❡ ♣❡r♠✐t❡♠ ❝r✐❛r ❛❧❣♦r✐t♠♦s ❤í❜r✐❞♦s ❛tr❛✈és ❞❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ❞✐❢❡r❡♥t❡s ❝♦♥❝❡✐t♦s ❞❡r✐✈❛❞♦s ❞❛s ❤❡✉✲ ríst✐❝❛s ❝❧áss✐❝❛s ❝♦♠♦ ❛ ✐♥t❡❧✐❣ê♥❝✐❛ ❛rt✐✜❝✐❛❧✱ ❡✈♦❧✉çã♦ ❜✐♦❧ó❣✐❝❛✱ ❡♥tr❡ ♦✉tr♦s✳

❆❧❣✉♠❛s ♠❡t❛❤❡✉ríst✐❝❛s ❛♠♣❧❛♠❡♥t❡ ❞✐❢✉♥❞✐❞❛s sã♦✿ ❙✐♠✉❧❛t❡❞ ❆♥♥❡❛❧✐♥❣ ❬❱❛♥ ▲❛❛r❤♦✈❡♥ ✫ ❆❛rts✱ ✶✾✽✼❪✱ ❆❧❣♦r✐t♠♦s ●❡♥ét✐❝♦s ❬❉❛✈✐s ✫ ▼✐t❝❤❡❧❧✱ ✶✾✾✶❪✱ ●❘❆❙P

✸_{❆❧❣♦r✐t♠♦s ❞❡t❡r♠✐♥íst✐❝♦s r❡s♦❧✈❡♠ ♦ ♣r♦❜❧❡♠❛ ❝♦♠ ✉♠❛ ❞❡❝✐sã♦ ❡①❛t❛ ❡ s❡♠♣r❡ r❡t♦r♥❛♠ ✉♠❛}