Alinhamentos e comparação de sequências

❆❧✐♥❤❛♠❡♥t♦s

❡ ❝♦♠♣❛r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s

❋r❛♥❝✐s❝♦ ❊❧♦✐ ❙♦❛r❡s ❞❡ ❆r❛✉❥♦

❚❡s❡ ❛♣r❡s❡♥t❛❞❛

❛♦

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

❞❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦

♣❛r❛

♦❜t❡♥çã♦ ❞♦ tít✉❧♦

❞❡

❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s

Pr♦❣r❛♠❛✿ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦sé ❆✉❣✉st♦ ❘❛♠♦s ❙♦❛r❡s

❆❧✐♥❤❛♠❡♥t♦s

❡ ❝♦♠♣❛r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s

❊st❛ ✈❡rsã♦ ❞❡✜♥✐t✐✈❛ ❞❛ t❡s❡ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ r❡❛❧✐③❛❞❛ ♣♦r ❋r❛♥❝✐s❝♦ ❊❧♦✐ ❙♦❛r❡s ❞❡ ❆r❛✉❥♦ ❡♠ ✷✹ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✷✳

❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿

• Pr♦❢✳ ❉r✳ ❏♦sé ❆✉❣✉st♦ ❘❛♠♦s ❙♦❛r❡s ✭Pr❡s✐❞❡♥t❡✮ ✕ ■▼❊ ✲ ❯❙P • Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❊❞✉❛r❞♦ ❋❡rr❡✐r❛ ✕ ■▼❊ ✲ ❯❙P

• Pr♦❢✳ ❉r✳ ▲✉✐③ ❈❛r❧♦s ❞❛ ❙✐❧✈❛ ❘♦③❛♥t❡ ✕ ❈▼❈❈ ✲ ❯❋❆❇❈ • Pr♦❢✳ ❉r✳ ❩❛♥♦♥✐ ❉✐❛s ✕ ■❈ ✲ ❯◆■❈❆▼P

❆❣r❛❞❡❝✐♠❡♥t♦s

▼✉✐t♦s ♣❛r❡♥t❡s ❡ ❛♠✐❣♦s ❢♦r❛♠ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ♥♦ss♦ ❞♦✉t♦r❛❞♦✳ ❈❡rt❛♠❡♥t❡ ❝❛❞❛ ✉♠ ❞❡ss❡s s❛❜❡ ❞❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❡ss❡ ♣r♦❝❡ss♦✳ ❙❡ ❡✉ ❡sq✉❡❝❡r ❞❡ r❡❢❡r❡♥❝✐❛r ❛❧❣✉é♠ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ♣❡r❞♦❡✲♠❡ ♣♦✐s ❛ ♠❡♠ór✐❛ às ✈❡③❡s ❢❛❧❤❛✳

❆❣r❛❞❡ç♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ à ♠✐♥❤❛ ❢❛♠í❧✐❛✿ à ■♦♥❡✱ ♠✐♥❤❛ ❡s♣♦s❛ ❡ ❛♦s ♠❡✉s ✜❧❤♦s ●❛❜r✐❡❧✱ ❙❛r❛❤ ❡ P❡❞r♦ q✉❡ ♠❡ ❞❡r❛♠ ❛♠♦r ❡ ❛♣♦✐♦ ❧♦❣íst✐❝♦ ❞✉r❛♥t❡ t♦❞♦s ♦s ❛♥♦s ❞❡ ♠❡✉ ❞♦✉t♦r❛❞♦❀ ❡ ❛♦ ❏♦sé ❆✉❣✉st♦✱ ♠❡✉ ♦r✐❡♥t❛❞♦r ❡ ❛♠✐❣♦✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ s♦❜r❡♥❛t✉r❛❧✳ ❈❡rt❛♠❡♥t❡✱ s❡♠ ♦ ❛♣♦✐♦ ❞❛ ❢❛♠í❧✐❛ ❡ ❛ ♣❛❝✐ê♥❝✐❛ ❞♦ ❏♦sé ❆✉❣✉st♦✱ ❡ss❡ tr❛❜❛❧❤♦ ♥ã♦ t❡r✐❛ t❡r♠✐♥❛❞♦✳

❆❣r❛❞❡ç♦ ❛♦s ✐r♠ã♦ ❡ ❝✉♥❤❛❞♦s✱ s♦❜r✐♥❤♦s ❡ s♦❜r✐♥❤❛s✱ s♦❣r♦ ❡ s♦❣r❛ ♣❡❧♦ ❛♣♦✐♦ ♠♦r❛❧✱ ❡♠ ♣❛rt✐❝✉❧❛r à ♠✐♥❤❛ ❝✉♥❤❛❞❛ ❈❧á✉❞✐❛ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ t❛♠❜é♠ ♣❡❧❛s ❝❛r♦♥❛s ❞♦ ❡ ♣❛r❛ ♦ ■▼❊✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ à ♠✐♥❤❛ ❢❛♠í❧✐❛ q✉❡ ❡st♦✉ r❡❞❡s❝♦❜r✐♥❞♦ ❡♠ ❈❛♠♣♦ ●r❛♥❞❡ ♣❡❧❛ ❛❝♦❧❤✐❞❛ ❡ ♣❡❧♦ ❝❛r✐♥❤♦ r❡❝❡❜✐❞♦ q✉❛♥❞♦ ❝❤❡❣✉❡✐ ♥❛q✉❡❧❛ ❝✐❞❛❞❡✿ ♠❡✉ t✐♦ ❏❛✐r ❡ ♠❡✉s ♣r✐♠♦s ❆✐❧②✱ ❆✐s❡ ❡ ❋❡r♥❛♥❞♦✱ ❜❡♠ ❝♦♠♦ ♦s r❡s♣❡❝t✐✈♦s ❝♦♠♣❛♥❤❡✐r♦s ❡ ❞❡s❝❡♥❞❡♥t❡s✳

P❡❧❛ t♦r❝✐❞❛ ❡ ♣❡❧❛ ❛♠✐③❛❞❡✱ ❛❣r❛❞❡ç♦ ❛♦s ❛♠✐❣♦s ❞♦ ❝♦❧é❣✐♦ ❙ã♦ ▲✉ís❀ ❛♦s ❝♦❧❡❣❛s ❞❡ ❏❛❜♦t✐❝❛❜❛❧ ✐♥❝❧✉✐♥❞♦ ❡♠ ♣❛rt✐❝✉❧❛r ♦s ✐r♠ã♦s ❞❡ ❝♦r❛çã♦ ❞❛ r❡♣ú❜❧✐❝❛ ❆♠♦r✐❜✉♥❞❛❀ ❛♦s ❛♠✐❣♦s ❝♦♠ q✉❡♠ tr❛❜❛❧❤❡✐✱ P❛✉❧♦ ❙ér❣✐♦ ♥❛ ❯♥✐s❛✱ ❍✐r❛t❛✱ ❏♦②❝❡✱ ◆❡❧s♦♥ ❡ ❙❤✐r❧❡② ♥♦ ❙❡♥❛❝✱ ❡ ❆♥❛ ▲ú❝✐❛✱ ❈❧á✉❞✐❛ ▼❡❧♦✱ ●❛❜✐r✉✱ ❍❛♠✐❧t♦♥ ❡ ❍❡❧❡♥❛ ♥❛ ▼❡t♦❞✐st❛❀ ❛♦s ❛t✉❛✐s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❞❛ ❯❋▼❙✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❛ ❋❆❈❖▼✲✷ ❈❛r❧♦s ❍✐❣❛✱ ❋á❜✐♦ ■❛✐♦♥❡ ❡ ❱❛❣♥❡r P❡❞r♦tt✐ q✉❡ ❛❝♦♠♣❛♥❤❛r❛♠ ♦s ú❧t✐♠♦s ♠♦♠❡♥t♦s ❞❡ss❡ tr❛❜❛❧❤♦❀ ❛♦s ❛♠✐❣♦s ❞❡ ❞♦✉t♦r❛❞♦ ❆❧❡①❛♥❞r❡✱ ➪❧✈❛r♦✱ ❈❛♦✱ ❈❤❛r❧✐❡ ❇r♦✇♥✱ ❋á❜✐♦ ❱✐❞✉❛♥✐✱ ❏❛✐r ❉♦♥❛❞❡❧❧✐✱ ▼❛r❝♦ ❆✉ré❧✐♦✱ ▼ár✐♦ ▲❡st♦♥✱ ❘♦❣ér✐♦ ❡ ❙❛✐❞✱ ❡ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ■▼❊ ❈♦❡❧❤♦✱ ❈r✐st✐♥❛✱ ❏♦sé ❆✉❣✉st♦✱ ◆❛♠✐✱ P❛✉❧♦ ❋❡♦✜❧♦✛✱ ❨♦s❤✐❤❛r✉ ❡ ❨♦s❤✐❦♦ q✉❡ ♠❡ ❡♥s✐♥❛r❛♠ ♠✉✐t♦✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛♦ Pr♦❢❡ss♦r ❈❛r❧✐♥❤♦s ❞♦ ■▼❊ q✉❡ ❡st❡✈❡ ♣r❡s❡♥t❡ ❡♠ t♦❞❛s ❛s ❡t❛♣❛s ❞❡ ❛✈❛❧✐❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ ❝♦♠ s✉❣❡stõ❡s ♠♦t✐✈❛❞♦r❛s ❡ ❡♥r✐q✉❡❝❡❞♦r❛s✳

❆❣r❛❞❡ç♦ ❛✐♥❞❛ à ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞✉r❛♥t❡ ♦ ❞♦✉t♦r❛❞♦✳

❊❧ó✐ ❆r❛ú❥♦

✐✐✐

❘❡s✉♠♦

❆r❛✉❥♦✱ ❋✳ ❊✳ ❙✳ ❆❧✐♥❤❛♠❡♥t♦s ❡ ❝♦♠♣❛r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s✳ ✷✵✶✷✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✷✳

❆ ❝♦♠♣❛r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s ✜♥✐t❛s é ✉♠❛ ❢❡rr❛♠❡♥t❛ q✉❡ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♠ ✈ár✐❛s ár❡❛s✳ ❈♦♠♣❛r❛♠♦s s❡q✉ê♥❝✐❛s ✐♥❢❡r✐♥❞♦ q✉❛✐s sã♦ ❛s ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ❞❡ s✉❜st✐t✉✐çã♦✱ ✐♥s❡rçã♦ ❡ r❡♠♦çã♦ ❞❡ sí♠❜♦❧♦s q✉❡ tr❛♥s❢♦r♠❛♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ✉♠❛ ♦✉tr❛✳ ❆s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ sã♦ ❡str✉t✉r❛s ❧❛r❣❛♠❡♥t❡ ✉t✐❧✐③❛❞❛s ❡ q✉❡ ❞❡✜♥❡♠ ✉♠ ❝✉st♦ ♣❛r❛ ❝❛❞❛ t✐♣♦ ❞❡ ♦♣❡r❛çã♦ ❞❡ ❡❞✐çã♦✳ ❯♠❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ γ é ✐♥❞❡①❛❞❛ ♣❡❧♦s sí♠❜♦❧♦s ❞♦ ❛❧❢❛❜❡t♦✳ ❆ ❡♥tr❛❞❛ ❞❡ γ ♥❛ ❧✐♥❤❛ a✱ ❝♦❧✉♥❛ b ♠❡❞❡ ♦ ❝✉st♦ ❞❛ ♦♣❡r❛çã♦

❞❡ ❡❞✐çã♦ ♣❛r❛ s✉❜st✐t✉✐r ♦ sí♠❜♦❧♦ a ♣❡❧♦ sí♠❜♦❧♦ b✳ ❆s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ ✐♥❞✉③❡♠

❢✉♥çõ❡s q✉❡ ❛tr✐❜✉❡♠ ✉♠❛ ♣♦♥t✉❛çã♦ ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦✳ ❆❧❣✉♠❛s ❞❡ss❛s ❢✉♥çõ❡s ♣❛r❛ ❛ ❝♦♠♣❛r❛çã♦ ❞❡ ❞✉❛s ❡ ❞❡ ✈ár✐❛s s❡q✉ê♥❝✐❛s sã♦ ❡st✉❞❛❞❛s ♥❡st❛ t❡s❡✳

◗✉❛♥❞♦ ❝❛❞❛ sí♠❜♦❧♦ ❞❡ ❝❛❞❛ s❡q✉ê♥❝✐❛ é ❡❞✐t❛❞♦ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③ ♣❛r❛ tr❛♥s❢♦r♠❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ♦✉tr❛✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ❡str✉t✉r❛ ❝♦♥❤❡❝✐❞❛ ♣♦r ❛❧✐♥❤❛♠❡♥t♦✳ ❉❡s❝r❡✈❡♠♦s ✉♠❛ ❡str✉t✉r❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ q✉❡ ♥ã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❡ ❞❡s❝r❡✈❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ót✐♠❛ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ✉s❛♥❞♦ ✉♠ ❛❧❣♦r✐t♠♦ ❝♦♥❤❡❝✐❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ót✐♠♦✳

❈♦♥s✐❞❡r❛♥❞♦ três ❞✐❢❡r❡♥t❡s ❢✉♥çõ❡s ✐♥❞✉③✐❞❛s ❞❡ ♣♦♥t✉❛çã♦✱ ❝❛r❛❝t❡r✐③❛♠♦s✱ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✱ ❛ ❝❧❛ss❡ ❞❛s ♠❛tr✐③❡s ♣❛r❛ ❛s q✉❛✐s ❛s ❢✉♥çõ❡s ✐♥❞✉③✐❞❛s ❞❡ ♣♦♥t✉❛çã♦ sã♦ ♠étr✐❝❛s ♥❛s s❡q✉ê♥❝✐❛s✳

❉❛❞❛s ❞✉❛s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ γ ❡ δ✱ ❞✐③❡♠♦s q✉❡ ❡❧❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s ♣❛r❛ ✉♠❛ ❞❛❞❛ ❢✉♥çã♦ q✉❡ é ✐♥❞✉③✐❞❛ ♣♦r ✉♠❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ ❡ q✉❡ ❛✈❛❧✐❛ ❛ q✉❛❧✐❞❛❞❡ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r ❞♦✐s ❛❧✐♥❤❛♠❡♥t♦s A ❡ B✱ ✈❛❧❡ ♦ s❡❣✉✐♥t❡✿ ♦ ❛❧✐♥❤❛♠❡♥t♦ A é ✏♠❡❧❤♦r✑ ❞♦ q✉❡ ♦ ❛❧✐♥❤❛♠❡♥t♦ B ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♠❛tr✐③ γ s❡ ❡ s♦♠❡♥t❡ s❡ A é ✏♠❡❧❤♦r✑ ❞♦ q✉❡ ♦ ❛❧✐♥❤❛♠❡♥t♦ B ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♠❛tr✐③ δ✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❞❡t❡r♠✐♥❛♠♦s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ s❡❥❛♠ ❡q✉✐✈❛❧❡♥t❡s✳

❋✐♥❛❧♠❡♥t❡✱ ❞❡✜♥✐♠♦s três ♥♦✈♦s ❝r✐tér✐♦s ♣❛r❛ ♣♦♥t✉❛r ❛❧✐♥❤❛♠❡♥t♦s ❞❡ ✈ár✐❛s s❡q✉ê♥✲ ❝✐❛s✳ ❚♦❞♦s ♦s ❝r✐tér✐♦s ❝♦♥s✐❞❡r❛♠ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛❧✐♥❤❛♠❡♥t♦ ❛❧é♠ ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ♣♦r ❡❧❡ r❡♣r❡s❡♥t❛❞❛s✳ P❛r❛ ❝❛❞❛ ✉♠ ❞♦s ❝r✐tér✐♦s ❞❡✜♥✐❞♦s✱ ♣r♦♣♦♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♠♦str❛♠♦s s❡r ◆P✲❝♦♠♣❧❡t♦✳

✈✐

❆❜str❛❝t

❆r❛✉❥♦✱ ❋✳ ❊✳ ❙✳ ❆❧✐❣♥♠❡♥t ❛♥❞ ❝♦♠♣❛r✐s♦♥ ♦❢ s❡q✉❡♥❝❡s✳ ✷✵✶✷✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✷✳

❈♦♠♣❛r✐s♦♥ ♦❢ ✜♥✐t❡ s❡q✉❡♥❝❡s ✐s ❛ t♦♦❧ ✉s❡❞ t♦ s♦❧✈❡ ♣r♦❜❧❡♠s ✐♥ s❡✈❡r❛❧ ❛r❡❛s✳ ■♥ ♦r❞❡r t♦ ❝♦♠♣❛r❡ s❡q✉❡♥❝❡s✱ ✇❡ ✐♥❢❡r ✇❤✐❝❤ ❛r❡ t❤❡ ❡❞✐t ♦♣❡r❛t✐♦♥s ♦❢ s✉❜st✐t✉t✐♦♥✱ ✐♥s❡rt✐♦♥ ❛♥❞ ❞❡❧❡t✐♦♥ ♦❢ s②♠❜♦❧s t❤❛t tr❛♥s❢♦r♠ ♦♥❡ s❡q✉❡♥❝❡ ✐♥t♦ ❛♥♦t❤❡r✳ ❙❝♦r✐♥❣ ♠❛tr✐❝❡s ❛r❡ ❛ ✇✐❞❡❧② ✉s❡❞ str✉❝t✉r❡ t♦ ❞❡✜♥❡ ❛ ❝♦st ❢♦r ❡❛❝❤ t②♣❡ ♦❢ ❡❞✐t ♦♣❡r❛t✐♦♥✳ ❆ s❝♦r✐♥❣ ♠❛tr✐①γ ✐s ✐♥❞❡①❡❞ ❜② s②♠❜♦❧s ♦❢ ❛♥ ❛❧♣❤❛❜❡t✳ ❚❤❡ ❡♥tr② ✐♥ γ ✐♥ r♦✇ a ❛♥❞ ❝♦❧✉♠♥ b ♠❡❛s✉r❡s t❤❡ ❝♦st ♦❢ t❤❡

❡❞✐t ♦♣❡r❛t✐♦♥ ❢♦r r❡♣❧❛❝✐♥❣ s②♠❜♦❧ a ❜② s②♠❜♦❧ b✳ ❙❝♦r✐♥❣ ♠❛tr✐❝❡s ✐♥❞✉❝❡ ❢✉♥❝t✐♦♥s t❤❛t

❛ss✐❣♥ ❛ s❝♦r❡ ❢♦r ❛ s❡t ♦❢ ❡❞✐t ♦♣❡r❛t✐♦♥s✳ ❙♦♠❡ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❢♦r ❝♦♠♣❛r✐♥❣ t✇♦ ❛♥❞ ♠✉❧t✐♣❧❡ s❡q✉❡♥❝❡s ❛r❡ st✉❞✐❡❞ ✐♥ t❤✐s t❤❡s✐s✳

■❢ ❡❛❝❤ s②♠❜♦❧ ✐s ❡❞✐t❡❞ ❡①❛❝t❧② ♦♥❝❡ ❢♦r tr❛♥s❢♦r♠✐♥❣ ❛ s❡q✉❡♥❝❡ ✐♥t♦ ❛♥♦t❤❡r✱ t❤❡ s❡t ♦❢ ❡❞✐t ♦♣❡r❛t✐♦♥s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ str✉❝t✉r❡ ❝❛❧❧❡❞ ❛❧✐❣♥♠❡♥t✳ ❲❡ ❞❡s❝r✐❜❡ ❛ str✉❝t✉r❡ t♦ r❡♣r❡s❡♥t t❤❡ s❡t ♦❢ ❡❞✐t ♦♣❡r❛t✐♦♥s t❤❛t ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ❝♦♥✈❡♥t✐♦♥❛❧ ❛❧✐❣♥♠❡♥t ❛♥❞ ✇❡ ❞❡s✐❣♥ ❛♥ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ t❤❡ ❝♦st ♦❢ ❛♥ ♦♣t✐♠❛❧ s❡q✉❡♥❝❡ ♦❢ ❡❞✐t ♦♣❡r❛t✐♦♥s ❜② ✉s✐♥❣ ❛ ❦♥♦✇♥ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ t❤❡ ❝♦st ♦❢ ❛♥ ♦♣t✐♠❛❧ ❛❧✐❣♥♠❡♥t✳

❈♦♥s✐❞❡r✐♥❣ t❤r❡❡ ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ ✐♥❞✉❝❡❞ s❝♦r✐♥❣ ❢✉♥❝t✐♦♥s✱ ✇❡ ❝❤❛r❛❝t❡r✐③❡✱ ❢♦r ❡❛❝❤ ♦♥❡ ♦❢ t❤❡♠✱ t❤❡ ❝❧❛ss ♦❢ ♠❛tr✐❝❡s ❢♦r ✇❤✐❝❤ t❤❡ ✐♥❞✉❝❡❞ s❝♦r✐♥❣ ❢✉♥❝t✐♦♥s ❛r❡ ♠❡tr✐❝s ♦♥ s❡q✉❡♥❝❡s✳

●✐✈❡♥ t✇♦ s❝♦r✐♥❣ ♠❛tr✐❝❡s γ ❛♥❞ δ✱ ✇❡ s❛② t❤❡② ❛r❡ ❡q✉✐✈❛❧❡♥t ❢♦r ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ t❤❛t ✐s ✐♥❞✉❝❡❞ ❜② ❛ s❝♦r✐♥❣ ♠❛tr✐① ❛♥❞ t❤❛t ❡✈❛❧✉❛t❡s t❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛❧✐❣♥♠❡♥t ✐❢✱ ❢♦r ❛♥② t✇♦ ❛❧✐❣♥♠❡♥tsA❛♥❞B ♦❢ t✇♦ s❡q✉❡♥❝❡s✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❛❧✐❣♥♠❡♥tA✐s ✏❜❡tt❡r✑ t❤❛♥ B ❝♦♥s✐❞❡r✐♥❣ s❝♦r✐♥❣ ♠❛tr✐① γ ✐❢ ❛♥❞ ♦♥❧② ✐❢ A ✐s ✏❜❡tt❡r✑ t❤❛♥ B ❝♦♥s✐❞❡r✐♥❣ s❝♦r✐♥❣ ♠❛tr✐① δ✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ❞❡t❡r♠✐♥❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r s❝♦r✐♥❣ ♠❛tr✐❝❡s t♦ ❜❡ ❡q✉✐✈❛❧❡♥t✳

❋✐♥❛❧❧②✱ ✇❡ ❞❡✜♥❡ t❤r❡❡ ♥❡✇ ❝r✐t❡r✐❛ ❢♦r s❝♦r✐♥❣ ❛❧✐❣♥♠❡♥ts ♦❢ s❡✈❡r❛❧ s❡q✉❡♥❝❡✳ ❊✈❡r② ❝r✐t❡r✐♦♥ ❝♦♥s✐❞❡rs t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❛❧✐❣♥♠❡♥t ❛♥❞ t❤❡ ❡❞✐t ♦♣❡r❛t✐♦♥s r❡♣r❡s❡♥t❡❞ ❜② ✐t✳ ❆♥ ❛❧❣♦r✐t❤♠ ❢♦r ❡❛❝❤ ❝r✐t❡r✐♦♥ ✐s st✉❞✐❡❞ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✐s s❤♦✇♥ t♦ ❜❡ ◆P✲❝♦♠♣❧❡t❡✳

❑❡②✇♦r❞s✿ ♠❡tr✐❝✱ ❡q✉✐✈❛❧❡♥t ♠❛tr✐❝❡s✱ ♥♦r♠❛❧✐③❡❞ ❛❧✐❣♥♠❡♥t ❝♦st✱ ❡❞✐t ❞✐st❛♥❝❡✱ ❡①t❡♥❞❡❞ ❛❧✐❣♥♠❡♥t✱ s❡q✉❡♥❝❡ ❛❧✐❣♥♠❡♥t✱ ♠✉❧t✐♣❧❡ s❡q✉❡♥❝❡ ❛❧✐❣♥♠❡♥ts✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✶✳✷ ❈♦♥tr✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✷ ❈♦♥❝❡✐t♦s ✼

✷✳✶ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✶✳✶ ❆❧❢❛❜❡t♦ ❡ s❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✶✳✷ ❆❧✐♥❤❛♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✶✳✸ ▼❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷✳✶✳✹ ❉✐❣r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✶✳✺ Pr♦❜❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✶✳✻ ❋❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✷ ❆❧✐♥❤❛♠❡♥t♦ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✷✳✶ ❈r✐tér✐♦vA_γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷✳✷ ❈r✐tér✐♦vN_γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✸ ❆❧✐♥❤❛♠❡♥t♦ ❞❡ ✈ár✐❛s s❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✷✳✸✳✶ ❆❧❣♦r✐t♠♦s ❡①❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✷✳✸✳✷ ❈♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸ ❆❧✐♥❤❛♠❡♥t♦ ❊st❡♥❞✐❞♦ ✸✼

✹ ▼❛tr✐③❡s q✉❡ ✐♥❞✉③❡♠ ♠étr✐❝❛s ✹✸

✹✳✶ ❉✐stâ♥❝✐❛ ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✹✳✷ ❉✐stâ♥❝✐❛ ♥♦r♠❛❧✐③❛❞❛ ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✹✳✸ ❉✐stâ♥❝✐❛ ❞❡ ❡❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻

✹✳✹ ❈♦♠❡♥tár✐♦ ✜♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

✺ ▼❛tr✐③❡s ❡q✉✐✈❛❧❡♥t❡s ✼✸

✺✳✶ ❖♣❡r❛çõ❡s ⊗❡ ⊕ ♥❛s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹

✺✳✷ ❈♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ γ ∼δ ❡♠ B ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

① ❙❯▼➪❘■❖

✻ ❆❧✐♥❤❛♠❡♥t♦ ❞❡ ✈ár✐❛s s❡q✉ê♥❝✐❛s ✽✸

✻✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

✻✳✷ ❆❧❣♦r✐t♠♦s ❡①❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹

✻✳✷✳✶ P♦♥t✉❛çã♦✲SP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✻✳✷✳✷ ❈r✐tér✐♦V1

γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

✻✳✷✳✸ ❈r✐tér✐♦V2

γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

✻✳✷✳✹ ❈r✐tér✐♦V3

γ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

✻✳✸ ❈♦♠♣❧❡①✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷

✻✳✹ ❆❧❣♦r✐t♠♦ ❞❡ ❆♣r♦①✐♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽

✻✳✹✳✶ v✲❡str❡❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾

✻✳✹✳✷ ❉❡s❞♦❜r❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵

✻✳✹✳✸ ❆❧✐♥❤❛♠❡♥t♦ ❝♦♠♣❛tí✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷

✻✳✹✳✹ ❆❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❆ ❝♦♠♣❛r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s ✜♥✐t❛s é ✉♠❛ ❢❡rr❛♠❡♥t❛ q✉❡ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♠ ✈ár✐❛s ár❡❛s t❛✐s ❝♦♠♦ ❜✐♦❧♦❣✐❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❬❙▼✾✼❪✱ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ t❡①t♦s ❬❆●✾✼❪✱ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♣❛❞rõ❡s ❬▼❱✾✸❪✱ ❡t❝✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♠♣❛r❛r ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♠ ♦✉tr❛s ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r q✉❛✐s sã♦ ❛s ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ❞❡ ✐♥s❡rçã♦✱ r❡♠♦çã♦ ❡ s✉❜st✐t✉✐çã♦ q✉❡ tr❛♥s❢♦r♠❛♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ♦✉tr❛s✳

❖ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ s❡q✉ê♥❝✐❛s ✜♥✐t❛s é ✉♠❛ ❡str✉t✉r❛ q✉❡ ♣❡r♠✐t❡ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ q✉❡ tr❛♥s❢♦r♠❛♠ ✉♠❛ s❡q✉ê♥❝✐❛ ♥❛s ❞❡♠❛✐s✳ ❉❡♣❡♥❞❡♥❞♦ ❞❛ ❛♣❧✐❝❛çã♦ ❡ ❞❛ ❢♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛✱ ❞✉❛s ♦✉ ✈ár✐❛s s❡q✉ê♥❝✐❛s ♣♦❞❡♠ s❡r ❝♦♠♣❛r❛❞❛s✳ ❈♦♠♣❛r❛♠♦s ❞✉❛s s❡q✉ê♥❝✐❛s q✉❛♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡r❡♠♦s ✐❞❡♥t✐✜❝❛r s❡ ❞♦✐s ❢r❛❣♠❡♥t♦s ❞❡ ❉◆❆ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♦r✐❣❡♠✳ ❈♦♠♣❛r❛♠♦s ✈ár✐❛s s❡q✉ê♥❝✐❛s q✉❛♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡r❡♠♦s r❡♣r❡s❡♥t❛r r❡❣✐õ❡s ❛❧t❛♠❡♥t❡ ❝♦♥s❡r✈❛❞❛s ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦t❡í♥❛s✳ ❯♠ ❛❧✐✲ ♥❤❛♠❡♥t♦ q✉❡ r❡♣r❡s❡♥t❛ ❛ ❝♦♠♣❛r❛çã♦ ❞❡ ❞✉❛s ♦✉ ♠❛✐s s❡q✉ê♥❝✐❛s é ♦❜t✐❞♦ ❝♦❧♦❝❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠❜❛✐①♦ ❞❛ ♦✉tr❛ ❞❡ ♠♦❞♦ q✉❡ ♦s sí♠❜♦❧♦s ✏❡❞✐t❛❞♦s✑ ✜q✉❡♠ ❡♠ ✉♠❛ ♠❡s♠❛ ❝♦❧✉♥❛✳ ❯♠ sí♠❜♦❧♦ ❡s♣❡❝✐❛❧ é ✉t✐❧✐③❛❞♦ ❞❡ ♠♦❞♦ ❛ ♣r❡❡♥❝❤❡r ❡s♣❛ç♦s ❡♠ ❜r❛♥❝♦ q✉❛♥❞♦ ❛❧✐♥❤❛♠♦s ❛s s❡q✉ê♥❝✐❛s✳ ❆ss✐♠✱

 

a b c c

a b c b

b b c

 ,

 

a b c c a b c b b b c

  ❡

 

a b c c

a b c b

b b c

 

sã♦ ❛❧✐♥❤❛♠❡♥t♦s ❞❡ (abcc,abcb,bbc)✳

❯♠❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ é ❞❡✜♥✐❞❛ ♣❛r❛ ❛tr✐❜✉✐r ✉♠❛ ♣♦♥t✉❛çã♦ ❛ ❝❛❞❛ ❛❧✐♥❤❛♠❡♥t♦✳ ❆ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ✉t✐❧✐③❛❞❛ é ♦ ❝r✐tér✐♦ ♣❛r❛ ♣♦♥t✉❛r ✉♠ ❛❧✐♥❤❛♠❡♥t♦✱ ❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ ✉♠❛ k✲✉♣❧❛ ❞❡ s❡q✉ê♥❝✐❛s ✜♥✐t❛s S ❝✉❥❛ ♣♦♥t✉❛çã♦ ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ é ♠í♥✐♠❛ é ❞✐t♦ ❛❧✐♥❤❛♠❡♥t♦ ót✐♠♦ ❞❡ S ♣❛r❛ ♦ ❝r✐tér✐♦ ❝♦♥s✐❞❡r❛❞♦✳

❙❡❥❛ S ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s k✲✉♣❧❛s ❞❡ s❡q✉ê♥❝✐❛s ✜♥✐t❛s✳ ❉✐③❡♠♦s q✉❡ f : S → R é

✉♠❛ ❢✉♥çã♦ ót✐♠❛ ♣❛r❛ ✉♠ ❞❛❞♦ ❝r✐tér✐♦ s❡ f(S) é ❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ót✐♠♦

❞❡S ∈ S✳

◆❡st❛ t❡s❡✱ ❡st✉❞❛♠♦s ❛❧❣✉♥s ❝r✐tér✐♦s ♣❛r❛ ♣♦♥t✉❛r ❛❧✐♥❤❛♠❡♥t♦s ❞❡ ❞✉❛s ❡ ❞❡ ✈ár✐❛s s❡q✉ê♥❝✐❛s✳ ❊st✉❞❛♠♦s t❛♠❜é♠ ✉♠ ❝r✐tér✐♦ ♣❛r❛ ♣♦♥t✉❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ q✉❡ tr❛♥s❢♦r♠❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ♦✉tr❛ ♠❛s✱ ❝♦♠♦ ✈❡r❡♠♦s✱ ♥ã♦ ♣♦ss✉✐ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ♣❛r❛ r❡♣r❡s❡♥tá✲❧♦✳

✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✶

✶✳✶ ❖❜❥❡t✐✈♦s

❙❡❥❛ Σ ✉♠ ❛❧❢❛❜❡t♦ ❡Σ = Σ∪ { }✱ ♦♥❞❡ ♦ sí♠❜♦❧♦ 6∈ Σé ✉t✐❧✐③❛❞♦ ♣❛r❛ r❡♣r❡s❡♥t❛r

✐♥s❡rçõ❡s ❡ r❡♠♦çõ❡s✳ ❉❡♥♦t❛♠♦s ♣♦rΣ∗ _{♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s ✜♥✐t❛s ❢♦r♠❛❞❛s}

❝♦♠ ❡❧❡♠❡♥t♦s ❞❡ Σ✳

❯♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ❞❡ ✐♥s❡rçã♦✱ r❡♠♦çã♦ ❡ ❞❡ s✉❜st✐t✉✐çã♦ q✉❡ tr❛♥s❢♦r♠❛♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ♦✉tr❛✳ ◆❡ss❡ s❡♥t✐❞♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s é ❡①❛t❛♠❡♥t❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ss❛s ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦✳ P♦r ❡①❡♠♣❧♦✱ ♦ ❛❧✐♥❤❛♠❡♥t♦

a b a

b a a b

r❡♣r❡s❡♥t❛ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ aba ❡♠ baab ❛tr❛✈és ❞❛s s✉❜st✐t✉✐çõ❡s ❞♦ ♣r✐♠❡✐r♦ ❡ ❞♦

ú❧t✐♠♦ a ❞❡ aba♣♦r b ❡ a r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ r❡♠♦çã♦ ❞♦ b ❡ ❛ ✐♥s❡rçã♦ ❛♣r♦♣r✐❛❞❛ ❞❡ ✉♠ a ❡ ✉♠ b ❡♠ aba ❝♦♠♣❧❡t❛♠ ❛ tr❛♥s❢♦r♠❛çã♦✳

❉✉❛s s❡q✉ê♥❝✐❛s ♣♦❞❡♠ s❡r ❛❧✐♥❤❛❞❛s ❞❡ ✈ár✐❛s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s✳ ❯♠❛ ❞❛❞❛ ❡str✉t✉r❛ ❝♦♥❤❡❝✐❞❛ ♣♦r ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❛✈❛❧✐❛r ❛ q✉❛❧✐❞❛❞❡ ❞❡ ✉♠ ❛❧✐♥❤❛✲ ♠❡♥t♦ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s✳ ❯♠❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ γ ♣❛r❛ Σ é ✉♠❛ ♠❛tr✐③ ❞❡ ♥ú♠❡r♦s

r❡❛✐s ✐♥❞❡①❛❞❛ ♥❛s ❧✐♥❤❛s ❡ ♥❛s ❝♦❧✉♥❛s ♣♦r ❡❧❡♠❡♥t♦s ❞❡ Σ ✱ ♦✉ s❡❥❛✱ s❡ a,b sã♦ sí♠❜♦❧♦s

❡♠ Σ ✱ γa→b ❞❡♥♦t❛ ❛ ❡♥tr❛❞❛ ❞❡γ ♥❛ ❧✐♥❤❛ a❡ ❝♦❧✉♥❛ b✳ ❙❡ a,b∈Σ✱ ❡♥tã♦ γa→b é ♦ ❝✉st♦ ❞❡ s✉❜st✐t✉✐çã♦ ❞♦ sí♠❜♦❧♦ a ♣❡❧♦ sí♠❜♦❧♦ b✱ γa→ é ♦ ❝✉st♦ ❞❡ r❡♠♦çã♦ ❞♦ sí♠❜♦❧♦ a ❡

γ →b é ♦ ❝✉st♦ ❞❡ ✐♥s❡rçã♦ ❞♦ sí♠❜♦❧♦b✳ ❆s ♦♣❡r❛çõ❡s ❞❡ s✉❜st✐t✉✐çã♦✱ ✐♥s❡rçã♦ ❡ r❡♠♦çã♦ s♦❜r❡ ✉♠❛ s❡q✉ê♥❝✐❛ sã♦ ❝❤❛♠❛❞❛s ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦✳ ❖ ✈❛❧♦r ❞❡γ → é ✐♥❞❡✜♥✐❞♦✳ ❆

❢✉♥çã♦ vA_γ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ❛❧✐♥❤❛♠❡♥t♦_A ❛ ♣♦♥t✉❛çã♦_vA_γ(A) q✉❡ é ❛ s♦♠❛ ❞♦s ❝✉st♦s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ r❡♣r❡s❡♥t❛❞❛s ❡♠A✳ ❆ ❢✉♥çã♦ ót✐♠❛ ♣❛r❛ ❡ss❡ ❝r✐tér✐♦ éoptA_γ✳ ❆ ❢✉♥çã♦ optA_γ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞✐stâ♥❝✐❛ ❞❡ ▲❡✈❡♥s❤t❡✐♥ ❬▲❡✈✻✻❪ s❡✱ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s

a 6= b ❡♠ Σ ✱ t❡♠♦s γa→a = 0 ❡ γa→b = 1✳ ❊♥tr❡t❛♥t♦✱ ♥❡♠ s❡♠♣r❡ ❛ ❢✉♥çã♦ optAγ ♣♦❞❡

s❡r ❝❤❛♠❛❞❛ ❞❡ ❞✐stâ♥❝✐❛✱ ♦✉ s❡❥❛✱ ♥❡♠ s❡♠♣r❡ optA_γ é ✉♠❛ ♠étr✐❝❛ ❡♠ _Σ∗✳ ❙❡❧❧❡rs ❬❙❡❧✼✹❪ ♠♦str♦✉ q✉❡ s❡ γ é ✉♠❛ ♠étr✐❝❛ ❡♠ Σ ✱ ❡♥tã♦ optA_γ t❛♠❜é♠ é ✉♠❛ ♠étr✐❝❛ ❡♠ _Σ∗✳ ❉❡✲ t❡r♠✐♥❛♠♦s ❬❆❙✵✻❪ ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ❞❛ ♠❛tr✐③ γ ♣❛r❛ q✉❡ optA_γ s❡❥❛ ♠étr✐❝❛ ❡♠ Σ∗✳ ▼♦str❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ s❡ ❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ é

γ =

a b c

a 0 1 3 1

b 1 0 1 1

c 4 1 0 1

1 1 1

, ✭✶✳✶✮

❡♥tã♦ optA_γ é ✉♠❛ ♠étr✐❝❛ ❡♠ _Σ∗✱ ❡♠❜♦r❛ _γ ♥ã♦ s❡❥❛ ♠étr✐❝❛ ❡♠ _Σ ♣♦✐s _{γa→c 6= γc→a} ❡ γa→c 6≤γa→ +γ →c✳

◆❛✈❛rr♦ ❬◆❛✈✵✻❪ q✉❡st✐♦♥♦✉ s❡ ♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ♣♦❞❡r✐❛ s❡r ❡st❡♥❞✐❞♦ ❝❛r❛❝t❡r✐③❛♥❞♦ ❛ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝❧❛ss❡ ❞❡ ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ γ ♦♥❞❡ optA_γ é ✉♠❛ q✉❛s✐♠étr✐❝❛ ❡♠ Σ∗_{✳ ❈♦♠♦ q✉❛s✐♠étr✐❝❛ ♣♦ss✉✐✱ ❡①❝❡t♦ ♣❡❧❛ s✐♠❡tr✐❛✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠❛ ♠étr✐❝❛✱ ♦}

q✉❡st✐♦♥❛♠❡♥t♦ ❞❡ ◆❛✈❛rr♦ ♠♦t✐✈♦✉ ❛ ✐♥✈❡st✐❣❛çã♦ ✐♥❞✐✈✐❞✉❛❧✱ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ♣r♦♣r✐❡✲ ❞❛❞❡s P ❞❡ ✉♠❛ ♠étr✐❝❛✱ ❞❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ❞❡ γ ♣❛r❛ optA_γ ♣♦ss✉✐r ❛ ♣r♦♣r✐❡❞❛❞❡ P ❡♠ Σ∗_✳

❙❡ γ ♥ã♦ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ❡♠ Σ ♦✉✱ ♣❛r❛ ❛❧❣✉♠a∈Σ✱

✶✳✶ ❖❇❏❊❚■❱❖❙ ✸

t❡♥❤❛ ❝♦♠♦ ✜♥❛❧✐❞❛❞❡ ❡ss❛ r❡♣r❡s❡♥t❛çã♦ ❡ ❛ ❝❤❛♠❛♠♦s ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ❡st❡♥❞✐❞♦✳ ❊♥q✉❛♥t♦ ❝❛❞❛ ❝♦❧✉♥❛ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♥✈❡♥❝✐♦♥❛❧ r❡♣r❡s❡♥t❛ ✉♠❛ ú♥✐❝❛ ♦♣❡r❛çã♦ ❞❡ ❡❞✐çã♦✱ ❡♠ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❡st❡♥❞✐❞♦ ❝❛❞❛ ❝♦❧✉♥❛ r❡♣r❡s❡♥t❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦✳

P♦r ❡①❡♠♣❧♦✱ _

   a b a c a b a b    

r❡♣r❡s❡♥t❛ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❡st❡♥❞✐❞♦ ❞❡ (aac,babb)✳ ◆❡ss❡ ❡①❡♠♣❧♦✱ ❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ r❡✲

♣r❡s❡♥t❛ q✉❡a❢♦✐ s✉❜st✐t✉í❞♦ ♣♦rb❀ ❛ s❡❣✉♥❞❛ ❝♦❧✉♥❛ r❡♣r❡s❡♥t❛ q✉❡a♥ã♦ s♦❢r❡✉ ❛❧t❡r❛çã♦❀

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

♣♦r b❀ ❡ ❛ q✉❛rt❛ ❝♦❧✉♥❛ r❡♣r❡s❡♥t❛ q✉❡ a ❢♦✐ ✐♥s❡r✐❞♦ ❡ ❞❡♣♦✐s s✉❜st✐t✉í❞♦ ♣♦r b✳ ❆ss✐♠

❝♦♠♦ ♥♦ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♥✈❡♥❝✐♦♥❛❧✱ ❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❡st❡♥❞✐❞♦ é ❛ s♦♠❛ ❞♦s ❝✉st♦s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ♣♦r ❡❧❡ r❡♣r❡s❡♥t❛❞❛s✳ ❖ ✈❛❧♦r ❞❛ ❢✉♥çã♦ optE_γ q✉❡ r❡❝❡❜❡ ❝♦♠♦ ❛r❣✉♠❡♥t♦ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ ❞❡ s❡q✉ê♥❝✐❛s(s, t)é ❛ ♣♦♥t✉❛çã♦ ❞♦ ❛❧✐♥❤❛♠❡♥t♦ ❡st❡♥✲

❞✐❞♦ ❞❡ (s, t)❞❡ ♠❡♥♦r ♣♦♥t✉❛çã♦✳ ❯♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❡ss❡ ❝r✐tér✐♦ é ❡st✉❞❛❞♦ ♥❡st❛ t❡s❡ ❡✱

❛ss✐♠ ❝♦♠♦ ♣❛r❛ ❛ ❢✉♥çã♦ optA_γ✱ ❡st✉❞❛♠♦s✱ ♣❛r❛ ❝❛❞❛ ♣r♦♣r✐❡❞❛❞❡ _P ❞❡ ✉♠❛ ♠étr✐❝❛✱ ❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ❞❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ γ ♣❛r❛ ❛ ❢✉♥çã♦ optE_γ ♣♦ss✉✐r ❛ ♣r♦♣r✐❡❞❛❞❡ P✳

❱♦❧t❛♥❞♦ ❛♦ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s✱ ✉♠ ♦✉tr♦ ❛s♣❡❝t♦ ❝♦♥s✐❞❡r❛❞♦ ♥❛ ♣♦♥t✉❛çã♦ é ♦ ❢❛t♦ ❞❡ q✉❡optA_γ♣♦❞❡ ♥ã♦ ❞❡s❝r❡✈❡r ❛❞❡q✉❛❞❛♠❡♥t❡ ♦ ❣r❛✉ ❞❡ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ❞✉❛s s❡q✉ê♥❝✐❛s✳ ❙✉♣♦♥❤❛✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ❞✉❛s s❡q✉ê♥❝✐❛s s ❡ t ❞✐❢❡r❡♠ ❛♣❡♥❛s ❡♠ ✉♠ sí♠❜♦❧♦✳ ❆ ❞✐❢❡r❡♥ç❛ é r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ú♥✐❝❛ ♦♣❡r❛çã♦ ❞❡ ❡❞✐çã♦✳ ❙❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ sí♠❜♦❧♦s ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s s❡q✉ê♥❝✐❛s é ♠✉✐t♦ ❣r❛♥❞❡ ❝♦♠♣❛r❛❞❛ ❝♦♠ ♦ ❝✉st♦ ❞❛ ♦♣❡r❛çã♦ ❞❡ ❡❞✐çã♦ ❡♥✈♦❧✈✐❞❛✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❛s s❡q✉ê♥❝✐❛s sã♦ ♣r❛t✐❝❛♠❡♥t❡ ✐❣✉❛✐s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ sí♠❜♦❧♦s ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s s❡q✉ê♥❝✐❛s é ♣❡q✉❡♥❛✱ ❡♥tã♦ ❛s s❡q✉ê♥❝✐❛s ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞❛s ♠✉✐t♦ ❞✐❢❡r❡♥t❡s✳ ▲❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛ ♥❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦✱ ▼❛r③❛❧ ❡ ❱✐❞❛❧ ❬▼❱✾✸❪ ❞❡✜♥✐r❛♠ ♦ ❝r✐tér✐♦ vN_γ q✉❡ é ❛ ♣♦♥t✉❛çã♦ ♥♦r♠❛❧✐③❛❞❛ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ✭❝♦♥✈❡♥❝✐♦♥❛❧✮✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡♥❞♦ A✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s✱vN_{γ(A) = v}A_γ(A)/|A|✱ ♦♥❞❡ _|A| é ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❡A✳ ❆ ❢✉♥çã♦ ót✐♠❛ ♣❛r❛ ♦ ❝r✐tér✐♦ vN_γ é_optN_γ✱ ♦✉ s❡❥❛✱ _optN_γ é ❛ ♣♦♥t✉❛çã♦✲vN_γ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ ♠❡♥♦r ♣♦♥t✉❛çã♦✲vN_γ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s✳

❉✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ ❝r✐tér✐♦ vA_γ ♣❛r❛ ♣♦♥t✉❛r ❛❧✐♥❤❛♠❡♥t♦s✱ _γ s❡r ✉♠❛ ♠étr✐❝❛ ❡♠ _Σ ♥ã♦ ✐♠♣❧✐❝❛ q✉❡ optN_γ s❡❥❛ ✉♠❛ ♠étr✐❝❛ ❡♠ _Σ∗✳ ❊ss❛ ♦❜s❡r✈❛çã♦ ❢♦✐ ❢❡✐t❛ ♣♦r ▼❛r③❛❧ ❡ ❱✐❞❛❧ ❬▼❱✾✸❪ ❛ ♣❛rt✐r ❞❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦

γ =

a b

a 0 5 5

b 5 0 1

5 1

.

◆♦t❡ q✉❡ γ é ✉♠❛ ♠étr✐❝❛ ❡♠Σ ✱ ♠❛s optN_γ ♥ã♦ é ♠étr✐❝❛ ❡♠ _Σ∗ ❞❡s❞❡ q✉❡ optN_{γ(a,b) = min}

vN_γ

a b

, vN_γ

a b

, vN_γ

a b = 3 > 17 6 = 1 2+ 7

3 =vNγ

a a b

+vN_γ

a b b

✹ ■◆❚❘❖❉❯➬➹❖ ✶✳✷

❨✉❥✐❛♥ ❡ ❇♦ ❬❨❇✵✼❪ ♦❜s❡r✈❛r❛♠ q✉❡ é ✉♠ ♣r♦❜❧❡♠❛ ❡♠ ❛❜❡rt♦ ❞❡❝✐❞✐r s❡ ✉♠❛ ♠❛tr✐③ γ ✐♥❞✉③ optN_γ ♠étr✐❝❛ ❡♠ _Σ∗✳ ❊♠ ❬❆❙✵✻❪ r❡s♦❧✈❡♠♦s ❡ss❡ ♣r♦❜❧❡♠❛ ♠♦str❛♥❞♦ ❛s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ❡ ♥❡❝❡ssár✐❛s ❞❡ γ ♣❛r❛ optN_γ s❡r ✉♠❛ ♠étr✐❝❛ ❡♠ _Σ∗✳

◗✉❛♥❞♦ ✉s❛♠♦s ♦ ❝r✐tér✐♦ vA_γ ❡ ❝❛❧❝✉❧❛♠♦s ❛ ♣♦♥t✉❛çã♦ ❞❡ ❞♦✐s ❛❧✐♥❤❛♠❡♥t♦s_A ❡_B ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s ❡ vA_γ(A)≤vA_γ(B)✱ ❡♥t❡♥❞❡♠♦s q✉❡✱ ♣❛r❛ ♦ ❝r✐tér✐♦_vA_γ✱ ♦ ❛❧✐♥❤❛♠❡♥t♦ _A é ✏♠❡❧❤♦r✑ ✭♦✉ ♣❡❧♦ ♠❡♥♦s ♥ã♦ é ♣✐♦r✮ ❞♦ q✉❡ ♦ ❛❧✐♥❤❛♠❡♥t♦B✳ ❊ss❛ ✏❝❧❛ss✐✜❝❛çã♦✑ ❞❡♣❡♥❞❡ ❞❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ γ✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ γ ❡ δ t❛✐s q✉❡

γ =

a b

a 0 1 1

b 1 0 1

1 1

e δ=

a b

a 0 1 2

b 1 0 2

2 2

❡ ♦s s❡❣✉✐♥t❡s ❛❧✐♥❤❛♠❡♥t♦s ❞❡(aba,bab)✿

A=

a b a b a b

❡ B =

a b a b a b

❙❡ ✉t✐❧✐③❛♠♦s ❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦γ✱ ❡♥tã♦B é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ✏♠❡❧❤♦r✑ ❞♦ q✉❡ A ♣♦✐s

vA_{γ(B) = 1 + 0 + 0 + 1 = 2<3 = 1 + 1 + 1 =v}A_γ(A), ♠❛s s❡ ✉t✐❧✐③❛♠♦s δ✱ ❡♥tã♦ é A q✉❡ é ❝♦♥s✐❞❡r❛❞♦ ✏♠❡❧❤♦r✑ ♣♦✐s

vA_{δ(A) = 1 + 1 + 1 = 3<4 = 2 + 0 + 0 + 2 =v}A_δ(B).

❊♥tr❡t❛♥t♦✱ ♥ã♦ é ✈❡r❞❛❞❡ q✉❡ s❡ γ 6=δ✱ ❡♥tã♦ ❡①✐st❡♠ s❡q✉ê♥❝✐❛ss, t ∈ Σ∗ _{❡ ❛❧✐♥❤❛♠❡♥t♦s}

A ❡ B ❞❡ (s, t) t❛✐s q✉❡ A é ♠❡❧❤♦r q✉❡ B s❡ ❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ é γ ❡ B é ♠❡❧❤♦r q✉❡ A s❡ ❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ é δ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♣♦❞❡ ❛❝♦♥t❡❝❡r ❞❡ γ 6=δ ❡

vA_γ(A)≤vA_γ(B) s❡ ❡ s♦♠❡♥t❡ s❡ _vA_δ(A)≤vA_δ(B) ✭✶✳✷✮ ♣❛r❛ ❝❛❞❛ ♣❛r ❞❡ ❛❧✐♥❤❛♠❡♥t♦s A ❡ B ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s✳ ❙❡ ✈❛❧❡ ✭✶✳✷✮✱ ❞✐③❡♠♦s q✉❡ ❛s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ γ ❡ δ sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ◆❡st❛ t❡s❡✱ ❝❛r❛❝t❡r✐③❛♠♦s ❛s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ ❡q✉✐✈❛❧❡♥t❡s ♣❛r❛ ✉♠ ❝♦♥❥✉♥t♦ ❛♠♣❧♦ ❞❡ ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦✳

✶✳✸ ❈❖◆❚❘■❇❯■➬Õ❊❙ ✺

✶✳✷ ❈♦♥tr✐❜✉✐çõ❡s

❆s ♣r✐♥❝✐♣❛✐s ❝♦♥tr✐❜✉✐çõ❡s ❞❡st❛ t❡s❡ ❡ q✉❡ ❢♦r❛♠ ❞✐s❝✉t✐❞❛s ♥❛ s❡çã♦ ❛♥t❡r✐♦r sã♦ r❡s✉✲ ♠✐❞❛s ❛ s❡❣✉✐r✳

✭❛✮ ❉❡✜♥✐♠♦s ❛❧✐♥❤❛♠❡♥t♦ ❡st❡♥❞✐❞♦ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s✱ q✉❡ é ✉♠❛ ❡str✉t✉r❛ ♠❛✐s ❣❡r❛❧ ❞♦ q✉❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❡ q✉❡ ✉t✐❧✐③❛♠♦s ♣❛r❛ r❡♣r❡s❡♥t❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ q✉❡ tr❛♥s❢♦r♠❛♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ♦✉tr❛✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ ♦ ❝r✐tér✐♦ optE_γ ♣❛r❛ ♣♦♥t✉❛r ❛❧✐♥❤❛♠❡♥t♦s ❡st❡♥❞✐❞♦s ❡ ❞❡s❝r❡✈❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❝♦♠♣✉t❛r optE_γ ❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❡st❡♥❞✐❞♦ ót✐♠♦✳

✭❜✮ ❉❡s❝r❡✈❡♠♦s ❡ ❛♠♣❧✐❛♠♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡♠ ❬❆❙✵✻❪ ❝❛r❛❝t❡r✐③❛♥❞♦✱ ♣❛r❛ ❝❛❞❛ ♣r♦♣r✐❡❞❛❞❡ P ❞❡ ✉♠❛ ♠étr✐❝❛✱ ❛ ❝❧❛ss❡ ❞❛s ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦γ ♣❛r❛ ❛s q✉❛✐s ❛ ❢✉♥çã♦ ✐♥❞✉③✐❞❛optA_γ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡_P ❡♠_Σ∗❀ ❡ ❢❛③❡♠♦s ♦ ♠❡s♠♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛s ❢✉♥çõ❡s optN_γ ❡_optE_γ✳

✭❝✮ ▼♦str❛♠♦s ❝♦♠♦ ✐❞❡♥t✐✜❝❛r ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦ ❡q✉✐✈❛❧❡♥t❡s ♣❛r❛ ✉♠❛ ❛♠♣❧❛ ❝❧❛ss❡ ❞❡ ♠❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦✳

✭❞✮ ❉❡✜♥✐♠♦s V1

γ✱ Vγ2 ❡ V3γ✱ q✉❡ sã♦✱ ❝♦♠♦ ✈❡r❡♠♦s✱ três ♥♦✈♦s ❝r✐tér✐♦s ♣❛r❛ ❛ ♣♦♥t✉❛çã♦

❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ ✈ár✐❛s s❡q✉ê♥❝✐❛s✳ ❊❧❡s ❧❡✈❛♠ ❡♠ ❝♦♥t❛ ♥ã♦ s♦♠❡♥t❡ ❛ s♦♠❛ ❞♦s ❝✉st♦s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦ ❡♠ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ♠❛s t❛♠❜é♠ ♦ s❡✉ ❝♦♠♣r✐♠❡♥t♦✳ P❛r❛ ❝❛❞❛ ✉♠ ❞❡ss❡s ❝r✐tér✐♦s✱ ❞❡s❝r❡✈❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ❡①❛t♦ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ♣♦♥✲ t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ót✐♠♦ ❡ ♠♦str❛♠♦s q✉❡ ❛ ✈❡rsã♦ ❞❡ ❞❡❝✐sã♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ é ◆P✲❝♦♠♣❧❡t❛✳

❉❡s❝r❡✈❡♠♦s ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ ♣♦♥t✉❛çã♦ ❛♣r♦①✐♠❛❞❛ ❞❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ót✐♠♦ ❞❡ ✈ár✐❛s s❡q✉ê♥❝✐❛s ♣❛r❛ ♦ ❝r✐tér✐♦ ♣♦♥t✉❛çã♦✲SP q✉❛♥❞♦ ❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦γ✐♥❞✉③ ♠étr✐❝❛ ♥❛ ❢✉♥çã♦optA_γ✳ ❊ss❡ ❛❧❣♦r✐t♠♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ t❛♠❜é♠ ❝♦♠♦ ❛❧❣♦r✐t♠♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ ♣♦♥t✉❛çã♦ ❛♣r♦①✐♠❛❞❛ ❞❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ót✐♠♦ ♣❛r❛ ♦ ❝r✐tér✐♦V2

γ q✉❛♥❞♦ ❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦

γ ✐♥❞✉③ ♠étr✐❝❛ ❡♠optN_γ✳

✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦

❈❛♣ít✉❧♦ ✷

❈♦♥❝❡✐t♦s

◆❡st❡ ❝❛♣ít✉❧♦ ❞✐ss❡rt❛♠♦s s♦❜r❡ ♦s ❝♦♥❝❡✐t♦s ✉t✐❧✐③❛❞♦s ♥❡st❛ t❡s❡✳ ❉❡s❝r❡✈❡♠♦s ♦s ♣r✐♥✲ ❝✐♣❛✐s ❛❧❣♦r✐t♠♦s ❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦s ♣r♦❜❧❡♠❛s ❡st✉❞❛❞♦s ❡ q✉❡ sã♦ ♠❡♥❝✐♦♥❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳ ◆❛ ❙❡çã♦ ✷✳✶ ❢❛③❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❜ás✐❝❛s q✉❡ sã♦ ✉t✐❧✐③❛❞❛s ♥❛s ❞❡♠❛✐s s❡çõ❡s ❡ ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ ◆❛s ❙❡çõ❡s ✷✳✷ ❡ ✷✳✸ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ♥♦ss❛ ♣❡sq✉✐s❛ ❜✐❜❧✐♦❣rá✜❝❛ s♦❜r❡ ❛❧✐♥❤❛♠❡♥t♦s ❞❡ ❞✉❛s ❡ ❞❡ ✈ár✐❛s s❡q✉ê♥❝✐❛s✳

✷✳✶ Pr❡❧✐♠✐♥❛r❡s

✷✳✶✳✶ ❆❧❢❛❜❡t♦ ❡ s❡q✉ê♥❝✐❛s

❯♠ ❛❧❢❛❜❡t♦ Σ é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ♥ã♦✲✈❛③✐♦ ❞❡ sí♠❜♦❧♦s✳ ❉❡♥♦t❛♠♦s ✉♠❛ s❡q✉ê♥❝✐❛

✜♥✐t❛ s s♦❜r❡ Σ ♣♦r s(1)s(2). . . s(n)✱ ♦♥❞❡ s(j) ∈ Σ✳ ❉✐③❡♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ s✱

❞❡♥♦t❛❞♦ ♣♦r|s|✱ én✳ ❉❡♥♦t❛♠♦s ❛ s❡q✉ê♥❝✐❛s(p)s(p+1). . . s(q)♣♦rs(p . . . q)✱ ❡ ❛ s❡q✉ê♥❝✐❛

✈❛③✐❛✱ q✉❡ é ❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✐❣✉❛❧ ❛ ③❡r♦✱ ♣♦r ǫ✳ ❆ s❡q✉ê♥❝✐❛ an é ❛ s❡q✉ê♥❝✐❛

❢♦r♠❛❞❛ ♣♦rnsí♠❜♦❧♦sa✳ ❉❡♥♦t❛♠♦s ❛ s❡q✉ê♥❝✐❛ ❢♦r♠❛❞❛ ♣❡❧❛ ❝♦♥❝❛t❡♥❛çã♦ ❞❛s s❡q✉ê♥❝✐❛s

s ❡ t ♣♦r st✳

✷✳✶✳✷ ❆❧✐♥❤❛♠❡♥t♦s

❙❡❥❛ Σ = Σ∪ { }✱ ♦♥❞❡ 6∈ Σ✳ ❈❤❛♠❛♠♦s ❡s♣❛ç♦ ♦ sí♠❜♦❧♦ ✳ ❊❧❡ é ✉s❛❞♦ ♣❛r❛

r❡♣r❡s❡♥t❛r ✉♠❛ ✐♥s❡rçã♦ ♦✉ ✉♠❛ r❡♠♦çã♦ ❡♠ ♦♣❡r❛çõ❡s ❞❡ ❡❞✐çã♦✳ ❙❡❥❛ S = (s1, . . . , sk)

✉♠❛ k✲✉♣❧❛ ❞❡ ❡❧❡♠❡♥t♦s ❡♠ Σ∗_{✳ ❯♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ S é ✉♠❛ k✲✉♣❧❛ A = (s}′

1, . . . , s′k) ❞❡

❡❧❡♠❡♥t♦s ❡♠ Σ∗✱ ♦♥❞❡

✭❛✮ ❝❛❞❛ s❡q✉ê♥❝✐❛ s′

h é ♦❜t✐❞❛ ✐♥s❡r✐♥❞♦✲s❡ ❡s♣❛ç♦s ❡♠ sh✱

✭❜✮ |s′

h|=|s′i| ♣❛r❛ ❝❛❞❛ h, i ❡

✭❝✮ ♥ã♦ ❡①✐st❡j t❛❧ q✉❡ s′

1(j) =. . .=s′k(j) = ✳

◗✉❛♥❞♦ ♦ ❛❧❢❛❜❡t♦ ❞♦s sí♠❜♦❧♦s q✉❡ ❢♦r♠❛♠ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ k✲✉♣❧❛ é Σ ✱ ♣r❡❢❡r✐♠♦s

❡s❝r❡✈ê✲❧❛ ❡♥tr❡ ❝♦❧❝❤❡t❡s ✏[✑ ❡ ✏]✑✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♠♦s ♣❛rê♥t❡s❡s ✏(✑ ❡ ✏)✑ ♣❛r❛ ❡s✲

❝r❡✈❡r k✲✉♣❧❛s ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❢♦r♠❛❞♦s ♣♦r sí♠❜♦❧♦s ❡♠ Σ✳ ❊♥tã♦✱ ♣r❡❢❡r✐♠♦s ❡s❝r❡✈❡r [s′

1, . . . , s′k] ❡♠ ✈❡③ ❞❡ (s′1, . . . , s′k) ♣❛r❛ ❞❡♥♦t❛r ♦ ❛❧✐♥❤❛♠❡♥t♦ ❢♦r♠❛❞♦ ♣❡❧❛s s❡q✉ê♥❝✐❛s

s′₁, . . . , s′k✳ ❉✐③❡♠♦s q✉❡ ❛k✲✉♣❧❛ ❞❡ sí♠❜♦❧♦s[s′1(j), . . . , s′k(j)]é ❛ ❝♦❧✉♥❛j ❞♦ ❛❧✐♥❤❛♠❡♥t♦ [s′

1, . . . , s′k] ❡ ❞❡♥♦t❛♠♦s ❛ ❝♦❧✉♥❛ j ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ A ♣♦rA(j)✳ ❙❡

A=[. . . , s′h, . . . , s′i, . . .],

✽ ❈❖◆❈❊■❚❖❙ ✷✳✶

❞✐③❡♠♦s q✉❡ ♦ ♣❛r ❞❡ sí♠❜♦❧♦s [s′

h(j), s′i(j)] ❛❧✐♥❤❛ ❡♠ A ♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱ q✉❡ s′h(j) ❡

s′

i(j) ❡stã♦ ❛❧✐♥❤❛❞♦s ❡♠ A s❡ ❛ ♦r❞❡♠ ❡stá s✉❜❡♥t❡♥❞✐❞❛✱ ❡ q✉❡|A|=|s′i| é ♦ ❝♦♠♣r✐♠❡♥t♦

❞♦ ❛❧✐♥❤❛♠❡♥t♦ A✳

❙ã♦ ❝♦♥❤❡❝✐❞♦s ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❛♥t❡s ♣❛r❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦✳ ❋❛t♦ ✶✳ ❙❡❥❛ A ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ (s1, . . . , sk)✳ ❊♥tã♦✱ maxi{|si|} ≤ |A| ≤P_i|si|✳

Pr♦✈❛✳ ❙✉♣♦♥❤❛ q✉❡ ié ✉♠ ✐♥t❡✐r♦ ❡ 1≤i≤k✳ ❖❜s❡r✈❡ q✉❡ ♦s sí♠❜♦❧♦s ❞❡si ❞❡✈❡♠ ❡st❛r

❡♠ ❝♦❧✉♥❛s ❞✐st✐♥t❛s ❡♠ A✳ ▲♦❣♦✱ A ❞❡✈❡ t❡r ♣❡❧♦ ♠❡♥♦s |si| ❝♦❧✉♥❛s✳ ❈♦♠♦ ❛ ♦❜s❡r✈❛çã♦

✈❛❧❡ ♣❛r❛ q✉❛❧q✉❡r i✱ t❡♠♦s q✉❡ maxi{|si|} ≤ |A|✳

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❛❧✐♥❤❛♠❡♥t♦✱ ❝❛❞❛ ❝♦❧✉♥❛ ❞❡ A ❞❡✈❡ t❡r ♣❡❧♦ ♠❡♥♦s ✉♠ sí♠❜♦❧♦ ❞❡ ❛❧❣✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ s1, . . . , sk✳ ❈♦♠♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ sí♠❜♦❧♦s ❡♠ (s1, . . . , sk) é

P

i|si|✱

s❡❣✉❡ q✉❡ |A| ≤P_i|si|✳

❙❡❣✉❡ ❞♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ q✉❡ maxi{|si|} ≤ |A| ≤

P

i|si|✳

❯♠ ❛❧✐♥❤❛♠❡♥t♦ t❛♠❜é♠ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♥♦ ❢♦r♠❛t♦ ♠❛tr✐❝✐❛❧ ❝♦❧♦❝❛♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛ s♦❜r❡ ♦✉tr❛✳ ❆ss✐♠✱ ♦s ❛❧✐♥❤❛♠❡♥t♦s

[aaa ,ab , cac] ❡ [ aaa ,ab , ca c]

❞❡(aaa,ab,cac)♣♦❞❡♠ s❡r ✈✐s✉❛❧✐③❛❞♦s ♣♦r

 

a a a a b

c a c

  ❡

 

a a a a b

c a c

 .

❙❡❥❛♠ I = {i1, . . . , im} ⊆ {1, . . . , k} t❛❧ q✉❡ i1 < . . . < im ❡ A = [s′1, . . . , s′k] ✉♠

❛❧✐♥❤❛♠❡♥t♦ ❞❡ S = (s1, . . . , sk)✳ ❊s❝r❡✈❡♠♦s SI ♣❛r❛ ❞❡♥♦t❛r ❛ m✲✉♣❧❛ (si1, . . . , sim)✳ ❖ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ SI ✐♥❞✉③✐❞♦ ♣♦r A é ♦ ❛❧✐♥❤❛♠❡♥t♦ AI ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞♦ ❛❧✐♥❤❛♠❡♥t♦ A

❝♦♥s✐❞❡r❛♥❞♦ s♦♠❡♥t❡ ❛s s❡q✉ê♥❝✐❛s ❝♦rr❡s♣♦♥❞❡♥t❡s às s❡q✉ê♥❝✐❛s ❡♠ SI ❡✱ ❞❛ ❡str✉t✉r❛

r❡s✉❧t❛♥t❡✱ ❡❧✐♠✐♥❛♥❞♦ ❛s ❝♦❧✉♥❛s ♦♥❞❡ t♦❞♦s ♦s sí♠❜♦❧♦s sã♦ ✐❣✉❛✐s ❛ ✳ ◆♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱

a a a a b

é ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ (aaa,ab) ✐♥❞✉③✐❞♦ ♣♦r [aaa ,ab , cac] ❞❡(aaa,ab,cac)✳

❉❡♥♦t❛♠♦s ♣♦r AS ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❛❧✐♥❤❛♠❡♥t♦s ❞❡S✳

✷✳✶✳✸ ▼❛tr✐③❡s ❞❡ ♣♦♥t✉❛çã♦

❈❛❞❛ ❝r✐tér✐♦ ❞❡ ♣♦♥t✉❛çã♦ ❛q✉✐ ❡st✉❞❛❞♦ t❡♠ ✉♠❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ ❛ss♦❝✐❛❞❛ à s✉❛ ❞❡✜♥✐çã♦✳ ❯♠❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ é ✉♠❛ ♠❛tr✐③ q✉❡ ♣♦ss✉✐ ♥ú♠❡r♦s r❡❛✐s ❝♦♠♦ ❡♥tr❛❞❛s ❡ q✉❡ sã♦ ✐♥❞❡①❛❞❛s ♥❛s ❧✐♥❤❛s ❡ ♥❛s ❝♦❧✉♥❛s ♣♦r ❡❧❡♠❡♥t♦s ❡♠ Σ ✳ P❛r❛ a,b ∈ Σ ❡ ✉♠❛

✷✳✶ P❘❊▲■▼■◆❆❘❊❙ ✾

✷✳✶✳✹ ❉✐❣r❛❢♦s

❯♠ ❞✐❣r❛❢♦ D❝♦♥s✐st❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ V(D)❞❡ ❡❧❡♠❡♥t♦s ❝❤❛♠❛❞♦s ✈ért✐❝❡s✱ ✉♠

❝♦♥❥✉♥t♦ E(D) ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ✈ért✐❝❡s ❝❤❛♠❛❞♦s ❛r❝♦s ❡ ✉♠❛ ❢✉♥çã♦

custo:E(D)→R.

❖ ❛r❝♦ (v, w) ∈ E(D) é ❞❡♥♦t❛❞♦ ♣♦r v → w ❡ ❞✐③❡♠♦s q✉❡ v → w s❛✐ ❞♦ ✈ért✐❝❡ v ❡ q✉❡ ❡♥tr❛ ♥♦ ✈ért✐❝❡ w✳ ❉✐③❡♠♦s q✉❡ ♦ ❝✉st♦ ❞♦ ❛r❝♦ e é custo(e) ❡✱ s♦❜r❡❝❛rr❡❣❛♥❞♦ ❛ ❢✉♥çã♦

custo✱ ❞❡✜♥✐♠♦s ♦ ❝✉st♦ ❞♦ ❞✐❣r❛❢♦ D ❝♦♠♦ custo(D) = P_e∈E(D)custo(e)✳

❙❡❥❛♠ v0, v1, . . . , vn ∈V(D) ❡e1, e2, . . . , en ∈E(D)✱ t❛✐s q✉❡ ei =vi−1 →vi ♣❛r❛ ❝❛❞❛ i✳

❉✐③❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❛r❝♦s P = (e1, . . . , en)é ✉♠ ❝❛♠✐♥❤♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ n ❞❡v0 ❛ vn✱ q✉❡ ❝❛❞❛ ❛r❝♦ ei ❛♣❛r❡❝❡ ❡♠ P✱ ❡ q✉❡ v0, v1, . . . , vn ❡ e1, e2, . . . , en sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

♦s ✈ért✐❝❡s ❡ ♦s ❛r❝♦s ❞♦ ❝❛♠✐♥❤♦ P✳ ❖ ❝✉st♦ ❞♦ ❝❛♠✐♥❤♦ P é custo(P) = Pn_i=1custo(ei)✳

❙❡ ♥ã♦ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ P′ ❞❡ v0 ❛ vn t❛❧ q✉❡ custo(P′) < custo(P)✱ ❞✐③❡♠♦s q✉❡ P é

✉♠ ❝❛♠✐♥❤♦ ót✐♠♦ ❞❡ v0 ❛ vn✳ ❙❡ v0 = vn✱ ❞✐③❡♠♦s q✉❡ P é ✉♠ ❝✐❝❧♦✳ ❙❡ P é ✉♠ ❝✐❝❧♦ ❡

custo(P)<0✱ ❞✐③❡♠♦s q✉❡ P é ✉♠ ❝✐❝❧♦ ♥❡❣❛t✐✈♦✳ ❙❡ ♦s ❛r❝♦s ❡♠ ✉♠ ❝✐❝❧♦P sã♦ ❞♦✐s ❛ ❞♦✐s ❞✐st✐♥t♦s✱ ❞✐③❡♠♦s q✉❡P é ✉♠ ❝✐❝❧♦ s✐♠♣❧❡s✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛r❝♦s{e1, . . . , en}

❞❡✜♥❡ ♦ ❝✐❝❧♦ P s❡P é ✉♠ ❝✐❝❧♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ n✱ é s✐♠♣❧❡s ❡ ❝❛❞❛ei ❛♣❛r❡❝❡ ❡♠ P✳

❙❡❥❛♠ P = (e1, . . . , en)❡Q= (f1, . . . , fm)❝❛♠✐♥❤♦s ❞❡v ❛z ❡ ❞❡ z ❛w r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ❞❡♥♦t❛♠♦s ❛ ❝♦♥❝❛t❡♥❛çã♦ ❞♦s ❝❛♠✐♥❤♦s P ❡ Q ♣♦r

P Q= (e1, e2, . . . , en, f1, f2, . . . , fm)

q✉❡ é ✉♠ ❝❛♠✐♥❤♦ ❞❡ v ❛ w✳

❉✐③❡♠♦s q✉❡ ✉♠ ❞✐❣r❛❢♦ D é ❡✉❧❡r✐❛♥♦ s❡✱ ♣❛r❛ ❝❛❞❛ ✈ért✐❝❡ v ∈ V(D)✱ ❛s q✉❛♥t✐❞❛❞❡s

❞❡ ❛r❝♦s q✉❡ ❡♥tr❛♠ ❡ q✉❡ s❛❡♠ ❞❡ v sã♦ ✐❣✉❛✐s✳ ❯♠ r❡s✉❧t❛❞♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦ ❬●P❇+_{✵✷❪ é}

♦ s❡❣✉✐♥t❡✳

❚❡♦r❡♠❛ ✷✳ ❯♠ ❞✐❣r❛❢♦ D é ❡✉❧❡r✐❛♥♦ s❡ ❡ s♦♠❡♥t❡ s❡ ❡①✐st❡ ✉♠❛ ♣❛rt✐çã♦ ❞❡ E(D) ♦♥❞❡

❝❛❞❛ ❝♦♥❥✉♥t♦ ❞❛ ♣❛rt✐çã♦ ❞❡✜♥❡ ✉♠ ❝✐❝❧♦✳

P❛r❛ ✉♠❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ γ✱ ♦ ❞✐❣r❛❢♦D(γ) é t❛❧ q✉❡

V(D) = Σ ,

E(D) = (Σ ×Σ )\ {( , )} ❡

custo(v →w) = γv→w ♣❛r❛ ❝❛❞❛ v →w∈E(D).

✷✳✶✳✺ Pr♦❜❧❡♠❛s

❯♠❛ ✐♥stâ♥❝✐❛ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ❡♠ ❣❡r❛❧ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❡q✉ê♥❝✐❛s✳ ❙❡♥❞♦ ✉♠ ❝♦♥❥✉♥t♦✱ ❛ ♦r❞❡♠ ❞❛s s❡q✉ê♥❝✐❛s ♥❛ ✐♥stâ♥❝✐❛ ♥ã♦ é ✐♠♣♦rt❛♥t❡✳ ■♠♣❧✐❝✐t❛♠❡♥t❡ ♥❡ss❡s ❝❛s♦s ❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ ❞❡✈❡ s❡r s✐♠étr✐❝❛✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❡ss❡ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r q✉❛♥❞♦ ❛ss✉♠✐♠♦s q✉❡ ✉♠❛ ✐♥stâ♥❝✐❛ é ✉♠❛ ❧✐st❛ ♦r❞❡♥❛❞❛ ❞❡ s❡q✉ê♥❝✐❛s ❡ q✉❡ ❛s ♠❛tr✐③❡s ✉t✐❧✐③❛❞❛s ♣♦❞❡♠ ♥ã♦ s❡r s✐♠étr✐❝❛s✳ ❈♦♥s✐❞❡r❛♠♦s ❡♠ ♥♦ss❛s ❞❡✜♥✐çõ❡s ❛ ❢♦r♠✉❧❛çã♦ ♠❛✐s ❣❡r❛❧✳

✶✵ ❈❖◆❈❊■❚❖❙ ✷✳✷

s❡q✉ê♥❝✐❛s s ❡ t✳ P❛r❛ ♦✉tr♦s ❝r✐tér✐♦s ❡st✉❞❛❞♦s ♥❡st❡ t❡①t♦✱ ✐♥❝❧✉s✐✈❡ ❛q✉❡❧❡s ✉t✐❧✐③❛❞♦s ♣❛r❛ ✈ár✐❛s s❡q✉ê♥❝✐❛s✱ ✉♠❛ ❛❞❛♣t❛çã♦ ó❜✈✐❛ ❞❡ss❡ ♠ét♦❞♦ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ❞❡ ♣♦♥t✉❛çã♦ ót✐♠❛ ❝♦♥s✉❧t❛♥❞♦ ✉♠❛ t❛❜❡❧❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛✳ ❊♥tã♦✱ ❞❡s❞❡ q✉❡ ♦s ❛❧❣♦r✐t♠♦s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❣❡r❛❧ ❝♦♥str♦❡♠ t❛❜❡❧❛s ❞❡ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛✱ ♦♣t❛♠♦s ♥❡st❡ t❡①t♦ ❡♠ ❞❡✜♥✐r ❡ tr❛t❛r ♦s ♣r♦❜❧❡♠❛s ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ❝♦♠♦ ✉♠ ♣r♦❜❧❡♠❛ ♣❛r❛ ❡♥❝♦♥tr❛r s♦♠❡♥t❡ ❛ ♣♦♥t✉❛çã♦ ❞❡ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ót✐♠♦ ❞❡ ✉♠❛ k✲✉♣❧❛ ❞❡ s❡q✉ê♥❝✐❛s✱ ♦♠✐t✐♥❞♦ ❞❡t❛❧❤❡s ❞❡ ❝♦♠♦ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ót✐♠♦ ♣♦❞❡ s❡r ♦❜t✐❞♦✳

❆❧é♠ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦✱ t❛♠❜é♠ tr❛t❛♠♦s ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❞❡❝✐sã♦ ♣❛r❛ ❞✐s✲ ❝✉t✐r r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡✳ ◆❡st❡ ❝❛s♦✱ ❞❡♥♦t❛♠♦s ♣♦r P(I)❛ r❡s♣♦st❛ ♦✉ s♦❧✉çã♦ ❞❡

✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐sã♦ P ♣❛r❛ ❛ ✐♥stâ♥❝✐❛ I✱ ♦✉ s❡❥❛✱ P(I)∈ {❙✐♠,◆ã♦}✳

✷✳✶✳✻ ❋❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦

❙❡❥❛♠ I ♦ ❝♦♥❥✉♥t♦ ❞❡ ✐♥stâ♥❝✐❛s ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛çã♦✱OPT_(I) ✉♠ ♥ú♠❡r♦ ❛ss♦❝✐❛❞♦ ❛ ❝❛❞❛ ✐♥stâ♥❝✐❛I ∈I❡A_(I)≥OPT_(I)✉♠ ♥ú♠❡r♦ ❝♦♠♣✉t❛❞♦ ♣♦r ✉♠ ❛❧❣♦r✐t♠♦

A ❝♦♠ ❡♥tr❛❞❛ _I✳ ❉✐③❡♠♦s q✉❡ A é ✉♠❛ _α✲❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s❡ A_(I)≤α·OPT_(I),

♣❛r❛ t♦❞❛ ✐♥stâ♥❝✐❛I✱ ♦♥❞❡α é ✉♠ ♥ú♠❡r♦ ❝♦♥st❛♥t❡✳ ❉✐③❡♠♦s t❛♠❜é♠ q✉❡ α é ✉♠❛ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ A✳

✷✳✷ ❆❧✐♥❤❛♠❡♥t♦ ❞❡ ❞✉❛s s❡q✉ê♥❝✐❛s

✷✳✷✳✶ ❈r✐tér✐♦

v

A

❙❡❥❛♠ s, t∈Σ∗✱ n =|s|, m=|t|✳ ❯♠ ❝r✐tér✐♦ s✐♠♣❧❡s ♣❛r❛ ♣♦♥t✉❛r ✉♠ ❛❧✐♥❤❛♠❡♥t♦ ✉s❛ ❛ ❢✉♥çã♦ vA_γ✳ P❛r❛ ✉♠ ❛❧✐♥❤❛♠❡♥t♦ _[s′_{, t}′_]❞❡ _{(s, t)} t❡♠♦s q✉❡

vA_γ([s′_{, t}′_{]) =}

|[s′,t′]|

X

j=1

γs′_(j)→t′_(j). ❉✐③❡♠♦s q✉❡ vA_γ([s′_{, t}′_])é ❛ ♣♦♥t✉❛çã♦✲_vA_γ ❞♦ ❛❧✐♥❤❛♠❡♥t♦_[s′_{, t}′_]✳

❆ ❢✉♥çã♦ ót✐♠❛ ♣❛r❛ ❡ss❡ ❝r✐tér✐♦ é optA_γ✱ ♦✉ s❡❥❛✱ _optA_{γ(s, t) = minA∈A}

{s,t}{vAγ(A)}✳ ❯♠ ❛❧✐♥❤❛♠❡♥t♦A❞❡(s, t)t❛❧ q✉❡vA_{γ(A) = opt}A_{γ(s, t)}é ❝❤❛♠❛❞♦ ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ❆✲ót✐♠♦ ❞❡(s, t)✳ ❖ ♣r♦❜❧❡♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ é

Pr♦❜❧❡♠❛ ✶✳ ❆P❙✿ ❆❧✐♥❤❛♠❡♥t♦ ❞❡ ✉♠ ♣❛r ❞❡ s❡q✉ê♥❝✐❛s

❉❛❞♦s s, t∈Σ∗_{✱ ❞❡t❡r♠✐♥❛roptAγ(s, t)✱ ♦♥❞❡γ é ✉♠❛ ♠❛tr✐③ ❞❡ ♣♦♥t✉❛çã♦ ✜①❛✳}

❯♠ ❛❧❣♦r✐t♠♦ ✐♥❣ê♥✉♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❞♦ Pr♦❜❧❡♠❛ ❆P❙ ❝❛❧❝✉❧❛ ❛s ♣♦♥t✉❛çõ❡s✲vA_γ ❞❡ t♦❞♦s ♦s ❛❧✐♥❤❛♠❡♥t♦s ❡♠ A{s,t} ❡ ❞❡✈♦❧✈❡ ❛ ♠❡♥♦r ♣♦♥t✉❛çã♦ ❝♦♠♣✉t❛❞❛✳ ❖ ♣r♦❜❧❡♠❛ ❝♦♠

❡ss❛ ❡str❛té❣✐❛ é q✉❡✱ ❡♠ ❣❡r❛❧✱ ♦ ✈❛❧♦r ❞❡ |A{s,t}| é ♠✉✐t♦ ❣r❛♥❞❡✳ ❊st✐♠❛♠♦s ❛ s❡❣✉✐r ❡ss❡

✈❛❧♦r s✉♣♦♥❞♦ q✉❡ n=|s|=|t|✳

Pr✐♠❡✐r❛♠❡♥t❡ ❝♦♥t❛♠♦s q✉❛♥t♦s ❛❧✐♥❤❛♠❡♥t♦s ❞✐❢❡r❡♥t❡s [s′_{, t}′_{]❞❡ ❝♦♠♣r✐♠❡♥t♦2n−j}

❡①✐st❡♠ ♣❛r❛ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ j✳ ❈♦♠♦ |[s′, t′]| = 2n−j✱ t❡♠♦s q✉❡ |s′_{|= |t}′_{| = 2n−j✱ ♦}

q✉❡ ✐♠♣❧✐❝❛✱ s❡♥❞♦ n=|s|=|t|✱ q✉❡ ❛s s❡q✉ê♥❝✐❛s s′ _{❡ t}′ _{♣♦ss✉❡♠✱ ❝❛❞❛ ✉♠❛✱n−j ❡s♣❛ç♦s✳}

❙❡♥❞♦|s′_{|= 2n−j✱ s❡❣✉❡ q✉❡ ❡①✐st❡♠} 2n−j n−j

♠♦❞♦s ❞✐❢❡r❡♥t❡s ❞❡ ❞❡✜♥✐r ❛ ♣♦s✐çã♦ ❞♦sn−j ❡s♣❛ç♦s ❡♠ s′_{✳ ❯♠❛ ✈❡③ ❢❡✐t♦ ✐st♦✱ ❞❡s❞❡ q✉❡ |t}′_{| = 2n−j ❡ q✉❡ ♥❡♥❤✉♠❛ ❝♦❧✉♥❛ ❞❡[s}′_{, t}′_]

♣♦❞❡ ♣♦ss✉✐r ❞♦✐s ❝❛r❛❝t❡r❡s ✐❣✉❛✐s ❛ ❡s♣❛ç♦✱ ❤á (2n−j)−(n−j)

n−j

✷✳✷ ❆▲■◆❍❆▼❊◆❚❖ ❉❊ ❉❯❆❙ ❙❊◗❯✃◆❈■❆❙ ✶✶

❞❡✜♥✐r ❛ ♣♦s✐çã♦ ❞♦s n −j ❡s♣❛ç♦s ♥❛ s❡q✉ê♥❝✐❛ t′_{✳ ▲♦❣♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛❧✐♥❤❛♠❡♥t♦s}

[s′_{, t}′_{]❞✐st✐♥t♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ 2n−j é}

2n−j n−j

·

n n−j

= (2n−j)! (n−j)!n! ·

n! (n−j)!j! =

(2n−j)! (n−j)!(n−j)!j!.

P♦rt❛♥t♦✱ ❝♦♥s✐❞❡r❛♥❞♦ j = 0,1, . . . , n✱ s❡❣✉❡

|A{s,t}| =

n

X

j=0

(2n−j)!

(n−j)!(n−j)!j!. ✭✷✳✶✮

❖ s❡❣✉✐♥t❡ ❢❛t♦ ❡st❛❜❡❧❡❝❡ ✉♠ ❧✐♠✐t❡ ✐♥❢❡r✐♦r ♣❛r❛ ♦ ✈❛❧♦r ❞❡ ✭✷✳✶✮✳ ❋❛t♦ ✸✳ ❙❡❥❛♠ s, t∈Σ ❝♦♠ n=|s|=|t|✳ ❊♥tã♦✱ |A{s,t}|= Ω(4n)✳

Pr♦✈❛✳ P❛r❛ ♠♦str❛r q✉❡|A{s,t}|= Ω(4n)✱ ❜❛st❛ ♠♦str❛r q✉❡ ♣❛r❛ n ≥1✱ t❡♠♦s q✉❡

(2n−1)! (n−1)!(n−1)! ≥

1 4

4n,

❞❡s❞❡ q✉❡ (2n−1)!/((n−1)!(n−1)!) é ❛♣❡♥❛s ✉♠❛ ❞❛s ♣❛r❝❡❧❛s ❞❛ ❡①♣r❡ssã♦ ❡♠ ✭✷✳✶✮✳

❆ ♣r♦✈❛ é ♣♦r ✐♥❞✉çã♦ ❡♠ n✳ ◆♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡✱ ♣❛r❛ n= 1✱ t❡♠♦s (2−1)!

(1−1)!(1−1)! = 1 =

1 4

41. ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ n >1 ❡ q✉❡

(2(n−1)−1)!

((n−1)−1)!((n−1)−1)! ≥

1 4

4n−1. ▲♦❣♦✱

(2n−1)!

(n−1)!(n−1)! =

(2n−1)(2n−2) (n−1)(n−1)

(2(n−1)−1)! ((n−1)−1)!((n−1)−1)!

≥ (2n−1)(2n−2)

(n−1)(n−1)

1 4

4n−1 _{✭✷✳✷✮}

≥ (2n−2)(2n−2)

(n−1)(n−1)

1 4

4n−1 _{✭✷✳✸✮}

=

1 4

4n_,

♦♥❞❡ ✭✷✳✷✮ s❡❣✉❡ ❞❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ❡ ❞❡ n >1✱ ❡ ✭✷✳✸✮ s❡❣✉❡ ❞❡ n >1✳ ❊♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ✉s❛r ✉♠ ❛❧❣♦r✐t♠♦ q✉❡ ✉s❡ ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛ ❜✉s❝❛r ✉♠ ❛❧✐✲ ♥❤❛♠❡♥t♦ ❆✲ót✐♠♦ ✈❡r✐✜❝❛♥❞♦ t♦❞♦s ♦s ♣♦ssí✈❡✐s ❛❧✐♥❤❛♠❡♥t♦s✱ ♣❡❧♦ ❋❛t♦ ✸✱ ❣❛st❛ t❡♠♣♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛ ♥♦ ❝❛s♦ ❣❡r❛❧✱ ♦ q✉❡ t♦r♥❛ ❛ ❡①❡❝✉çã♦ ❞❡ss❡ ❛❧❣♦r✐t♠♦ ✐♠♣r❛t✐❝á✈❡❧✳

◆❡❡❞❧❡♠❛♥ ❡ ❲✉♥s❝❤ ❬◆❲✼✵❪ ❞❡s❝r❡✈❡♠ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ♣r♦❣r❛♠❛çã♦ ❞✐♥â♠✐❝❛ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❆P❙✳ P❛r❛ ♠♦str❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ❛❧❣♦r✐t♠♦✱ ❝♦♥s✐❞❡r❡ ❛s s❡q✉ê♥❝✐❛s s, t∈Σ❝♦♠n, msí♠❜♦❧♦s ❝❛❞❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡A=[s′_{(1). . . s}′_{(N), t}′_{(1). . . t}′_(N)]✉♠