❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦
❯♠ ❛❧❣♦r✐t♠♦ ❛❧❣é❜r✐❝♦ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❛ ❉✐stâ♥❝✐❛
❞❡ ❚r❛♥s♣♦s✐çã♦ ❡♠ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s
▲✉✐③ ❆✉❣✉st♦ ●❛r❝✐❛ ❞❛ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❝♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ ❡♠ ■♥❢♦r♠át✐❝❛
❖r✐❡♥t❛❞♦r❛
Pr♦❢✳❛ ❉r✳❛ ▼❛r✐❛ ❊♠í❧✐❛ ▼✳ ❚✳ ❲❛❧t❡r
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ✖ ❯♥❇ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ▼❡str❛❞♦ ❡♠ ■♥❢♦r♠át✐❝❛
❈♦♦r❞❡♥❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ P❡③③✉♦❧ ❏❛❝♦❜✐
❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❝♦♠♣♦st❛ ♣♦r✿
Pr♦❢✳❛ ❉r✳❛ ▼❛r✐❛ ❊♠í❧✐❛ ▼✳ ❚✳ ❲❛❧t❡r ✭❖r✐❡♥t❛❞♦r❛✮ ✖ ❈■❈✴❯♥❇ Pr♦❢✳ ❉r✳ ◆♦r❛✐ ❘♦♠❡✉ ❘♦❝❝♦ ✖ ▼❆❚✴❯♥❇
Pr♦❢✳ ❉r✳ ❩❛♥♦♥✐ ❉✐❛s ✖ ■❈✴❯♥✐❝❛♠♣
❈■P ✖ ❈❛t❛❧♦❣❛çã♦ ■♥t❡r♥❛❝✐♦♥❛❧ ♥❛ P✉❜❧✐❝❛çã♦
❙✐❧✈❛✱ ▲✉✐③ ❆✉❣✉st♦ ●❛r❝✐❛ ❞❛✳
❯♠ ❛❧❣♦r✐t♠♦ ❛❧❣é❜r✐❝♦ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❛ ❉✐stâ♥❝✐❛ ❞❡ ❚r❛♥s♣♦s✐çã♦ ❡♠ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s ✴ ▲✉✐③ ❆✉❣✉st♦ ●❛r❝✐❛ ❞❛ ❙✐❧✈❛✳ ❇r❛sí❧✐❛ ✿ ❯♥❇✱ ✷✵✶✸✳
✶✸✸ ♣✳ ✿ ✐❧✳ ❀ ✷✾✱✺ ❝♠✳
❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✖ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❇r❛sí❧✐❛✱ ✷✵✶✸✳
✶✳ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s✱ ✷✳ Pr♦❜❧❡♠❛ ❞❛ ❉✐stâ♥❝✐❛ ❞❡ ❚r❛♥s♣♦s✐çã♦✱ ✸✳ ➪❧❣❡❜r❛✱ ✹✳ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✱ ✺✳ ❆❧❣♦r✐t♠♦s ❞❡ ❛♣r♦①✐♠❛çã♦❀
❈❉❯ ✵✵✹
❊♥❞❡r❡ç♦✿ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
❈❛♠♣✉s ❯♥✐✈❡rs✐tár✐♦ ❉❛r❝② ❘✐❜❡✐r♦ ✖ ❆s❛ ◆♦rt❡ ❈❊P ✼✵✾✶✵✲✾✵✵
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦
❯♠ ❛❧❣♦r✐t♠♦ ❛❧❣é❜r✐❝♦ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❛ ❉✐stâ♥❝✐❛
❞❡ ❚r❛♥s♣♦s✐çã♦ ❡♠ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s
▲✉✐③ ❆✉❣✉st♦ ●❛r❝✐❛ ❞❛ ❙✐❧✈❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❝♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ ❡♠ ■♥❢♦r♠át✐❝❛
Pr♦❢✳❛ ❉r✳❛ ▼❛r✐❛ ❊♠í❧✐❛ ▼✳ ❚✳ ❲❛❧t❡r ✭❖r✐❡♥t❛❞♦r❛✮ ❈■❈✴❯♥❇
Pr♦❢✳ ❉r✳ ◆♦r❛✐ ❘♦♠❡✉ ❘♦❝❝♦ Pr♦❢✳ ❉r✳ ❩❛♥♦♥✐ ❉✐❛s
▼❆❚✴❯♥❇ ■❈✴❯♥✐❝❛♠♣
Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ P❡③③✉♦❧ ❏❛❝♦❜✐ ❈♦♦r❞❡♥❛❞♦r ❞♦ ▼❡str❛❞♦ ❡♠ ■♥❢♦r♠át✐❝❛
❉❡❞✐❝❛tór✐❛
❆ ♠❡✉s ♣❛✐s✱ P❛✉❧♦ ❡ ▼❛r✐❛✱ q✉❡ ♥✉♥❝❛ ♠❡❞✐r❛♠ ❡s❢♦rç♦s ♣❛r❛ q✉❡ s❡✉s ✜❧❤♦s t✐✈❡ss❡♠ ✉♠❛ ❜♦❛ ❡❞✉❝❛çã♦✳
❆❣r❛❞❡❝✐♠❡♥t♦s
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ❡s♣♦s❛✱ ❏✉❧✐❛♥❛ ❈❛♣❡❧❧❛✱ ♣❡❧♦ ❛♣♦✐♦✱ ✐♥❝❡♥t✐✈♦ ❡ ♣❛❝✐ê♥❝✐❛ ❞✉r❛♥t❡ t♦❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳
➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ Pr♦❢❡ss♦r❛ ▼❛r✐❛ ❊♠í❧✐❛✱ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡ s♦❜r❡t✉❞♦ ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❡♠ ♠❡ ❛❥✉❞❛r✱ ♠❡s♠♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s✳
❆♦ Pr♦❢❡ss♦r ◆♦r❛í ❘♦❝❝♦✱ ♣❡❧♦s ❝♦♥s❡❧❤♦s ❡ ♣❡❧❛s ✐♥ú♠❡r❛s ❧✐çõ❡s ♣❛rt✐❝✉❧❛r❡s ❞❡ ➪❧❣❡❜r❛ q✉❡ ❡✉ t✐✈❡ ♦ ♣r✐✈✐❧é❣✐♦ ❞❡ t❡r✳
❘❡s✉♠♦
❊♠ ❇✐♦❧♦❣✐❛ ❈♦♠♣✉t❛❝✐♦♥❛❧✱ ❡✈❡♥t♦s ♠✉t❛❝✐♦♥❛✐s ❛❢❡t❛♥❞♦ ❣r❛♥❞❡s ♣♦rçõ❡s ❞❡ ✉♠ ❣❡✲ ♥♦♠❛ sã♦ ❡st✉❞❛❞♦s ♥❛ ár❡❛ ❞❡ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ ❛ tr❛♥s♣♦s✐çã♦ é ✉♠ ❡✈❡♥t♦ ♠✉t❛❝✐♦♥❛❧ q✉❡ tr♦❝❛ ❞❡ ♣♦s✐çã♦ ❞♦✐s ❜❧♦❝♦s ❝♦♥tí❣✉♦s ❞❡ ❣❡♥❡s ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦✳ ❊st❡ ❡✈❡♥t♦ ❣❡r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ✭P❉❚✮✱ q✉❡ ❝♦♥✲ s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ tr❛♥s♣♦s✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ tr❛♥s❢♦r♠❛r ✉♠ ❝r♦♠♦ss♦♠♦ ❡♠ ♦✉tr♦✳ ❘❡❝❡♥t❡♠❡♥t❡✱ ❢♦✐ ♠♦str❛❞♦ q✉❡ ♦ P❉❚ é N P✲❞✐❢í❝✐❧✳
◆❛ ❧✐t❡r❛t✉r❛✱ ♠✉✐t♦s ❛❧❣♦r✐t♠♦s ❢♦r❛♠ ♣r♦♣♦st♦s ♣❛r❛ r❡s♦❧✈❡r ❡st❡ ♣r♦❜❧❡♠❛✱ s❡❣✉✐♥❞♦ ❛❜♦r❞❛❣❡♥s ❞✐❢❡r❡♥t❡s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❢♦r♠❛❧✐s♠♦ ❛❧❣é❜r✐❝♦ ♣r♦♣♦st♦ ♣♦r ▼❡✐❞❛♥✐s ❡ ❉✐❛s✱ ♣❛r❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❝r♦♠♦ss♦♠♦s ❡ tr❛♥s♣♦s✐çõ❡s✱ ❡ r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s ♣❛r❛ ♣r♦♣♦r ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ r❛③ã♦ 2♣❛r❛
♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦✳
❊♠❜♦r❛ ❡①✐st❛♠ ❛❧❣♦r✐t♠♦s ❝♦♠ r❛③ã♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♠❡❧❤♦r❡s✱ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ é t❡ór✐❝❛✱ ♣♦✐s ♣r♦♣õ❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ❞❡ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛✲ çõ❡s✱ ❞❡s✈✐♥❝✉❧❛❞❛ ❞♦ ❢♦r♠❛❧✐s♠♦ ❝❧áss✐❝♦ ❇❛❢♥❛ ❡ P❡✈③♥❡r✳ ➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ ♥♦ss♦ ❛❧❣♦r✐t♠♦ s✐♠✉❧❛✱ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧✱ ❛ s♦❧✉çã♦ ❜❛s❡❛❞❛ ❡♠ ❣r❛❢♦ ❞❡ ❝✐❝❧♦s ❞❡ ❇❛❢♥❛ ❡ P❡✈③✲ ♥❡r✳ ◆♦ss❛ s♦❧✉çã♦ ♣♦❞❡rá s❡r ❛✉t♦♠❛t✐③❛❞❛ ❡♠ ♣❛rt❡✱ ❡ ❛❝r❡❞✐t❛♠♦s q✉❡ ✐♥❞✐❝❛ ❝❛♠✐♥❤♦s ♥♦✈♦s✱ q✉❡ ♣♦ss✐❜✐❧✐t❛rã♦ t❛♥t♦ ❞✐♠✐♥✉✐r ❛ r❛③ã♦ ❞❡ ❛♣r♦①✐♠❛çã♦ q✉❛♥t♦ ♦❜t❡r ✉♠❛ ♦✉✲ tr❛ ♣r♦✈❛ ✉s❛♥❞♦ r❡s✉❧t❛❞♦s ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s ♣❛r❛ ♠♦str❛r q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ é N P✲❞✐❢í❝✐❧✳
❖ ❛❧❣♦r✐t♠♦ ♣r♦♣♦st♦ ❢♦✐ ✐♠♣❧❡♠❡♥t❛❞♦ ♥❛ ❧✐♥❣✉❛❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦ ❏❛✈❛✳ ❯t✐❧✐✲ ③❛♠♦s ✉♠ s✐st❡♠❛ ❞❡ á❧❣❡❜r❛ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ❝❤❛♠❛❞♦ ●❆P✱ ♣❛r❛ ❝♦♠♣✉t❛r ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♣❡r♠✉t❛çõ❡s✳ ❖ ❛❧❣♦r✐t♠♦ ❢♦✐ ❛✉❞✐t❛❞♦ ♥❛ ❢❡rr❛♠❡♥t❛ ●❘❆❆✉✱ ♦ q✉❡ ♣❡r♠✐✲ t✐✉ ❛ ❝♦♠♣❛r❛çã♦ ❞❡ t♦❞❛s ❛s ❞✐stâ♥❝✐❛s ❞❡ tr❛♥s♣♦s✐çõ❡s ❞❛❞❛s ♣♦r ♥♦ss♦ ❛❧❣♦r✐t♠♦✱ ♣❛r❛ t♦❞❛s ❛s ♣❡r♠✉t❛çõ❡s ❞❡ t❛♠❛♥❤♦ 2 ❛té 11✱ ❝♦♠ ♦s ✈❛❧♦r❡s ❡①❛t♦s✳ ❖s r❡s✉❧t❛❞♦s ❞❡ss❛
❛✉❞✐t♦r✐❛ ❢♦r❛♠ ❝♦♠♣❛r❛❞♦s ❝♦♠ ♦✉tr♦s ❡♥❝♦♥tr❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s✱ Pr♦❜❧❡♠❛ ❞❛ ❉✐stâ♥❝✐❛ ❞❡ ❚r❛♥s♣♦s✐çã♦✱ ➪❧❣❡✲ ❜r❛✱ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✱ ❆❧❣♦r✐t♠♦s ❞❡ ❛♣r♦①✐♠❛çã♦❀
❆❜str❛❝t
■♥ ❝♦♠♣✉t❛t✐♦♥❛❧ ❜✐♦❧♦❣②✱ ♠✉t❛t✐♦♥❛❧ ❡✈❡♥ts ❛✛❡❝t✐♥❣ ❧❛r❣❡ ♣♦rt✐♦♥s ♦❢ ❛ ❣❡♥♦♠❡ ❛r❡ st✉❞✐❡❞ ✐♥ ❣❡♥♦♠❡ r❡❛rr❛♥❣❡♠❡♥ts✳ P❛rt✐❝✉❧❛r❧②✱ tr❛♥s♣♦s✐t✐♦♥ ✐s ❛ ♠✉t❛t✐♦♥❛❧ ❡✈❡♥t t❤❛t ❝❤❛♥❣❡s t✇♦ ❝♦♥t✐❣✉♦✉s ❜❧♦❝❦s ♦❢ ❣❡♥❡s ✐♥s✐❞❡ ❛ s✐♥❣❧❡ ❝❤r♦♠♦s♦♠❡✳ ❚❤✐s ❡✈❡♥t ❣❡♥❡r❛t❡s t❤❡ ♣r♦❜❧❡♠ ♦❢ tr❛♥s♣♦s✐t✐♦♥ ❞✐st❛♥❝❡✱ ✇❤✐❝❤ ✐s t♦ ✜♥❞ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ tr❛♥s♣♦s✐✲ t✐♦♥s tr❛♥s❢♦r♠✐♥❣ ❛ ❝❤r♦♠♦s♦♠❡ ✐♥t♦ ❛♥♦t❤❡r✳ ❘❡❝❡♥t❧②✱ t❤✐s ♣r♦❜❧❡♠ ✇❛s ♣r♦✈❡❞ t♦ ❜❡ N P✲❤❛r❞✳
■♥ t❤❡ ❧✐t❡r❛t✉r❡ ♠❛♥② ❛❧❣♦r✐t❤♠s ✇❡r❡ ♣r♦♣♦s❡❞ t♦ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠✱ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s✳ ■♥ t❤❡ ♣r❡s❡♥t ✇♦r❦✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ❛❧❣❡❜r❛✐❝ ❢♦r♠❛❧✐s♠ ❢♦r ❝❤r♦♠♦s♦♠❡ ♠♦❞❡❧✐♥❣ ❛♥❞ tr❛♥s♣♦s✐t✐♦♥s ♣r♦♣♦s❡❞ ❜② ▼❡✐❞❛♥✐s ❛♥❞ ❉✐❛s✱ ❛♥❞ ❝❧❛ss✐❝ r❡✲ s✉❧ts ♦❢ P❡r♠✉t❛t✐♦♥ ●r♦✉♣s t♦ s✉❣❣❡st ❛2✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ tr❛♥s♣♦s✐t✐♦♥
❞✐st❛♥❝❡ ♣r♦❜❧❡♠✳
❆❧t❤♦✉❣❤ t❤❡r❡ ❛r❡ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s✱ t❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤✐s ✇♦r❦ ✐s t❤❡ ♣r♦♣♦s✐t✐♦♥ ♦❢ ❛ s♦❧✉t✐♦♥ t♦ t❤❡ tr❛♥s♣♦s✐t✐♦♥ ❞✐st❛♥❝❡ ♣r♦❜❧❡♠ ✉s✐♥❣ ♦♥❧② ❦♥♦✇♥ r❡s✉❧ts ♦❢ P❡r♠✉t❛t✐♦♥ ●r♦✉♣s ❚❤❡♦r②✱ ❞✐ss♦❝✐❛t❡❞ ❢r♦♠ t❤❡ ❝❧❛ss✐❝ ❢♦r♠❛❧✐s♠ ♣r♦♣♦s❡❞ ❜② ❇❛❢♥❛ ❛♥❞ P❡✈③♥❡r✳ ■t ✐s ✇♦rt❤ ♥♦t✐♥❣ t❤❛t ♦✉r ❛❧❣♦r✐t❤♠ s✐♠✉❧❛t❡s✱ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ t❤❡ s♦❧✉t✐♦♥ ❜❛s❡❞ ♦♥ ❇❛❢♥❛ ❛♥❞ P❡✈③♥❡r✬s ❝②❝❧❡ ❣r❛♣❤✳ ❖✉r s♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❛✉t♦♠❛t❡❞ ✐♥ ♣❛rt✱ ❛♥❞ ✇❡ ❜❡❧✐❡✈❡ t❤❛t ✐t ✐♥❞✐❝❛t❡s ♥❡✇ ✇❛②s t❤❛t ❡♥❛❜❧❡ t♦ ❞❡❝r❡❛s❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ❛♥❞ t♦ ❛❝❤✐❡✈❡ ❛♥♦t❤❡r ♣r♦♦❢✱ ✉s✐♥❣ r❡s✉❧ts ♦❢ P❡r♠✉t❛t✐♦♥ ●r♦✉♣s✱ t♦ s❤♦✇ t❤❛t t❤❡ ♣r♦❜❧❡♠ ♦❢ tr❛♥s♣♦s✐t✐♦♥ ❞✐st❛♥❝❡ ✐s N P✲❤❛r❞✳
❚❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠ ✇❛s ✐♠♣❧❡♠❡♥t❡❞ ✉s✐♥❣ ❏❛✈❛ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡✳ ❲❡ ❤❛✈❡ ✉s❡❞ ❛ ❝♦♠♣✉t❡r ❛❧❣❡❜r❛ s②st❡♠✱ ❝❛❧❧❡❞ ●❆P✱ t♦ ❝♦♠♣✉t❡ ♦♣❡r❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ♣❡r♠✉t❛✲ t✐♦♥s✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✇❛s ❛❧s♦ ❛✉❞✐t❡❞ ✐♥ ●❘❆❆✉ t♦♦❧✱ ✇❤✐❝❤ ❛❧❧♦✇❡❞ t❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ ❛❧❧ tr❛♥s♣♦s✐t✐♦♥ ❞✐st❛♥❝❡s ❣✐✈❡♥ ❜② ♦✉r ❛❧❣♦r✐t❤♠✱ ❢♦r ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ♦❢ s✐③❡ 2 t♦ 11✱
✇✐t❤ t❤❡ ❡①❛❝t ✈❛❧✉❡s✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤✐s ❛✉❞✐t ✇❡r❡ ❝♦♠♣❛r❡❞ ✇✐t❤ ♦t❤❡rs ❢♦✉♥❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳
❑❡②✇♦r❞s✿ ●❡♥♦♠❡ ❘❡❛rr❛♥❣❡♠❡♥t✱ ❚r❛♥s♣♦s✐t✐♦♥ ❉✐st❛♥❝❡ Pr♦❜❧❡♠✱ ❆❧❣❡❜r❛✱ P❡r♠✉✲ t❛t✐♦♥ ❣r♦✉♣s✱ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s❀
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷ ▲❡✈❛♥t❛♠❡♥t♦ ❜✐❜❧✐♦❣rá✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ▼♦t✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❖❜❥❡t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❖r❣❛♥✐③❛çã♦ ❞♦s ❝❛♣ít✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s ✶✶
✷✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✶ ●r✉♣♦ ❆❧t❡r♥❛❞♦ An ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✷ Pr♦❞✉t♦s ♠✐♥✐♠❛✐s ❞❡ 2✲❝✐❝❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✸ ❋❛t♦r❛çõ❡s ❞❡ ✉♠n✲❝✐❝❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸ ❋♦r♠❛❧✐s♠♦s ❡♠♣r❡❣❛❞♦s ♥♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ✶✾
✸✳✶ ❋♦r♠❛❧✐s♠♦ ❝❧áss✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶✳✷ ▲✐♠✐t❡ ✐♥❢❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✶✳✸ ▲✐♠✐t❡ s✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ❋♦r♠❛❧✐s♠♦ ❛❧❣é❜r✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷✳✷ ▲✐♠✐t❡ ✐♥❢❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✹ ❆❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ 2 ❜❛s❡❛❞♦ ❡♠ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s ✸✸
✹✳✶ ▲✐♠✐t❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✷ ❯♠❛ ❞❡♠♦♥str❛çã♦ ❝♦♥str✉t✐✈❛ ❞❡ ❧✐♠✐t❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡
tr❛♥s♣♦s✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✸ ❆❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ r❛③ã♦2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹✳✹ ❆♥á❧✐s❡ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✺ ■♠♣❧❡♠❡♥t❛çã♦ ✹✽
✺✳✶ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✷ ❆✉❞✐t♦r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✻ ❈♦♥❝❧✉sõ❡s ✺✷
❘❡❢❡rê♥❝✐❛s ✺✹
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❊✈❡♥t♦ ❞❡ r❡✈❡rsã♦ ❛t✉❛♥❞♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❊✈❡♥t♦ ❞❡ tr❛♥s♣♦s✐çã♦ ❛t✉❛♥❞♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ❊✈❡♥t♦ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s ❛t✉❛♥❞♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹ ❊✈❡♥t♦ ❉❈❏ s✐♠✉❧❛♥❞♦ ✉♠❛ r❡✈❡rsã♦ ✭❛❞❛♣t❛❞♦ ❞❡ ❋❡rt✐♥ ❡t ❛❧✳ ❬✷✽❪✮✳ ✳ ✳ ✳ ✹ ✶✳✺ ❖♣❡r❛çã♦ ❝✉t ❞♦ ❡✈❡♥t♦ ❙❈❏ ❝❛✉s❛♥❞♦ ✉♠❛ ✜ssã♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦
❧✐♥❡❛r ❡ ❧✐♥❡❛r✐③❛♥❞♦ ✉♠ ❝r♦♠♦ss♦♠♦ ❝✐r❝✉❧❛r✳ ❆ ♦♣❡r❛çã♦ ❥♦✐♥ ❛❣❡ ❞❡ ❢♦r♠❛ ❝♦♥trár✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✻ ❯♠ ❡✈❡♥t♦ ❉❈❏ ❝♦rt❛♥❞♦ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r ❡ ❥✉♥t❛♥❞♦ ❞✉❛s ❡①tr❡♠✐✲
❞❛❞❡s ❞❡ ♠♦❞♦ ❛ ♣r♦❞✉③✐r ✉♠ ❝r♦♠♦ss♦♠♦ ❝✐r❝✉❧❛r ✭❛❞❛♣t❛❞♦ ❞❡ ❋❡rt✐♥ ❡t ❛❧✳ ❬✷✽❪✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✶ ●r❛❢♦ ❞♦ ♣r♦❞✉t♦ (3 7)(3 6)(2 7)(3 5)(1 7)(3 4) = (1 2 3 4 5 6 7)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸✳✶ ●r❛❢♦✲❞❡✲❝✐❝❧♦s ❞❛ ♣❡r♠✉t❛çã♦ π= (8 5 1 4 3 2 7 6)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✸✳✷ ❙❡q✉ê♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çõ❡s ♦r❞❡♥❛♥❞♦π = (4 3 2 1 5)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✸ G(π) ❝♦♥t❡♥❞♦ três ❝✐❝❧♦s ❡G(ρ(i, j, k)·π) ❝♦♥t❡♥❞♦ ✉♠ ❝✐❝❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸✳✹ G(π) ❡ G(ρ(i, j, k)·π) ❝♦♥t❡♥❞♦ ❞♦✐s ❝✐❝❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸✳✺ G(π) ❡ G(ρ(i, j, k)·π) ❝♦♥t❡♥❞♦ ✉♠ ❝✐❝❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸✳✻ G(π) ❝♦♥t❡♥❞♦ ✉♠ ❝✐❝❧♦ ❡G(ρ(i, j, k)·π) ❝♦♥t❡♥❞♦ três ❝✐❝❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸✳✼ ❯♠❛ tr❛♥s♣♦s✐çã♦ ❛❣✐♥❞♦ ❡♠ ❞♦✐s ❝✐❝❧♦s✱ ❝r✐❛♥❞♦ ✉♠ ❝✐❝❧♦ ♦r✐❡♥t❛❞♦✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✶ ❖❜t❡♥çã♦ ❞❛s ♣♦ssí✈❡✐s ❢♦r♠❛s∆′′′φµψ ❞❡ ∆✳ ❖s sí♠❜♦❧♦s d✱ e✱ f sã♦ t❛✐s
q✉❡ d6=a✱e /∈ {a, b} ❡f /∈ {a, b, c}✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✺✳✶ ❈❛♣t✉r❛ ❞❡ t❡❧❛ ❞♦ ❛♣❧✐❝❛t✐✈♦ ✐♠♣❧❡♠❡♥t❛❞♦✱ ♠♦str❛♥❞♦ ❛ ♦r❞❡♥❛çã♦ ❞♦ ❝r♦♠♦ss♦♠♦ (0,4,3,2,1,8,7,6,5)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✶✳✶ ❈♦♠♣❧❡①✐❞❛❞❡✱ ♠ét♦❞♦ ❡ ♠❡❧❤♦r r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ r❡❛rr❛♥❥♦ ❞❡ ❣❡♥♦♠❛s ✭❛❞❛♣t❛❞♦ ❞❡ ❋❡✐❥ã♦ ❡ ▼❡✐❞❛♥✐s ❬✷✺❪✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✺✳✶ ❘❡s✉❧t❛❞♦ ❞❛ ❛✉❞✐t♦r✐❛ ❞♦ ❛❧❣♦r✐t♠♦ ▲▼◆ ♥❛ ❢❡rr❛♠❡♥t❛ ●❘❆❆✉✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✷ ❘❡s✉❧t❛❞♦ ❞❛ ❛✉❞✐t♦r✐❛ ❞♦ ❛❧❣♦r✐t♠♦ ▲▼◆❤ ♥❛ ❢❡rr❛♠❡♥t❛ ●❘❆❆✉✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✸ ❈♦♠♣❛r❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ✐❣✉❛❧❞❛❞❡ ♦❜t✐❞♦s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ♣r♦♣♦st♦
♥❡st❡ tr❛❜❛❧❤♦ ❝♦♠ ♦✉tr♦s ❞✐s♣♦♥í✈❡✐s ♥❛ ❧✐t❡r❛t✉r❛✳ ❖s rót✉❧♦s ❞❛s ❝♦✲ ❧✉♥❛s ❞❛ t❛❜❡❧❛ sã♦✿ ❲❉▼ ✲ ❲❛❧t❡r✱ ❉✐❛s ❡ ▼❡✐❞❛♥✐s ❬✻✻❪ ✲ r❛③ã♦ 2.25✱
❍❛rt♠❛♥ ❬✸✻❪ ✲ r❛③ã♦ 1.5 ✭✐♠♣❧❡♠❡♥t❛❞♦ ♣♦r ❍♦♥❞❛ ❬✹✵❪✮✱ ❇P ✲ ❇❛❢♥❛ ❡
P❡✈③♥❡r ❬✻❪ ✲ r❛③ã♦1.5✭✐♠♣❧❡♠❡♥t❛❞♦ ♣♦r ❖❧✐✈❡✐r❛ ❬✺✻❪✮✱ ▲▼◆ ✭❡st❡ tr❛❜❛✲
❧❤♦✮ ❡ ▲▼◆❤ ✭❡st❡ tr❛❜❛❧❤♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❙❡❣✉♥❞♦ ❛ t❡♦r✐❛ ❞❛ ❡✈♦❧✉çã♦ ❞❡ ❉❛r✇✐♥ ❬✶✼❪✱ t♦❞❛s ❛s ❡s♣é❝✐❡s ❛t✉❛✐s ❞❡s❝❡♥❞❡♠ ❞❡ ❡s♣é✲ ❝✐❡s ❛♥❝❡str❛✐s q✉❡ s♦❢r❡r❛♠ ♠♦❞✐✜❝❛çõ❡s ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ❡ t♦❞❛ ❛ ❞✐✈❡rs✐❞❛❞❡ ❜✐♦❧ó❣✐❝❛ ❛t✉❛❧♠❡♥t❡ ❡①✐st❡♥t❡ ❞❡❝♦rr❡ ❞♦ ♣r♦❝❡ss♦ ❞❡ s❡❧❡çã♦ ♥❛t✉r❛❧✳ ❊♠ ❇✐♦❧♦❣✐❛ ▼♦❧❡❝✉❧❛r✱ ❡ss❛ t❡♦r✐❛ é ❧❡✈❛❞❛ ❡♠ ❝♦♥s✐❞❡r❛çã♦ q✉❛♥❞♦✱ ❛tr❛✈és ❞❡ ❝♦♠♣❛r❛çõ❡s✱ ❜✉s❝❛♠♦s r❡❧❛çõ❡s ❞❡ ❤♦♠♦❧♦❣✐❛ ❡♥tr❡ s❡q✉ê♥❝✐❛s ❣❡♥ô♠✐❝❛s ✭❢r❛❣♠❡♥t♦s ❞❡ ❉◆❆✮ ❞❡ ♦r❣❛♥✐s♠♦s ❞❡ ❡s♣é❝✐❡s ❞✐❢❡r❡♥t❡s✳ ❍♦♠♦❧♦❣✐❛s ♣♦❞❡♠ s❡r ✐♥❢❡r✐❞❛s ♣♦r ♠❡❞✐❞❛s ❝♦♠♦ s✐♠✐❧✐❛r✐❞❛❞❡ ❬✻✶❪✱ q✉❡ ✐❞❡♥✲ t✐✜❝❛♠ q✉ã♦ ✏♣❛r❡❝✐❞❛s✑ sã♦ ❞✉❛s s❡q✉ê♥❝✐❛s✳ ❖ ♣r❡ss✉♣♦st♦ ❜ás✐❝♦ é ♦ ❞❡ q✉❡ s❡q✉ê♥❝✐❛s ❣❡♥ô♠✐❝❛s ✏s✐♠✐❧❛r❡s✑ ❢♦r❛♠ ❝♦♥s❡r✈❛❞❛s ❞✉r❛♥t❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❡✈♦❧✉çã♦ ❡ ♣♦ss✐✈❡❧♠❡♥t❡ ❡①❡r❝❡♠ ♦ ♠❡s♠♦ ♣❛♣❡❧ ♥♦s ♠❡❝❛♥✐s♠♦s ❝❡❧✉❧❛r❡s✱ ♦✉ s❡❥❛✱ tê♠ ❛ ♠❡s♠❛ ❢✉♥çã♦ ❜✐♦❧ó❣✐❝❛✳ ◆❛s ú❧t✐♠❛s três ❞é❝❛❞❛s✱ ♣r♦❥❡t♦s ❞❡ s❡q✉❡♥❝✐❛♠❡♥t♦ ❣❡♥ô♠✐❝♦ ❬✷✻✱ ✺✺❪ ❡♠ t♦❞♦ ♦ ♠✉♥❞♦ ❣❡r❛r❛♠ ✉♠ ❡♥♦r♠❡ ✈♦❧✉♠❡ ❞❡ ❞❛❞♦s✳ ❆♣❡♥❛s ❝♦♠♦ ❡①❡♠♣❧♦✱ ♦ Pr♦❥❡t♦ ●❡♥♦♠❛ ❍✉♠❛♥♦ ❬✹✺❪ r❡s✉❧t♦✉ ♥♦ s❡q✉❡♥❝✐❛♠❡♥t♦ ❞❡ 3,2❜✐❧❤õ❡s ❞❡ ♣❛r❡s ❞❡ ❜❛s❡s✳ ❙❡r✐❛ ✐♥✈✐á✈❡❧
❝♦♠♣❛r❛r ✈♦❧✉♠❡s ❞❡ ❞❛❞♦s ❡♠ t❛❧ ❡s❝❛❧❛ s❡♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛✉t♦♠❛t✐③❛❞❛ ❡ ♣♦rt❛♥t♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s t♦r♥♦✉✲s❡ ✐♠♣r❡s❝✐♥❞í✈❡❧ à ♣❡sq✉✐s❛ ❡♠ ❇✐♦❧♦❣✐❛ ▼♦❧❡❝✉❧❛r✳ ◆❡ss❡ s❡♥t✐❞♦✱ ♣❛r❛ ❛✉①✐❧✐❛r ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ❢✉♥çõ❡s ❜✐♦❧ó❣✐❝❛s ❞❛s s❡q✉ê♥❝✐❛s ❣❡♥ô♠✐❝❛s✱ é ❢✉♥❞❛♠❡♥t❛❧ q✉❡ s❡❥❛♠ ❝r✐❛❞♦s ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦rr❡t♦s ❡ ❡✜❝✐❡♥t❡s ♣❛r❛ ❛ t❛r❡❢❛ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s✳ ❖s r❡s✉❧t❛❞♦s ❞❛s ❝♦♠♣❛r❛çõ❡s ✐♥ s✐❧✐❝♦ ♣♦t❡♥❝✐❛❧♠❡♥t❡ ✐♥❞✐❝❛♠ ❞✐r❡çõ❡s ❞❡ ♣❡sq✉✐s❛ ♣r♦♠✐ss♦r❛s ♣❛r❛ ♦s ❜✐ó❧♦❣♦s✱ q✉❡ ♣♦❞❡♠ ❡♥tã♦ ♣r♦❥❡t❛r s❡✉s ❡①♣❡r✐♠❡♥t♦s ✐♥ ✈✐tr♦ ❡ ✐♥ ✈✐✈♦ ❝♦♠ ❜❛s❡ ♥❡❧❡s✳
P❡sq✉✐s❛s ❡♠ ❇✐♦❧♦❣✐❛ ▼♦❧❡❝✉❧❛r ❡ ❛ ❝♦♥st❛♥t❡ ❜✉s❝❛ ♣♦r ♥♦✈❛s té❝♥✐❝❛s ❝♦♠♣✉t❛❝✐♦✲ ♥❛✐s✱ ❝❛♣❛③❡s ❞❡ ❛♥❛❧✐s❛r ❛ ❡♥♦r♠❡ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çõ❡s ❣❡♥ô♠✐❝❛s ❞✐s♣♦♥í✈❡✐s✱ sã♦ ❥✉st✐✜❝❛❞❛s✱ ♣♦✐s s❡✉s r❡s✉❧t❛❞♦s tê♠ ✉t✐❧✐③❛çã♦ ❡♠ ✈ár✐❛s ❛♣❧✐❝❛çõ❡s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ s♦❝✐❡❞❛❞❡✱ ❝♦♠♦ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ ♥♦✈❛s ❞r♦❣❛s✱ ❛✉♠❡♥t♦ ❞❛ ♣r♦❞✉t✐✈✐❞❛❞❡ ❛❣rí❝♦❧❛✱ ❝♦♠✲ ❜❛t❡ ❞❡ ♣r❛❣❛s ❡ ❞♦❡♥ç❛s ❡♠ ❡s♣é❝✐❡s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ♣r♦❞✉çã♦ ❛❧✐♠❡♥tí❝✐❛ ♦✉ ❛✐♥❞❛✱ ✐♥✈❡st✐❣❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❡✈♦❧✉çã♦ ❞❛s ❡s♣é❝✐❡s ✈✐✈❛s ❛t✉❛❧♠❡♥t❡✳
❖s ♠ét♦❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ✉t✐❧✐③❛❞♦s ♥❛ ♣❡sq✉✐s❛ ❣❡♥ô♠✐❝❛ ✉t✐❧✐③❛♠ t❡♦r✐❛s ❡st✉❞❛❞❛s ❡♠ ❞✐❢❡r❡♥t❡s s✉❜ár❡❛s ❞❡ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳ P♦❞❡♠♦s ❝✐t❛r✱ ❞❡♥tr❡ ❡❧❛s✿ ❆❧❣♦r✐t✲ ♠♦s ❬✷✱ ✻✹❪✱ ■♥t❡❧✐❣ê♥❝✐❛ ❆rt✐✜❝✐❛❧ ❬✹✻✱ ✺✽❪ ❡ ❙✐st❡♠❛s ❉✐str✐❜✉í❞♦s ❡ P❛r❛❧❡❧♦s ❬✺✾✱ ✻✵✱ ✻✷❪✳ ❊ss❡s ♠ét♦❞♦s ❜❛s❡✐❛♠✲s❡ ♥❛ ❝♦♠♣❛r❛çã♦ ❞❡ ❞❛❞♦s ❣❡♥ô♠✐❝♦s t❛♥t♦ ❡♠ ♥í✈❡❧ ❞❡ ♥✉❝❧❡✲ ♦tí❞❡♦s ❬✶✱ ✹✹❪✱ q✉❛♥t♦ ❡♠ ♥í✈❡❧ ❞❡ ❢r❛❣♠❡♥t♦s ❞❡ ❝r♦♠♦ss♦♠♦s ❬✸✱ ✺✱ ✻✱ ✷✹✱ ✸✺✱ ✻✺❪✳ ❆s ❝♦♠♣❛r❛çõ❡s sã♦ ❡❢❡t✉❛❞❛s ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❡✈❡♥t♦s q✉❡ ♣♦ss✐✈❡❧♠❡♥t❡ t❡♥❤❛♠ ♦❝♦rr✐❞♦ ♥❛ ♥❛t✉r❡③❛ ❬✷✶✱✺✹✱✺✼❪ ❡ q✉❡ ❝♦♥st✐t✉❡♠ ❢❛t♦r❡s ✐♠♣♦rt❛♥t❡s ♥♦ ♣r♦❝❡ss♦ ❡✈♦❧✉❝✐♦♥ár✐♦✳
◆❡st❛ ❞✐ss❡rt❛çã♦✱ ♦ ❢♦❝♦ é ❛ ❝♦♠♣❛r❛çã♦ ♥♦ ♥í✈❡❧ ❞❡ ❢r❛❣♠❡♥t♦s ❞❡ ❝r♦♠♦ss♦♠♦s✱ ♣♦✐s ❡❧❛ r❡♣r❡s❡♥t❛ ♦ ❡s❝♦♣♦ ♥♦ q✉❛❧ ❡stá ✐♥s❡r✐❞♦ ♦ ♣r♦❜❧❡♠❛ ❛q✉✐ ✐♥✈❡st✐❣❛❞♦✳
✶✳✶ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s
❊st❛ s❡çã♦ ❞❡st✐♥❛✲s❡ à ❛♣r❡s❡♥t❛çã♦ ❞❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s ❡ ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ ❜✐❜❧✐♦❣rá✜❝❛ s♦❜r❡ ❛ ár❡❛ ❬✷✽✱✻✶❪✳
❆ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ❞♦✐s ❝r♦♠♦ss♦♠♦s é r❡❛❧✐③❛❞❛ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❜❧♦❝♦s ❞❡ ❣❡♥❡s q✉❡ ♦❝♦rr❡♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❡♠ ❛♠❜♦s ❡ ❡♠ ❡✈❡♥t♦s ❞❡ ♠✉t❛çã♦ ❝♦♥❤❡❝✐❞♦s ♣❡❧♦s ❜✐ó❧♦❣♦s✱ q✉❡ ♠♦str❛♠ ✉♠ ❝❡♥ár✐♦ ♣❧❛✉sí✈❡❧ ❞❡ ❡✈♦❧✉çã♦✳ ❆ ❝♦♥str✉çã♦ ❞❡ t❛❧ ❝❡♥ár✐♦ ❝♦♥s✐❞❡r❛ ❛ ❍✐♣ót❡s❡ ❞❡ P❛r❝✐♠ô♥✐❛✱ ❛ q✉❛❧ s✉❣❡r❡ q✉❡ ❛s ❡s♣é❝✐❡s s✉r❣❡♠✱ ❛♦ ❧♦♥❣♦ ❞❛ ❡✈♦❧✉çã♦✱ ❛ ♣❛rt✐r ❞❡ ✉♠ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ♠✉t❛çõ❡s✳ P♦rt❛♥t♦✱ ♥❛ ár❡❛ ❞❡ ❘❡❛rr❛♥❥♦ ❞❡ ●❡✲ ♥♦♠❛s✱ ♦s ♣❡sq✉✐s❛❞♦r❡s tê♠ ❝♦♠♦ ♣r♦❜❧❡♠❛ ❜ás✐❝♦ ♦ ❞❡ ❞❡t❡r♠✐♥❛r ♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ❡✈❡♥t♦s ♠✉t❛❝✐♦♥❛✐s✱ ❞♦r❛✈❛♥t❡ ❝❤❛♠❛❞♦s ❞❡ ❡✈❡♥t♦s ❞❡ r❡❛rr❛♥❥♦ ♦✉ s✐♠♣❧❡s♠❡♥t❡ r❡❛rr❛♥❥♦s✱ ❛t✉❛♥❞♦ s♦❜r❡ ❢r❛❣♠❡♥t♦s ❞❡ ❝r♦♠♦ss♦♠♦s ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ tr❛♥s❢♦r♠❛r ✉♠ ❣❡♥♦♠❛ ❡♠ ♦✉tr♦✳
❋✐❣✉r❛ ✶✳✶✿ ❊✈❡♥t♦ ❞❡ r❡✈❡rsã♦ ❛t✉❛♥❞♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r✳
❉✐✈❡rs♦s ❡✈❡♥t♦s ❞❡ r❡❛rr❛♥❥♦ tê♠ s✐❞♦ ❡st✉❞❛❞♦s ♥❡st❛ ár❡❛✳ ❉❡♥tr❡ ♦s ♠❛✐s ✐♥✈❡st✐✲ ❣❛❞♦s✱ ♣♦❞❡♠♦s ❝✐t❛r três ❡✈❡♥t♦s ❛❣✐♥❞♦ ♥✉♠ ♠❡s♠♦ ❝r♦♠♦ss♦♠♦✿ r❡✈❡rsã♦✱ q✉❡ ❛t✉❛ ✏✐♥✈❡rt❡♥❞♦✑ ✉♠ ❢r❛❣♠❡♥t♦ ❞♦ ❝r♦♠♦ss♦♠♦ ✭❋✐❣✉r❛ ✶✳✶✮❀ tr❛♥s♣♦s✐çã♦✱ q✉❡ ✏❝♦rt❛✑ ✉♠ ❢r❛❣♠❡♥t♦ ❡ ♦ ✏❝♦❧❛✑ ❡♠ ♦✉tr♦ ❧♦❝❛❧ ✭❋✐❣✉r❛ ✶✳✷✮❀ ❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s q✉❡ ✏❝♦rt❛✑ ❡ ✏tr♦❝❛✑ ❞❡ ❧✉❣❛r ❞♦✐s ❢r❛❣♠❡♥t♦s ❞♦ ❝r♦♠♦ss♦♠♦ ✭❋✐❣✉r❛ ✶✳✸✮✳ P♦❞❡♠♦s ❝✐t❛r t❛♠❜é♠ ❉♦✉❜❧❡✲ ❈✉t✲❛♥❞✲❏♦✐♥ ✭❋✐❣✉r❛ ✶✳✹✮ ❡ ❙✐♥❣❧❡✲❈✉t✲❖r✲❏♦✐♥ ✭❋✐❣✉r❛ ✶✳✺✮✱ q✉❡ r❡♣r❡s❡♥t❛♠ ♦♣❡r❛çõ❡s ❞❡ r❡❛rr❛♥❥♦ ❣❡♥ér✐❝❛s✱ ❝❛♣❛③❡s ❞❡ s✐♠✉❧❛r ✈ár✐♦s t✐♣♦s ❞❡ ❡✈❡♥t♦s ❛t✉❛♥❞♦ s♦❜r❡ ✉♠ ♦✉ ♠❛✐s ❝r♦♠♦ss♦♠♦s✳
❈❛❞❛ ✉♠ ❞♦s ❡✈❡♥t♦s ❞❡ r❡❛rr❛♥❥♦✱ ❛ss✐♠ ❝♦♠♦ ❝❛❞❛ ✉♠❛ ❞❛s ❝♦♠❜✐♥❛çõ❡s ❡♥tr❡ ❡❧❡s✱ ❣❡r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ ❞✐❢❡r❡♥t❡ ❡ ✈ár✐♦s ♠ét♦❞♦s ❢♦r❛♠ ♣r♦♣♦st♦s ♣❛r❛ r❡s♦❧✈❡r ❝❛❞❛ ✉♠ ❞❡❧❡s✳
✶✳✶✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s
◆❡st❛ s❡çã♦✱ ♠♦str❛r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❝♦♠♦ ❝r♦♠♦ss♦♠♦s ❧✐♥❡❛r❡s sã♦ ✉s✉❛❧♠❡♥t❡ ♠♦❞❡✲ ❧❛❞♦s ❡♠ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❢♦r♠❛❧✐③❛çã♦ ❞♦s ❡✈❡♥t♦s ❞❡ r❡❛rr❛♥❥♦ ♠❡♥❝✐♦♥❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✳
❋✐❣✉r❛ ✶✳✷✿ ❊✈❡♥t♦ ❞❡ tr❛♥s♣♦s✐çã♦ ❛t✉❛♥❞♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r✳
❋✐❣✉r❛ ✶✳✸✿ ❊✈❡♥t♦ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s ❛t✉❛♥❞♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r✳
▼♦❞❡❧❛❣❡♠ ❞❡ ❝r♦♠♦ss♦♠♦s
❊♠ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s✱ ❝r♦♠♦ss♦♠♦s ♥♦r♠❛❧♠❡♥t❡ sã♦ ♠♦❞❡❧❛❞♦s ❝♦♠♦ ♣❡r♠✉t❛çõ❡s ❞❡ ♥ú♠❡r♦ ✐♥t❡✐r♦s✳ ❈❛❞❛ ♥ú♠❡r♦ r♦t✉❧❛ ✉♠ ❢r❛❣♠❡♥t♦✱ ❢♦r♠❛❞♦ ♣♦r ✉♠ ♦✉ ♠❛✐s ❜❧♦❝♦s ❞❡ ❣❡♥❡s ❞♦ ❝r♦♠♦ss♦♠♦✳ ❊✈❡♥t✉❛❧♠❡♥t❡✱ ❡ss❡s ♥ú♠❡r♦s ♣♦❞❡♠ ❡st❛r ❛❝♦♠♣❛♥❤❛❞♦s ❞❡ s✐♥❛✐s✱ ✐♥❞✐❝❛♥❞♦ q✉❡ ❛ ♦r✐❡♥t❛çã♦ ❞❡ ❝❛❞❛ ❢r❛❣♠❡♥t♦ é ❝♦♥❤❡❝✐❞❛✳ ❙❡ ❞♦✐s ❢r❛❣♠❡♥t♦s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♦r✐❡♥t❛çã♦✱ ❛ss♦❝✐❛♠♦s ❛ ❡❧❡s ♦ s✐♥❛❧ ✏✰✑✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❛ss♦❝✐❛♠♦s ❛♦ s❡❣✉♥❞♦ ❢r❛❣♠❡♥t♦ ♦ s✐♥❛❧ ✏✲✑✳
❆s ♣❡r♠✉t❛çõ❡s s❡♠ s✐♥❛✐s sã♦ r❡♣r❡s❡♥t❛❞❛s ♣♦r ❜✐❥❡çõ❡s s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ {1,2, . . . n}✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❡❧❡♠❡♥t❛r ♣❛r❛ ✉♠❛ ❜✐❥❡çã♦ é ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❞✉❛s ❧✐♥❤❛s
✭❝❛❞❛ πi ∈ {1,2, . . . n} ❡ 1≤i≤n✮✿
π = 1 2 . . . n
π1 π2 . . . πn
!
✳
P♦ré♠✱ ❡♠ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s✱ ❛ r❡♣r❡s❡♥t❛çã♦ ♠❛✐s ❝♦♠✉♠ é ❛ ❞❡ ✉♠❛ ❧✐♥❤❛✱ q✉❡ ❝♦♥s✐st❡ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❛♥t❡r✐♦r✱ ♣♦ré♠ ❝♦♠ ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ s✉♣r✐♠✐❞❛✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ♣❡r♠✉t❛çã♦ ❛♥t❡r✐♦r é r❡♣r❡s❡♥t❛❞❛ ❛♣❡♥❛s ❝♦♠♦ π = (π1π2. . . πn)✳
P♦r s✉❛ ✈❡③✱ ♣❡r♠✉t❛çõ❡s ❝♦♠ s✐♥❛✐s sã♦ r❡♣r❡s❡♥t❛❞❛s ❝♦♠♦ ❜✐❥❡çõ❡s s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ {−n,· · ·−2,−1,1,2, . . . , n}✳ ❆ s❡❣✉✐r✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❞✉❛s ❧✐♥❤❛s ❞❡ ✉♠❛ ♣❡r♠✉t❛çã♦ π ❝♦♠ s✐♥❛✐s ✭❝❛❞❛ πi ∈ {1,2, . . . n} ❡1≤i≤n✮✿
π = −n . . . −2 −1 1 2 . . . +n
−πn . . . −π2 −π1 π1 π2 . . . πn
!
✳
T T
t h h t t h t h h t t h
12 4 14 1 7 8
T T
t h h t t h h t h t t h
12 4 7 1 14 8
cut cut
join join
❋✐❣✉r❛ ✶✳✹✿ ❊✈❡♥t♦ ❉❈❏ s✐♠✉❧❛♥❞♦ ✉♠❛ r❡✈❡rsã♦ ✭❛❞❛♣t❛❞♦ ❞❡ ❋❡rt✐♥ ❡t ❛❧✳ ❬✷✽❪✮✳
❋✐❣✉r❛ ✶✳✺✿ ❖♣❡r❛çã♦ ❝✉t ❞♦ ❡✈❡♥t♦ ❙❈❏ ❝❛✉s❛♥❞♦ ✉♠❛ ✜ssã♦ ❡♠ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r ❡ ❧✐♥❡❛r✐③❛♥❞♦ ✉♠ ❝r♦♠♦ss♦♠♦ ❝✐r❝✉❧❛r✳ ❆ ♦♣❡r❛çã♦ ❥♦✐♥ ❛❣❡ ❞❡ ❢♦r♠❛ ❝♦♥trár✐❛✳
◆❛ r❡♣r❡s❡♥t❛çã♦ ♠❛✐s ✉t✐❧✐③❛❞❛✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛✱ ❛s ❝♦❧✉♥❛s ❞❡ ❞♦♠í♥✐♦ ♥❡❣❛t✐✈♦ t❛♠❜é♠ sã♦ s✉♣r✐♠✐❞❛s✱ ♣♦✐s π(−i) = −π(i)✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡
♣❡r♠✉t❛çõ❡s ❝♦♠ ♦✉ s❡♠ s✐♥❛✐s é s❡♠❡❧❤❛♥t❡✳
Pr♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ r❡✈❡rsã♦
❉❡✜♥✐♠♦s ✉♠❛ r❡✈❡rsã♦ ♥♦ ✐♥t❡r✈❛❧♦ [i, j]✱ ❝♦♠ i < j✱ ❞❡♥♦t❛❞❛ ♣♦r r[i, j]✱ ❛❣✐♥❞♦ s♦❜r❡
✉♠❛ ♣❡r♠✉t❛çã♦ π= (π1. . . πi−1πiπi+1. . . πj−1πjπj+1. . . πn)✱ ♣♦r
r[i, j]·π = (π1. . . πi−1πjπj−1. . . πi+1πiπj+1. . . πn)✱
♦♥❞❡ πk ❞❡♥♦t❛ ❛ ✐♥✈❡rsã♦ ❞♦ s✐♥❛❧ ❞❡ πk✳
❖ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ r❡✈❡rsã♦ é ❢♦r♠❛❧✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❞❛❞♦s ❞♦✐s ❝r♦♠♦ss♦♠♦s✱ r❡♣r❡s❡♥t❛❞♦s ♣❡❧❛s ♣❡r♠✉t❛çõ❡s π ❡ σ✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡
r❡✈❡rsã♦ ❡♥tr❡ π ❡ σ é ♦ ❞❡ ❡♥❝♦♥tr❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ r❡✈❡rsõ❡s r1, r2, . . . rρ t❛❧ q✉❡
rρ. . . r2·r1·π=σ✱ ♦♥❞❡ρé ♠í♥✐♠♦✳ ❙❡σ❝♦rr❡s♣♦♥❞❡ à ♣❡r♠✉t❛çã♦ ✐❞❡♥t✐❞❛❞❡(1 2 . . . n)✱
❝❤❛♠❛♠♦s ❡st❡ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❜❧❡♠❛ ❞❛ ♦r❞❡♥❛çã♦ ♣♦r r❡✈❡rsõ❡s✳ ❖ ♥ú♠❡r♦ ρ é
❝❤❛♠❛❞♦ ❞❡ ❞✐stâ♥❝✐❛ ❞❡ r❡✈❡rsã♦ ❡ ❞❡♥♦t❛♠♦s ❛ ❢✉♥çã♦ q✉❡ t♦♠❛ ❞♦✐s ❝r♦♠♦ss♦♠♦s✱
π ❡σ✱ ❡ ❞❡✈♦❧✈❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ r❡✈❡rsã♦ ❡♥tr❡ ❡❧❡s✱ ♣♦r dρ(π, σ)✳ ❙❡ σ é ❛ ✐❞❡♥t✐❞❛❞❡✱ ❡ss❛
❢✉♥çã♦ é ❞❡♥♦t❛❞❛ ❛♣❡♥❛s ♣♦r dρ(π)✳
❙❡ ❛s ♦r✐❡♥t❛çõ❡s ❞♦s ❜❧♦❝♦s ❞❡ ❣❡♥❡s ❞❡π sã♦ ❞❡s❝♦♥❤❡❝✐❞❛s✱ ❛ r❡✈❡rsã♦ ❛t✉❛ ❛♣❡♥❛s
✐♥✈❡rt❡♥❞♦ ❛ ♦r❞❡♠ ❞♦s ❜❧♦❝♦s ♥♦ ✐♥t❡r✈❛❧♦ [i, j]✳ ▲♦❣♦✱ ❛ ❞❡✜♥✐çã♦ ❞❛ r❡✈❡rsã♦ s♦❜r❡
♣❡r♠✉t❛çõ❡s s❡♠ s✐♥❛✐s é ❛♥á❧♦❣❛ à ❞❡✜♥✐çã♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❛♣r❡s❡♥t❛r✳
❊①❡♠♣❧♦ ✶✳✶✳✶✳ ❙❡❥❛♠ ❛s ♣❡r♠✉t❛çõ❡s ❝♦♠ s✐♥❛✐s π = (+1 − 5 + 4 −3 + 2) ❡
σ = (+1 + 2 + 3 + 4 + 5)✳ ❆ s❡q✉ê♥❝✐❛ ❞❡ r❡✈❡rsõ❡s r1 = r(5,5)✱ r2 = r(3,3) ❡
r3 =r(2,5)tr❛♥s❢♦r♠❛ π ❡♠ σ✳
r1·π=r(5,5)·(+1 −5 + 4 −3 +2)
r2·r1·π=r(3,3)·(+1 −5 +4 −3 −2)
r3·r2·r1·π=r(2,5)·(+1 −5 −4 −3 −2)
= (+1 + 2 + 3 + 4 + 5) =σ
❊①❡♠♣❧♦ ✶✳✶✳✷✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s ❛s ♠❡s♠❛s ♣❡r♠✉t❛çõ❡s ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♣♦ré♠ ❛❣♦r❛ s❡♠ s✐♥❛✐s✱ t❡r❡♠♦s q✉❡ π = (1 5 4 3 2) ❡ σ = (1 2 3 4 5)✳ ❆ r❡✈❡rsã♦ r(2,5) é
s✉✜❝✐❡♥t❡ ♣❛r❛ tr❛♥s❢♦r♠❛r π ❡♠ σ✳
Pr♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦
❖ ❡✈❡♥t♦ ❞❡ tr❛♥s♣♦s✐çã♦ ❢❛③ ❝♦♠ q✉❡ ❜❧♦❝♦s ❞❡ ❣❡♥❡s ❞❡ ✉♠ ❝r♦♠♦ss♦♠♦ s❡❥❛♠ ✏❝♦rt❛✲ ❞♦s✑ ❡ ❡♠ s❡❣✉✐❞❛ ✏❝♦❧❛❞♦s✑ ♥✉♠❛ ♦✉tr❛ ♣❛rt❡ ❞♦ ♠❡s♠♦ ❝r♦♠♦ss♦♠♦✳
❉❡ ♠❛♥❡✐r❛ ❢♦r♠❛❧✱ ✉♠❛ tr❛♥s♣♦s✐çã♦ ❛t✉❛♥❞♦ s♦❜r❡ ✉♠ ❝r♦♠♦ss♦♠♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ♣❡r♠✉t❛çã♦ π ❝♦♠ n ❜❧♦❝♦s ❞❡ ❣❡♥❡s é ❞❡✜♥✐❞❛ ❝♦♠♦ ρ(i, j, k)✱ ❝♦♠ 1≤ i < j ≤ n+ 1✱ 1≤k ≤n+ 1 ❡k /∈[i, j]✳ ❆ tr❛♥s♣♦s✐çã♦ ✏❝♦rt❛✑ ♦s ❡❧❡♠❡♥t♦s ❡♥tr❡ πi ❡πj−1 ❡ ♦s ✏❝♦❧❛✑
❡♥tr❡ πk−1 ❡ πk✱ ♦✉ s❡❥❛
ρ(i, j, k)·π =ρ(i, j, k)·(π1. . . πi−1 πi. . . πj−1 πj. . . πk−1 πk. . . πn)
=(π1. . . πi−1 πj. . . πk−1 πi. . . πj−1 πk. . . πn).
❘❡♣❛r❡ q✉❡ ❛ tr❛♥s♣♦s✐çã♦ t❡♠ ♦ ♠❡s♠♦ ❡❢❡✐t♦ ❞❡ tr♦❝❛r ❛s ♣♦s✐çõ❡s ❞❡ ❞♦✐s ❜❧♦❝♦s ❡♥tr❡
[i, j−1] ❡[j, k−1]✳
❖ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ é ❢♦r♠❛❧✐③❛❞♦ ❝♦♠♦ s❡❣✉❡✳ ❉❛❞♦s ❞♦✐s ❝r♦✲ ♠♦ss♦♠♦s✱ ♠♦❞❡❧❛❞♦s ♣❡❧❛s ♣❡r♠✉t❛çõ❡sπ ❡σ✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦✲
s✐çã♦ é ❡♥❝♦♥tr❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çõ❡s ρ1, ρ2, . . . ρt t❛❧ q✉❡ρt. . . ρ2·ρ1·π=σ
❡ t é ♠í♥✐♠♦✳ ❙❡ σ ❝♦rr❡s♣♦♥❞❡ à ♣❡r♠✉t❛çã♦ ✐❞❡♥t✐❞❛❞❡ (1 2 . . . n)✱ ❝❤❛♠❛♠♦s ❡st❡
♣r♦❜❧❡♠❛ ❞❡ ♣r♦❜❧❡♠❛ ❞❛ ♦r❞❡♥❛çã♦ ♣♦r tr❛♥s♣♦s✐çõ❡s✳ ❖ ♥ú♠❡r♦ t é ❝❤❛♠❛❞♦ ❞❡
❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ❡ ❛ ❢✉♥çã♦ q✉❡ ❛♣❧✐❝❛❞❛ ❛ ❞✉❛s ♣❡r♠✉t❛çõ❡s π ❡ σ✱ ❞❡✈♦❧✈❡
❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ❡♥tr❡ ❡❧❛s✱ é ❞❡♥♦t❛❞❛ ♣♦r dt(π, σ)✳ ❙❡σ é ❛ ✐❞❡♥t✐❞❛❞❡✱ ❡ss❛
❢✉♥çã♦ é ❞❡♥♦t❛❞❛ ❛♣❡♥❛s ♣♦r dt(π)✳
❊①❡♠♣❧♦ ✶✳✶✳✸✳ ❙❡❥❛♠ ❛s ♣❡r♠✉t❛çõ❡s π = (1 5 2 4 3) ❡ σ = (1 2 3 4 5)✳ ❆s tr❛♥s✲
♣♦s✐çõ❡s ρ1 = ρ(2,4,5) ❡ ρ2 = ρ(2,4,6)✱ ❛♣❧✐❝❛❞❛s ♥❡st❛ s❡q✉ê♥❝✐❛✱ tr❛♥s❢♦r♠❛♠ π ❡♠
σ✳
ρ1·π=ρ(2,4,5)·(1 5 2 4 3)
ρ2·ρ1·π=ρ(2,4,6)·(1 4 5 2 3)
= (1 2 3 4 5) =σ
Pr♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s
❖ ❡✈❡♥t♦ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ tr❛♥s♣♦s✐çã♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❡❧❛ tr♦❝❛ ❞♦✐s ❜❧♦❝♦s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♥tí❣✉♦s ❞❡ ✉♠ ❝r♦♠♦ss♦♠♦✳
P♦❞❡♠♦s ❢♦r♠❛❧✐③❛r ❛ tr♦❝❛✲❞❡✲❜❧♦❝♦s ❝♦♠♦ s❡❣✉❡✳ P❛r❛ ✉♠❛ ♣❡r♠✉t❛çã♦ π✱ ❛ tr♦❝❛
❞❡ ❜❧♦❝♦s β(i, j, k, l) ❝♦♠ 1 ≤ i < j < k < l ≤ n+ 1✱ ❛♣❧✐❝❛❞❛ ❛ π✱ tr♦❝❛ ♦ ✐♥t❡r✈❛❧♦
[i, j−1] ♣❡❧♦ ✐♥t❡r✈❛❧♦[k, l−1]✱ tr❛♥s❢♦r♠❛♥❞♦π ❡♠ β(i, j, k, l)·π✳
β(i, j, k, l)·π=β(i, j, k, l)·(π1. . . πi−1πi. . . πj−1πj. . . πk−1πk. . . πl−1πl. . . πn)
=(π1. . . πi−1πk. . . πl−1πj. . . πk−1πi. . . πj−1πl. . . πn).
❉❛❞♦s ❞♦✐s ❝r♦♠♦ss♦♠♦s✱ ♠♦❞❡❧❛❞♦s ♣❡❧❛s ♣❡r♠✉t❛çõ❡s π ❡ σ✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐s✲
tâ♥❝✐❛ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s é ❡♥❝♦♥tr❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s β1, β2, . . . βb
t❛❧ q✉❡ βb. . . β2 ·β1 ·π = σ ❡ b é ♠í♥✐♠♦✳ ❙❡ σ ❝♦rr❡s♣♦♥❞❡ à ♣❡r♠✉t❛çã♦ ✐❞❡♥t✐❞❛❞❡
(1 2 . . . n)✱ ❝❤❛♠❛♠♦s ❡st❡ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❜❧❡♠❛ ❞❛ ♦r❞❡♥❛çã♦ ♣♦r tr♦❝❛✲❞❡✲
❜❧♦❝♦s✳ ❖ ♥ú♠❡r♦ b é ❝❤❛♠❛❞♦ ❞❡ ❞✐stâ♥❝✐❛ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s ❡ ❛ ❢✉♥çã♦ q✉❡
❛♣❧✐❝❛❞❛ ❛ ❞✉❛s ♣❡r♠✉t❛çõ❡s π ❡ σ✱ ❞❡✈♦❧✈❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s ❡♥tr❡ ❡❧❛s✱ é
❞❡♥♦t❛❞❛ ♣♦r bid(π, σ)✳ ❙❡ σ é ❛ ✐❞❡♥t✐❞❛❞❡✱ ❡ss❛ ❢✉♥çã♦ é ❞❡♥♦t❛❞❛ ❛♣❡♥❛s ♣♦rbid(π)✳
❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❙❡❥❛♠ ❛s ♣❡r♠✉t❛çõ❡s π = (1 5 6 8 2 3 7 4) ❡ σ = (1 2 3 4 5 6 7 8)✳ ❆s
tr♦❝❛s✲❞❡✲❜❧♦❝♦s β(4,5,6,8) ❡ β(2,5,6,8)✱ ❛♣❧✐❝❛❞❛s ♥❡st❛ s❡q✉ê♥❝✐❛✱ tr❛♥s❢♦r♠❛♠ π ❡♠ σ✳
β1·π=β(4,5,6,8)·(1 5 6 8 2 3 7 4)
β2·β1·π=β(2,5,6,8)·(1 5 6 7 4 2 3 8)
= (1 2 3 4 5 6 7 8) =σ
Pr♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ ❉♦✉❜❧❡✲❈✉t✲❛♥❞✲❏♦✐♥
❯♠❛ ♦♣❡r❛çã♦ ❉♦✉❜❧❡✲❈✉t✲❛♥✲❏♦✐♥ ✭❉❈❏✮ ❛t✉❛ ❡♠ ❞✉❛s ❛❞❥❛❝ê♥❝✐❛sab❡cd❞❡ ✉♠ ❣❡♥♦♠❛
Π ❡ tr❛♥s❢♦r♠❛ ❡ss❛s ❛❞❥❛❝ê♥❝✐❛s ❡♠ ac ❡ bd ♦✉ ad ❡ bc✳ ❉✐③❡♠♦s q✉❡ ❉❈❏ ✏❝♦rt❛✑ ❛s
❛❞❥❛❝ê♥❝✐❛s ab❡ cd❡ ❛s ✏❥✉♥t❛✑ ❡♠ ac❡ bd✭♦✉ ad ❡ bc✮✳
◆❛ ❋✐❣✉r❛ ✶✳✻✱ ♠♦str❛♠♦s ✉♠ ❡✈❡♥t♦ ❉❈❏ ❝♦rt❛♥❞♦ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r ❡♠ ❞♦✐s ❧✉❣❛r❡s✳ ❊♠ s❡❣✉✐❞❛✱ ❞✉❛s ❡①tr❡♠✐❞❛❞❡s sã♦ ✉♥✐❞❛s ❞❡ ♠♦❞♦ ❛ ♣r♦❞✉③✐r ✉♠ ❝r♦♠♦ss♦♠♦ ❝✐r❝✉❧❛r✳
❯♠❛ ♦♣❡r❛çã♦ ❉❈❏ ♣♦❞❡✱ ❡♥tr❡ ♦✉tr♦s ❡✈❡♥t♦s✱ s✐♠✉❧❛r r❡✈❡rsõ❡s✳ ❉✉❛s ♦♣❡r❛çõ❡s ❉❈❏ ❝♦♥s❡❝✉t✐✈❛s ♣♦❞❡♠ s✐♠✉❧❛r tr❛♥s♣♦s✐çõ❡s ❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s✳
❚❛♥t♦ ❣❡♥♦♠❛s✱ ❝♦♠♦ ❝r♦♠♦ss♦♠♦s ❧✐♥❡❛r❡s✱ ♣♦❞❡♠ s❡r ❝♦♠♣❛r❛❞♦s ♥♦ â♠❜✐t♦ ❞❛s ♦♣❡r❛çõ❡s ❉❈❏✱ ♠❡s♠♦ q✉❛♥❞♦ ❛♣r❡s❡♥t❛♠ ♥ú♠❡r♦s ❞✐❢❡r❡♥t❡s ❞❡ ❝r♦♠♦ss♦♠♦s ❡ ❡①tr❡✲ ♠✐❞❛❞❡s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ♣❛r❛ ❉❈❏✱ ♥ã♦ ❢❛③ s❡♥t✐❞♦ ❛ ✐❞❡✐❛ ❞❡ ♦r❞❡♥❛çã♦✱ ❛♣❡♥❛s ❛ ✐❞❡✐❛ ❞❡ ❞✐stâ♥❝✐❛✳
❉❛❞♦s ❞♦✐s ❣❡♥♦♠❛s Π ❡ Γ✱ ❛ ❞✐stâ♥❝✐❛ ❞❡ ❉♦✉❜❧❡✲❈✉t✲❛♥❞✲❏♦✐♥ ❡♥tr❡ Π ❡ Γ✱
❞❡♥♦t❛❞❛ ♣♦r dcj(Π,Γ)✱ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ♦♣❡r❛çõ❡s ❉❈❏ ♥❡❝❡ssár✐❛s
♣❛r❛ s❡ tr❛♥s❢♦r♠❛r Π ❡♠ Γ✳
T T
t h h t t h t h h t t h
12 4 14 1 7 8
T T
t h h t t h h
t
h t
t h
12 4 14
1
7 8
cut
join jo
in cut
h
❋✐❣✉r❛ ✶✳✻✿ ❯♠ ❡✈❡♥t♦ ❉❈❏ ❝♦rt❛♥❞♦ ✉♠ ❝r♦♠♦ss♦♠♦ ❧✐♥❡❛r ❡ ❥✉♥t❛♥❞♦ ❞✉❛s ❡①tr❡♠✐❞❛❞❡s ❞❡ ♠♦❞♦ ❛ ♣r♦❞✉③✐r ✉♠ ❝r♦♠♦ss♦♠♦ ❝✐r❝✉❧❛r ✭❛❞❛♣t❛❞♦ ❞❡ ❋❡rt✐♥ ❡t ❛❧✳ ❬✷✽❪✮✳
Pr♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ ❙✐♥❣❧❡✲❈✉t✲♦r✲❏♦✐♥
❊♠ ❜✉s❝❛ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ r❡❛rr❛♥❥♦ q✉❡ ❢♦ss❡ ♦ ♠❛✐s ❡❧❡♠❡♥t❛r ♣♦ssí✈❡❧✱ ❋❡✐❥ã♦ ❡ ▼❡✐❞❛✲ ♥✐s ❬✷✹❪ ♣r♦♣✉s❡r❛♠ ✉♠ ♠♦❞❡❧♦ ❝❤❛♠❛❞♦ ❙✐♥❣❧❡✲❈✉t✲♦r✲❏♦✐♥ ✭❙❈❏✮✳ ❊ss❡ ♠♦❞❡❧♦ ❝♦♥s✐st❡ ❞❡ ❞✉❛s ♦♣❡r❛çõ❡s ❜ás✐❝❛s✿ ❝✉t✱ q✉❡ s✐❣♥✐✜❝❛ ✏❝♦rt❛r✑ ✉♠❛ ❛❞❥❛❝ê♥❝✐❛✱ ❡ ❥♦✐♥✱ q✉❡ s✐❣♥✐✜❝❛ ✏❥✉♥t❛r✑ ❞✉❛s ❡①tr❡♠✐❞❛❞❡s✳
❉❛❞♦s ❞♦✐s ❣❡♥♦♠❛s Π ❡Γ✱ ❛ ❞✐stâ♥❝✐❛ ❞❡ ❙✐♥❣❧❡✲❈✉t✲♦r✲❏♦✐♥ ❡♥tr❡ Π ❡ Γ✱ ❞❡♥♦✲
t❛❞❛ ♣♦r dSCJ(Π,Γ)✱ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ♦♣❡r❛çõ❡s ❙❈❏ ♥❡❝❡ssár✐❛s ♣❛r❛
s❡ tr❛♥s❢♦r♠❛r Π ❡♠ Γ✳
✶✳✶✳✷ ▲❡✈❛♥t❛♠❡♥t♦ ❜✐❜❧✐♦❣rá✜❝♦
P❛r❛ ❝❛❞❛ ❡✈❡♥t♦ ❞❡ r❡❛rr❛♥❥♦ ♠❡♥❝✐♦♥❛❞♦ ❛té ❛q✉✐✱ ✈❛♠♦s ❞✐s❝✉t✐r ❜r❡✈❡♠❡♥t❡ ✉♠❛ r❡✈✐sã♦ ❞♦s r❡s✉❧t❛❞♦s ❞✐s♣♦♥í✈❡✐s ♥❛ ❧✐t❡r❛t✉r❛✳
❘❡✈❡rsã♦
❖ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ r❡✈❡rsã♦ s❡♠ s✐♥❛✐s é N P✲❞✐❢í❝✐❧ ❡ ❛ s✉❛ ❞❡♠♦♥str❛çã♦ é ❞❡✈✐❞❛ ❛ ❈❛♣r❛r❛ ❬✶✸❪✳ ❙♦❧✉çõ❡s ❛♣r♦①✐♠❛❞❛s ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞❛s ♣♦r ❲❛tt❡rs♦♥ ❛t ❛❧✳ ❬✻✼❪✱ ❑❡❝❡❝✐♦❣❧✉ ❡ ❙❛♥❦♦✛ ❬✹✸❪✱ ❇❛❢♥❛ ❡ P❡✈③♥❡r ❬✺❪✱ ❈❤r✐st✐❡ ❬✶✺❪✱ ❡ ❇❡r♠❛♥✱ ❍❛♥♥❡♥❤❛❧❧✐ ❡ ❑❛r♣✐♥s❦✐ ❬✶✵❪✳ ❊st❡s ú❧t✐♠♦s ❢♦r♥❡❝❡r❛♠ ❛ ♠❡❧❤♦r ❛♣r♦①✐♠❛çã♦ ❛té ❛❣♦r❛ ❝♦♥❤❡❝✐❞❛✱ ❝♦♠ r❛③ã♦ ❞❡ 1.375✳
❊♠ r❡❧❛çã♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ r❡✈❡rsã♦ ❝♦♠ s✐♥❛✐s✱ ❍❛♥♥❡♥❤❛❧❧✐ ❡ P❡✈③♥❡r ❬✸✺❪ ❢♦r❛♠ ♦s ♣r✐♠❡✐r♦s ❛ ❢♦r♥❡❝❡r ✉♠❛ s♦❧✉çã♦ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✱ q✉❡ ❡①❡❝✉t❛ ❡♠ t❡♠♣♦ O(n4)✳ ❊st❡ ❛❧❣♦r✐t♠♦ ❢♦✐ ❛♣❡r❢❡✐ç♦❛❞♦ ♣♦r ❚❛♥♥✐❡r✱ ❇❡r❣❡r♦♥ ❡ ❙❛❣♦t ❬✻✸❪✱
q✉❡ ❢♦r♥❡❝❡r❛♠ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡♠ t❡♠♣♦ s✉❜q✉❛❞rát✐❝♦O(n3/2√logn)✳ ❆
♠❡❧❤♦r s♦❧✉çã♦✱ q✉❡ ❡①❡❝✉t❛ ❡♠ t❡♠♣♦ ❧✐♥❡❛r✱ é ❞❡✈✐❞❛ ❛ ❇❛❞❡r✱ ▼♦r❡t ❡ ❨❛♥ ❬✸❪✳
❚r❛♥s♣♦s✐çã♦
❇❛❢♥❛ ❡ P❡✈③♥❡r ❬✹❪ ♣r♦♣✉s❡r❛♠ ✉♠❛ ❡str✉t✉r❛ ❞❡♥♦♠✐♥❛❞❛ ❣r❛❢♦✲❞❡✲❝✐❝❧♦s ♣❛r❛ r❡♣r❡✲ s❡♥t❛çã♦ ❞❡ ❜r❡❛❦♣♦✐♥ts✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦s ♣♦♥t♦s ❞❡ r✉♣t✉r❛ ❞❛
❚❛❜❡❧❛ ✶✳✶✿ ❈♦♠♣❧❡①✐❞❛❞❡✱ ♠ét♦❞♦ ❡ ♠❡❧❤♦r r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ r❡✲ ❛rr❛♥❥♦ ❞❡ ❣❡♥♦♠❛s ✭❛❞❛♣t❛❞♦ ❞❡ ❋❡✐❥ã♦ ❡ ▼❡✐❞❛♥✐s ❬✷✺❪✮✳
❊✈❡♥t♦s ❞❡
r❡❛rr❛♥❥♦ ❈♦♠♣❧❡①✐❞❛❞❡ ❚✐♣♦ ❞❡ ❛❧❣♦r✐t♠♦ ❆✉t♦r❡s
❘❡✈❡rsã♦ s❡♠ s✐♥❛✐s N P✲❞✐❢í❝✐❧ ❆♣r♦①✐♠❛çã♦ 1.375 ❇❡r♠❛♥ ❡t ❛❧✳ ❬✾❪
❘❡✈❡rsã♦ ❝♦♠ s✐♥❛✐s O(n) ❊①❛t♦ ❇❛❞❡r ❡t ❛❧✳ ❬✸❪
❚r❛♥s♣♦s✐çã♦ N P✲❞✐❢í❝✐❧ ❆♣r♦①✐♠❛çã♦ 1.375 ❊❧✐❛s ❡ ❍❛rt♠❛♥ ❬✷✸❪
❚r❛♥s♣♦s✐çã♦
❞❡ ♣r❡✜①♦s ♥ã♦ ❝♦♥❤❡❝✐❞❛ ❆♣r♦①✐♠❛çã♦ 2 ❉✐❛s ❡ ▼❡✐❞❛♥✐s ❬✶✾❪
❚r♦❝❛✲❞❡✲❜❧♦❝♦s O(n2) ❊①❛t♦ ❈❤r✐st✐❡ ❬✶✹❪
❉♦✉❜❧❡✲❈✉t✲❛♥❞✲❏♦✐♥ O(n) ❊①❛t♦ ❨❛♥❝♦♣♦✉❧♦s
❡t ❛❧✳ ❬✻✽❪
❙✐♥❣❧❡✲❈✉t✲♦r✲❏♦✐♥ O(n) ❊①❛t♦ ❋❡✐❥ã♦ ❡ ▼❡✐❞❛♥✐s ❬✷✹❪
♦r❞❡♥❛çã♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛✳ ▲❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❝✐❝❧♦s ❞❡ss❛ ❡str✉t✉r❛✱ ♣r♦♣✉s❡r❛♠ ❛❧❣♦r✐t♠♦s ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ r❛③õ❡s 2✱ 1.75 ❡1.5✳
❈❤r✐st✐❡ ❬✶✻❪✱ ✉t✐❧✐③❛♥❞♦ ❛ ❡str✉t✉r❛ ♣r♦♣♦st❛ ♣♦r ❇❛❢♥❛ ❡ P❡✈③♥❡r✱ t❛♠❜é♠ ♣r♦♣ôs ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ r❛③ã♦ 1.5✱ ♣♦ré♠ ❞✐❢❡r❡♥t❡ ❞♦ ♣r♦♣♦st♦ ♣♦r ❇❛❢♥❛ ❡
P❡✈③♥❡r✳
❲❛❧t❡r✱ ❉✐❛s ❡ ▼❡✐❞❛♥✐s ❬✻✻❪ ❛♣r❡s❡♥t❛r❛♠ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ q✉❛❞rát✐❝♦ ❝♦♠ r❛③ã♦ 2.25✱ ❜❛s❡❛❞♦ ❡♠ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥t❡ ❞❛ ♣r♦♣♦st❛ ♣♦r ❇❛❢♥❛ ❡ P❡✈③♥❡r✳
▼❡✐❞❛♥✐s ❡ ❉✐❛s ❬✹✾❪ ❢♦r♠❛❧✐③❛r❛♠ ❛ t❡♦r✐❛ ❞❡ ❣r❛❢♦✲❞❡✲❝✐❝❧♦s ❞❡ ❇❛❢♥❛ ❡ P❡✈③♥❡r ✉t✐❧✐③❛♥❞♦ ❡❧❡♠❡♥t♦s ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✱ ♦ ❝❤❛♠❛❞♦ ❢♦r♠❛❧✐s♠♦ ❛❧✲ ❣é❜r✐❝♦ ❬✹✾❪✳
❍❛rt♠❛♥ ❬✸✼❪ ♣r♦♣ôs ✉♠❛ ✈❛r✐❛çã♦ ❞♦ ❣r❛❢♦✲❞❡✲❝✐❝❧♦s ❞❡ ❇❛❢♥❛ ❡ P❡✈③♥❡r✱ ❞♦ q✉❛❧ ❞❡r✐✈♦✉ ✉♠ ❛❧❣♦r✐t♠♦ ❝♦♠ r❛③ã♦ ❞❡ ❛♣r♦①✐♠❛çã♦ 1.5 ❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ t❡♠♣♦ O(n2)✳
P♦st❡r✐♦r♠❡♥t❡✱ ✉t✐❧✐③❛♥❞♦ ❡st❛ ❡str✉t✉r❛✱ ❊❧✐❛s ❡ ❍❛rt♠❛♥ ❬✷✸❪ ♣r♦♣✉s❡r❛♠ ✉♠ ❛❧❣♦r✐t♠♦ ❝♦♠ r❛③ã♦ ❞❡ ❛♣r♦①✐♠❛çã♦ 1.375 ❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ t❡♠♣♦O(n2)✳ ❊st❡ é ♦ ♠❡❧❤♦r r❡s✉❧✲
t❛❞♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛✳
❘❡❝❡♥t❡♠❡♥t❡✱ ❇✉❧t❡❛✉✱ ❋❡rt✐♥ ❡ ❘✉s✉ ❬✶✷❪ ❞❡♠♦♥str❛r❛♠ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ é N P✲❞✐❢í❝✐❧✳
❯♠❛ ✈❛r✐❛♥t❡ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ é ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ♣♦r ♣r❡✜①♦s✳ ❯♠❛ tr❛♥s♣♦s✐çã♦ ♣♦r ♣r❡✜①♦s é ✉♠❛ tr❛♥s♣♦s✐çã♦
ρ(i, j, k)✱ ♦♥❞❡ i = 1✳ ❊st❡ ♣r♦❜❧❡♠❛ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❉✐❛s ❡ ▼❡✐❞❛♥✐s ❬✶✾❪✱ ❛♦s q✉❛✐s
t❛♠❜é♠ é ❞❡✈✐❞♦ ♦ ♠❡❧❤♦r ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦✱ ❝♦♠ r❛③ã♦ 2 ❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡
t❡♠♣♦ O(n2)✳ ❆❧é♠ ❞✐ss♦✱ ❉✐❛s ❡ ▼❡✐❞❛♥✐s ♠♦str❛r❛♠ q✉❡ q✉❛❧q✉❡r tr❛♥s♣♦s✐çã♦ ♣♦❞❡
s❡r s✐♠✉❧❛❞❛ ❝♦♠ ❛té ❞✉❛s tr❛♥s♣♦s✐çõ❡s ♣♦r ♣r❡✜①♦s✳ P♦rt❛♥t♦✱ q✉❛❧q✉❡r ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ r❛③ã♦ k ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ é t❛♠❜é♠ ✉♠
❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ r❛③ã♦2k♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ♣♦r
♣r❡✜①♦s✳ ❆ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡st❡ ♣r♦❜❧❡♠❛ é ❞❡s❝♦♥❤❡❝✐❞❛✳
❚r♦❝❛✲❞❡✲❜❧♦❝♦s
❖ ♣r♦❜❧❡♠❛ ❞❛ ♦r❞❡♥❛çã♦ ♣♦r tr♦❝❛✲❞❡✲❜❧♦❝♦s ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❈❤r✐st✐❡ ❬✶✹❪✱ q✉❡ t❛♠✲ ❜é♠ ❢♦r♥❡❝❡✉ ✉♠❛ s♦❧✉çã♦ ❡♠ t❡♠♣♦ O(n)✳ ❙❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ tr♦❝❛✲❞❡✲❜❧♦❝♦s ❢♦r δ✱ ✉♠❛
s♦❧✉çã♦ ❛❧t❡r♥❛t✐✈❛ ♣r♦♣♦st❛ ♣♦r ▲✐♥ ❡t ❛❧✳ ❬✹✼❪ ❡①❡❝✉t❛ ❡♠ t❡♠♣♦ O(δn)✳ ❊st❛ s♦❧✉çã♦
❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ✉t✐❧✐③❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ❛❧❣é❜r✐❝♦ ❞❡ ▼❡✐❞❛♥✐s ❡ ❉✐❛s ❬✹✾❪✳ ❯♠❛ ♦✉tr❛ s♦❧✉çã♦ ❛❧t❡r♥❛t✐✈❛ ❢♦✐ ♣r♦✈✐❞❛ ♣♦r ❋❡♥❣ ❡ ❩❤✉ ❬✷✼❪✱ ❡ ❡①❡❝✉t❛ ❡♠ t❡♠♣♦ O(nlogn)✳
❉♦✉❜❧❡✲❈✉t✲❛♥❞✲❏♦✐♥
❊st❛ ♦♣❡r❛çã♦ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❨❛♥❝♦♣♦✉❧♦s ❡t ❛❧✳ ❬✻✽❪✱ ♦s q✉❛✐s ❢♦r♥❡❝❡r❛♠ ✉♠❛ s♦❧✉çã♦ ❡♠ t❡♠♣♦ q✉❛❞rát✐❝♦✳
P♦st❡r✐♦r♠❡♥t❡✱ ❇❡r❣❡r♦♥ ❡t ❛❧✳ ❬✼❪ ❡st❡♥❞❡r❛♠ ❛ t❡♦r✐❛ ❉❈❏✱ ❝r✐❛♥❞♦ ✉♠❛ ❡str✉t✉r❛ ❝❤❛♠❛❞❛ ❞❡ ❣r❛❢♦✲❞❡✲❛❞❥❛❝ê♥❝✐❛s✱ q✉❡ ❝♦♥s✐st❡ ♥✉♠ ❣r❛❢♦ ❣❡r❛❞♦ ♣❡❧❛ ✉♥✐ã♦ ❞❡ ❝❛♠✐♥❤♦s ❡ ❝✐❝❧♦s ✉s❛❞♦s ♥❛ ♠♦❞❡❧❛❣❡♠ ❞♦s ❝r♦♠♦ss♦♠♦s✳ ❊st❡s ❛✉t♦r❡s ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠ ❛❧❣♦r✐t♠♦ ❣✉❧♦s♦ ✉t✐❧✐③❛♥❞♦ ❡ss❛ ❡str✉t✉r❛✱ q✉❡ ❡①❡❝✉t❛ ❡♠ O(n)✳
▼✐r❛ ❡ ▼❡✐❞❛♥✐s ❬✺✸❪ t❛♠❜é♠ ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ s♦❧✉çã♦ ❧✐♥❡❛r ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✱ ✉t✐❧✐③❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ❛❧❣é❜r✐❝♦ ❞❡ ▼❡✐❞❛♥✐s ❡ ❉✐❛s ❬✹✾❪✳
❙✐♥❣❧❡✲❈✉t✲♦r✲❏♦✐♥
❊st❛ ♦♣❡r❛çã♦ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ r❡❝❡♥t❡♠❡♥t❡ ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦♠❛s✱ ❡ ♣♦r ❡♥q✉❛♥t♦ ❝♦♥t❛♠♦s ❛♣❡♥❛s ♦ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦ ❞❡ ❋❡✐❥ã♦ ❡ ▼❡✐❞❛♥✐s ❬✷✹❪✱ ❝✉❥❛ s♦❧✉çã♦ ❡①❡❝✉t❛ ❡♠ t❡♠♣♦ ❧✐♥❡❛r O(n)✳
✶✳✷ ▼♦t✐✈❛çã♦
❚❛♥t♦ q✉❛♥t♦ s❛❜❡♠♦s✱ ♣r❡❞♦♠✐♥❛♠ ♥❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛s ❡♠ ❘❡❛rr❛♥❥♦ ❞❡ ●❡♥♦✲ ♠❛s✱ ❝♦♥❝❡✐t♦s ❡ ❛r❣✉♠❡♥t♦s ❡ss❡♥❝✐❛❧♠❡♥t❡ ❣rá✜❝♦s✱ ❝❛r❡❝❡♥❞♦ ❞❡ ✉♠ ❢♦r♠❛❧✐s♠♦ ♠❛✐s r✐❣♦r♦s♦✳ ❈♦♠♦ ❞✐t♦ ❛♥t❡s✱ ▼❡✐❞❛♥✐s ❡ ❉✐❛s ❬✹✾❪ ♣r♦♣✉s❡r❛♠ ✉♠❛ t❡♦r✐❛ ❞❡ r❡❛rr❛♥❥♦s ♥♦✈❛✱ ❜❛s❡❛❞❛ ♥❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✱ ❝❤❛♠❛❞♦ ❞❡ ❢♦r♠❛❧✐s♠♦ ❛❧❣é❜r✐❝♦✳ ❖ ♠ét♦❞♦ ♣r♦♣♦st♦ ❝♦♥s✐st❡ ♥❛ ♠♦❞❡❧❛❣❡♠ ❞♦s ❝♦♥❝❡✐t♦s ❡ ❛r❣✉♠❡♥t♦s ❞❛s t❡♦r✐❛s ❝❧áss✐❝❛s ❞❡ r❡❛rr❛♥❥♦ ❡♠ ❞❡✜♥✐çõ❡s ❡ ❞❡♠♦♥str❛çõ❡s ❛❧❣é❜r✐❝❛s✱ ♥♦ s❡♥t✐❞♦ ❞❡ ♣❡r♠✐t✐r ❛ ❛♣❧✐❝❛çã♦ ❞✐r❡t❛ ❞❡ r❡s✉❧t❛❞♦s ❥á ❝♦♥❤❡❝✐❞♦s ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✳
❱ár✐♦s tr❛❜❛❧❤♦s ❛♣❧✐❝❛r❛♠ ❡st❛ t❡♦r✐❛ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛❧❣♦r✐t♠♦s ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ r❡❛rr❛♥❥♦ ❡s♣❡❝í✜❝♦s ♦✉ ❝♦♠❜✐♥❛çõ❡s ❞❡❧❡s✳ ❉❡♥tr❡ ❛❧❣✉♥s✱ ♣♦❞❡♠♦s ❝✐t❛r ❋❡✐❥ã♦ ❡ ▼❡✐❞❛♥✐s ❬✷✹❪✱ ❉✐❛s ❡ ▼❡✐❞❛♥✐s ❬✶✽✱ ✶✾❪✱ ▼✐r❛ ❡t ❛❧✳ ❬✺✵❪✱ ❡ ▼✐r❛ ❡ ▼❡✐❞❛♥✐s ❬✺✸❪✱ ▲✐♥ ❡t ❛❧✳ ❬✹✼❪✱ ❍✉❛♥❣ ❡t ❛❧✳ ❬✹✶❪✳ ❊♠❜♦r❛ ♦ ❢♦r♠❛❧✐s♠♦ ❛❧❣é❜r✐❝♦ t❡♥❤❛ s✐❞♦ ❡♠♣r❡❣❛❞♦ ♥❡ss❡s tr❛❜❛❧❤♦s✱ ❛s ❞❡✜♥✐çõ❡s ❡ ❛r❣✉♠❡♥t♦s ❞❛s t❡♦r✐❛s ❝❧áss✐❝❛s ❞❡ r❡❛rr❛♥❥♦ ❛✐♥❞❛ ❞❡t❡r♠✐♥❛♠ ❛ ❢♦r♠❛ ❝♦♠♦ ♦s ❛❧❣♦r✐t♠♦s sã♦ ❝♦♥str✉í❞♦s✳
✶✳✸ ❖❜❥❡t✐✈♦
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s r❡s✉❧t❛❞♦s ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✳
✶✳✹ ❖r❣❛♥✐③❛çã♦ ❞♦s ❝❛♣ít✉❧♦s
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❢❛r❡♠♦s ✉♠❛ r❡✈✐sã♦ s♦❜r❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❛ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s ❞❡ P❡r✲ ♠✉t❛çõ❡s ♥❡❝❡ssár✐♦s ❛♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s s♦❜r❡ ❢❛t♦r❛çã♦ ❞❡ ♣❡r♠✉t❛çõ❡s✳
❖ ❈❛♣ít✉❧♦ ✸ s❡rá ❞❡st✐♥❛❞♦ ❛ ✉♠❛ r❡✈✐sã♦ ❞♦s ❢♦r♠❛❧✐s♠♦s ❝❧áss✐❝♦ ❬✻❪ ❡ ❛❧❣é❜r✐❝♦ ❬✹✾✱ ✺✶✱✺✷❪✱ ✉t✐❧✐③❛❞♦s ♥❛ ❛❜♦r❞❛❣❡♠ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦✳
◆♦ ❈❛♣ít✉❧♦ ✹✱ ❞❡♠♦str❛r❡♠♦s✱ ✉s❛♥❞♦ r❡s✉❧t❛❞♦s ❞❡ ●r✉♣♦s ❞❡ P❡r♠✉t❛çõ❡s✱ ❧✐♠✐✲ t❡s ✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐stâ♥❝✐❛ ❞❡ tr❛♥s♣♦s✐çã♦✱ ❡ ♣r♦♣♦r❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ❝♦♠ r❛③ã♦ 2✳
◆♦ ❈❛♣ít✉❧♦ ✺✱ s❡rã♦ ❞✐s❝✉t✐❞♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❡❧❛ ✐♠♣❧❡♠❡♥t❛çã♦ ✉s❛♥❞♦ ●❆P ❬✸✷❪✱ ✉♠ s✐st❡♠❛ ❞❡ á❧❣❡❜r❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♣❛r❛ ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♣❡r♠✉t❛çõ❡s✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ♠❡❞✐❞❛s ♦❜t✐❞❛s ♣❡❧♦ ●❘❆❆✉ ❬✸✵❪✱ q✉❡ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ❞❡ ❛✉❞✐t♦r✐❛ ❞❡ ❛❧❣♦r✐t♠♦s ❞❡ r❡❛rr❛♥❥♦✳
P♦r ✜♠✱ ♦ ❈❛♣ít✉❧♦ ✻ tr❛rá ❛s ❝♦♥❝❧✉sõ❡s ❡ ♣r♦♣♦rá tr❛❜❛❧❤♦s ❢✉t✉r♦s✳
