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■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❈♦♠♣✉t✐♥❣ ❙♣❛♥♥✐♥❣ ❚r❡❡s ✐♥ ❛ ❙♦❝✐❛❧ ❈❤♦✐❝❡ ❈♦♥t❡①t

❆♥❞r❡❛s ❉❛r♠❛♥♥∗✱ ❈❤r✐st✐❛♥ ❑❧❛♠❧❡r∗ ❛♥❞ ❯❧r✐❝❤ P❢❡rs❝❤②∗∗

∗■♥st✐t✉t❡ ♦❢ P✉❜❧✐❝ ❊❝♦♥♦♠✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ●r❛③ ∗∗■♥st✐t✉t❡ ♦❢ ❙t❛t✐st✐❝s ❛♥❞ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❯♥✐✈❡rs✐t② ♦❢ ●r❛③

✷♥❞ ■♥t❡r♥❛t✐♦♥❛❧ ❲♦r❦s❤♦♣ ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ❙♦❝✐❛❧ ❈❤♦✐❝❡✱ ▲✐✈❡r♣♦♦❧ ✷✵✵✽

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

■♥tr♦❞✉❝t✐♦♥

❈♦♠❜✐♥❡ ❙♦❝✐❛❧ ❈❤♦✐❝❡ ❚❤❡♦r② ✇✐t❤ ❉✐s❝r❡t❡ ❖♣t✐♠✐③❛t✐♦♥

• ✐✈❡♥✿ ✐♥❞✐✈✐❞✉❛❧s✬ ♣r❡❢❡r❡♥❝❡s ♦✈❡r ❡❞❣❡s ♦❢ ❛ ❣r❛♣❤

❆✐♠✿ ❋✐♥❞ ❛ ✏s♦❝✐❛❧❧② ❜❡st✑ s♣❛♥♥✐♥❣ tr❡❡ ✐♥ t❤❡ ❣r❛♣❤ ❆♣♣❧✐❝❛t✐♦♥s✿

♦✐❧ ♣✐♣❡❧✐♥❡ ❝♦♥str✉❝t✐♦♥ ✇❛t❡r ♥❡t✇♦r❦ ❝♦♥str✉❝t✐♦♥ ✐♥ ❛ ✈✐❧❧❛❣❡

• r♦✉♣ r❛♥❦✐♥❣ ♦❢ ❡❞❣❡s ♠❛② ♦r ♠❛② ♥♦t ❛❧❧♦❝❛t❡ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s t♦

t❤❡ ❡❞❣❡s ▼❛✐♥ r❡s✉❧t✿

✶ s❡ts ♦❢ ❜❡st s♣❛♥♥✐♥❣ tr❡❡s ❢♦r t❤❡ ❞✐s❝✉ss❡❞ r❛♥❦✐♥❣ r✉❧❡s ❝♦✐♥❝✐❞❡ ✷ ❛ ❜❡st s♣❛♥♥✐♥❣ tr❡❡ ❢♦r ❡❛❝❤ ❝♦♥s✐❞❡r❡❞ r❛♥❦✐♥❣ r✉❧❡ ❝❛♥ ❜❡

❞❡t❡r♠✐♥❡❞ ❡✣❝✐❡♥t❧②

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❋♦r♠❛❧ ❋r❛♠❡✇♦r❦

❯♥❞✐r❡❝t❡❞ ●r❛♣❤ ● = (❱ , ❊) ❚ ⊆ ❊ ✐s ❛ s♣❛♥♥✐♥❣ tr❡❡ :⇐ ⇒ s✉❜❣r❛♣❤ (❱ , ❚) ♦❢ ● ✐s ❛❝②❝❧✐❝ ❛♥❞ ❝♦♥♥❡❝t❡❞ τ...s❡t ♦❢ s♣❛♥♥✐♥❣ tr❡❡s ♦❢ ● ✜♥✐t❡ s❡t ♦❢ ✐♥❞✐✈✐❞✉❛❧s ■ = {✶, ✷, ..., ❦}✱ ❧✐♥❡❛r ♦r❞❡rs P✐ ♦♥ ❊✱ ✶ ≤ ✐ ≤ ❦ π = (P✶, P✷, ..., P❦) ✐s ❛ ✭✈♦t❡r✮ ♣r❡❢❡r❡♥❝❡ ♣r♦✜❧❡ ❈♦♠♣❧❡t❡ ♦r❞❡r ♦♥ ❊✿ ❛s②♠♠❡tr✐❝ ♣❛rt ≻ ❛♥❞ s②♠♠❡tr✐❝ ♣❛rt ∼ ❈♦♠♣❧❡t❡ ♦r❞❡r ♦♥ τ✿ ❛s②♠♠❡tr✐❝ ♣❛rt ⊲ ❛♥❞ s②♠♠❡tr✐❝ ♣❛rt ⊲ ⊳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❉❡✜♥✐t✐♦♥ ▲❡t ❜❡ ❛ ❝♦♠♣❧❡t❡ ♦r❞❡r ♦♥ τ✳ ❚ ∈ τ ✐s ❛ ❜❡st tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ :⇐ ⇒ ∄❚ ′ ∈ τ : ❚ ′ ⊲ ❚ ❊①❛♠♣❧❡s ▲❡t ✇(❡) ∈ R ❢♦r ❛❧❧ ❡ ∈ ❊✳ ▼✐♥✐♠✉♠ ❙♣❛♥♥✐♥❣ ❚r❡❡ Pr♦❜❧❡♠ ✐s t♦ ❞❡t❡r♠✐♥❡ ❛ ❜❡st tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r❡❧❛t✐♦♥ ❚✶ ❚✷ :⇐ ⇒

❡∈❚✶ ✇(❡) ≤ ❡∈❚✷ ✇(❡)✳

▼❛①✐♠✉♠ ❙♣❛♥♥✐♥❣ ❚r❡❡ Pr♦❜❧❡♠✿ ✜♥❞ ❛ ❜❡st tr❡❡ ✇✳r✳t✳ r❡❧❛t✐♦♥ ❚✶ ❚✷ :⇐ ⇒

❡∈❚✶ ✇(❡) ≥ ❡∈❚✷ ✇(❡)✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❇❛s✐❝ ♦r❞❡rs ♦♥ t❤❡ ❡❞❣❡✲s❡t

❉❡✜♥✐t✐♦♥s ❇♦r❞❛✬s ♠❡t❤♦❞ ✭s❡❡ ❬❇❋✵✷❪✮✿ ■♥❞✐✈✐❞✉❛❧ ✐✬s ❇♦r❞❛ ❝♦✉♥t ♦❢ ❡❞❣❡ ❡ ✐s ❣✐✈❡♥ ❜② ❇✐(❡) := |{❢ ∈ ❊ : ❡P✐❢ }|✳ ❚❤❡ t♦t❛❧ ❇♦r❞❛ ❝♦✉♥t ♦❢ ❡❞❣❡ ❡ ✐s ❞❡✜♥❡❞ ❜② ❇(❡) :=

✐∈■ ❇✐(❡)✳ ❋♦r ❡, ❢ ∈ ❊ ✇❡ ❞❡✜♥❡ t❤❡

❇♦r❞❛✲♦r❞❡r ♦♥ ❊ ❜② ❡ ❜ ❢ :⇐ ⇒ ❇(❡) ≥ ❇(❢ )✳ ❙✐♠♣❧❡ ▼❛❥♦r✐t②✲♦r❞❡r ✭s❡❡ ❬❇❋✵✷❪✮✿ ▲❡t ❡, ❢ ∈ ❊✳ ❚❤❡♥ ✇❡ ❞❡✜♥❡ t❤❡ ❙✐♠♣❧❡ ▼❛❥♦r✐t②✲♦r❞❡r ♦♥ ❊ ❜② ❡ s♠ ❢ :⇐ ⇒ |{✐ ∈ ■ : ❡P✐❢ }| ≥ |{✐ ∈ ■ : ❢P✐❡}|✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❉❡✜♥✐t✐♦♥s ▲❡t ❡, ❢ ∈ ❊✳ ❋♦r ❛❧❧ ✐ ∈ ■✱ ♣❛rt✐t✐♦♥ ❡❞❣❡✲s❡t ❊ ✐♥t♦ ❛ s❡t ❙✐ ⊂ ❊ ♦❢ ❡❞❣❡s ✐♥❞✐✈✐❞✉❛❧ ✐ ❛♣♣r♦✈❡s ♦❢ ❛♥❞ ❛ s❡t ❊ \ ❙✐ ✐♥❞✐✈✐❞✉❛❧ ✐ ❞✐s❛♣♣r♦✈❡s ♦❢✳ ❆♣♣r♦✈❛❧✲♦r❞❡r ✭s❡❡ ❬❇❋✽✸❪✮✿ ❚❤❡ ❆♣♣r♦✈❛❧ ❝♦✉♥t ♦❢ ❡ ✐s ❞❡✜♥❡❞ ❜② ❆(❡) := |{✐ ∈ ■ : ❡ ∈ ❙✐}|✳ ❚❤❡ ❆♣♣r♦✈❛❧✲♦r❞❡r ❛ ✐s t❤❡♥ ❞❡✜♥❡❞ ❜② ❡ ❛ ❢ :⇐ ⇒ ❆(❡) ≥ ❆(❢ )✳ ❋♦r ❛❧❧ ✐ ∈ ■ t❤❡ s❡t ❙t

✐ := {❡ ∈ ❊|❡P✐❢ ∀❢ ∈ ❊ \ {❡}} r❡♣r❡s❡♥ts

✐♥❞✐✈✐❞✉❛❧ ✐✬s t♦♣ ❝❤♦✐❝❡ P❧✉r❛❧✐t②✲♦r❞❡r ✭s❡❡ ❬❘♦❜✾✶❪✮✿ ❚❤❡ P❧✉r❛❧✐t② ❝♦✉♥t ♦❢ ❡ ✐s P❧(❡) := |{✐ ∈ ■ : ❡ ∈ ❙t

✐ }|✳ ❚❤❡

P❧✉r❛❧✐t②✲♦r❞❡r ♣❧ ✐s ❞❡✜♥❡❞ ❜② ❡ ♣❧ ❢ :⇐ ⇒ P❧(❡) ≥ P❧(❢ )✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❇♦r❞❛✲✱ ❆♣♣r♦✈❛❧✲ ❛♥❞ P❧✉r❛❧✐t②✲♦r❞❡r ❛r❡ ✇❡❛❦ ♦r❞❡rs ♦♥ ❊ ✭❝♦♠♣❧❡t❡ ❛♥❞ tr❛♥s✐t✐✈❡✮✳ ❙✐♠♣❧❡ ▼❛❥♦r✐t②✲♦r❞❡r ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t tr❛♥s✐t✐✈❡ ⇒ ♣r❡❢❡r❡♥❝❡ ❝②❝❧❡s

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ

■❞❡❛✿ ❉❡r✐✈❡ ✇❡❛❦ ♦r❞❡rs ♦♥ τ ❢r♦♠ ✇❡❛❦ ♦r❞❡rs ♦♥ ❊ ❉❡✜♥✐t✐♦♥ ❋♦r ❚ ∈ τ ✇❡ ❞❡✜♥❡ t❤❡ ❇♦r❞❛ ❝♦✉♥t ♦❢ ❚ ❜② ❇(❚) :=

❡∈❚ ❇(❡)✳

❚❤❡♥ t❤❡ ❇♦r❞❛✲♦r❞❡r ❇ ♦♥ τ ✐s ❞❡✜♥❡❞ ❜② ❧❡tt✐♥❣✱ ❢♦r ❛❧❧ ❚✶, ❚✷ ∈ τ✱ ❚✶ ❇ ❚✷ :⇐ ⇒ ❇(❚✶) ≥ ❇(❚✷) . ❆♥❛❧♦❣♦✉s❧②✿ ❆♣♣r♦✈❛❧✲♦r❞❡r ❆ ♦♥ τ✱ P❧✉r❛❧✐t②✲♦r❞❡r P❧ ♦♥ τ✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

• ❡♥❡r❛❧ ❝♦♥❝❡♣t ♦❢ ❛ ❜❡st tr❡❡ ✇✳r✳t✳ ♦r❞❡rs✱ t❤❛t ❛r❡ ❜❛s❡❞ ♦♥ s✉♠♠✐♥❣

✉♣ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s ♦❢ t❤❡ ❡❞❣❡s✿ ❉❡✜♥✐t✐♦♥ ▲❡t τ ❜❡ t❤❡ s❡t ♦❢ s♣❛♥♥✐♥❣ tr❡❡s ♦❢ ● ❛♥❞ ❧❡t ❜❡ ❛ ✇❡❛❦ ♦r❞❡r ♦♥ ❊✳ ❆ tr❡❡ ▼ ∈ τ ✐s ❝❛❧❧❡❞ ♠❛①✲s♣❛♥♥✐♥❣ tr❡❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡✈❡r② ❡❞❣❡ ❢ = {✐, ❥}✱ ❢ / ∈ ▼✱ ❢ ❡ ❤♦❧❞s ❢♦r ❛❧❧ ❡ ∈ ▼ t❤❛t ❛r❡ ♣❛rt ♦❢ t❤❡ ✉♥✐q✉❡ s✐♠♣❧❡ ♣❛t❤ ❜❡t✇❡❡♥ ✐ ❛♥❞ ❥ ✐♥ ▼✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❘❡♠❛r❦s✿

✶ ❆❜♦✈❡ ❞❡✜♥✐t✐♦♥ ❣❡♥❡r❛❧✐③❡s t❤❡ ♣❛t❤ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡

♠❛①✐♠✉♠ s♣❛♥♥✐♥❣ tr❡❡ ♣r♦❜❧❡♠ st❛t❡❞ ✐♥ ❬❆▼❖✾✸❪

✷ ❆ ♠❛①✲s♣❛♥♥✐♥❣ tr❡❡ ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❡✣❝✐❡♥t❧② ❜② ❛ ❣r❡❡❞②

❛❧❣♦r✐t❤♠ ✭❡✳❣✳ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠✮

✸ ◆♦t❡ t❤❛t ❢♦r ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥ ❞♦❡s ♥♦t ♥❡❡❞ t♦ ❜❡ ❜❛s❡❞ ♦♥

♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s

✹ ❙✐♠♣❧❡✲▼❛❥♦r✐t② ♦r❞❡r ❞♦❡s ♥♦t ✜t ✐♥ t❤✐s ❝♦♥❝❡♣t

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❆❧t❡r♥❛t✐✈❡ ✐❞❡❛ t♦ r❛♥❦ t✇♦ tr❡❡s✿

✶ ❚❤♦s❡ ❡❞❣❡s t❤❛t ❛r❡ s✐♠✉❧t❛♥❡♦✉s❧② ❝♦♥t❛✐♥❡❞ ✐♥ ❜♦t❤ tr❡❡s s❤♦✉❧❞

♥♦t ♣❧❛② ❛ r♦❧❡✳ ❚❤✉s✱ ✇❡ s✐♠♣❧② r❡♠♦✈❡ t❤♦s❡ ❡❞❣❡s t❤❛t ❜♦t❤ tr❡❡s ❤❛✈❡ ✐♥ ❝♦♠♠♦♥✳

✷ ❘❛♥❦ tr❡❡s ❚✶, ❚✷ ∈ τ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s✉♠ ♦❢ ✇✐♥s ♦❢ ❡❞❣❡s ♦❢ ˜

❚✶ ❛❣❛✐♥st t❤♦s❡ ♦❢ ˜ ❚✷✱ ✇❤❡r❡ ˜ ❚✶ := ❚✶ \ ❚✷ ❛♥❞ ˜ ❚✷ := ❚✷ \ ❚✶ ❉❡✜♥✐t✐♦♥ ▲❡t ❜❡ ❛ ❝♦♠♣❧❡t❡ ♦r❞❡r ♦♥ ❊✳ ❋♦r ❡, ❢ ∈ ❊ ❧❡t s(❡, ❢ ) :=    ✶ ✐❢ ❡ ≻ ❢ ✵ ✐❢ ❡ ∼ ❢ −✶ ✐❢ ❡ ≺ ❢ ❜❡ t❤❡ s❝♦r❡ ♦❢ ❡ ✈❡rs✉s ❢ ✳ ❋♦r ❚✶, ❚✷ ∈ τ ✇❡ ❞❡✜♥❡ ❚✶ ❙ ❚✷ :⇐ ⇒

• ❛∈ ˜

❚✶

• ❜∈ ˜

❚✷

s(❛, ❜) ≥ ✵ .

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

t t t t ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❜ ❛ ❞ ❝ ✶ ✷ ✸ ❛ ❜ ❝ ❜ ❝ ❛ ❝ ❛ ❜ ❞ ❞ ❞

❊①❛♠♣❧❡

✸ s♣❛♥♥✐♥❣ tr❡❡s✿ ❚✶ := {❛, ❜, ❞}✱ ❚✷ := {❜, ❝, ❞} ❛♥❞ ❚✸ := {❛, ❝, ❞}✳ Pr❡❢❡r❡♥❝❡ ❝②❝❧❡ ❛ ≻s♠ ❜ ≻s♠ ❝ ≻s♠ ❛✳ ❲❡ ❣❡t ❚✶ \ ❚✷ = {❛} ❛♥❞ ❚✷ \ ❚✶ = {❝}✱ ❚✶ \ ❚✸ = {❜} ❛♥❞ ❚✸ \ ❚✶ = {❝}✱ ❚✷ \ ❚✸ = {❜} ❛♥❞ ❚✸ \ ❚✷ = {❛}✳ ❚❤✉s ✇❡ ❤❛✈❡ ❚✸ ⊲❙ ❚✷ ⊲❙ ❚✶ ⊲❙ ❚✸✳ ❍❡♥❝❡ ❛ ❜❡st tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ ❙ ❞♦❡s ♥♦t ❡①✐st ✐♥ t❤✐s ❡①❛♠♣❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

t t t t t t ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❜ ❛ ❞ ❝ ❡ ❢ ✶ ✷ ✸ ❝ ❢ ❜ ❛ ❡ ❢ ❜ ❞ ❛ ❞ ❝ ❞ ❡ ❛ ❡ ❢ ❜ ❝

❊①❛♠♣❧❡

❋♦r t❤❡ ❛❜♦✈❡ ❣r❛♣❤ t❤❡ ❙✐♠♣❧❡ ▼❛❥♦r✐t②✲♦r❞❡r ♦♥ ❊ = {❛, ❜, ❝, ❞, ❡, ❢ } ✐s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ ❛ ≻s♠ ❜ ❜ ≻s♠ ❞ ❝ ≻s♠ ❛ ❞ ≻s♠ ❡ ❢ ≻s♠ ❝ ❛ ≻s♠ ❞ ❜ ≻s♠ ❡ ❝ ≻s♠ ❜ ❡ ≻s♠ ❝ ❢ ≻s♠ ❞ ❛ ≻s♠ ❡ ❜ ≻s♠ ❢ ❞ ≻s♠ ❝ ❢ ≻s♠ ❛ ❢ ≻s♠ ❡ ❡❞❣❡ # ♦❢ ✐♥❢✳ ❡❞❣❡s ❢ ✹ ❛, ❜ ✸ ❝, ❞ ✷ ❡ ✶ ⇒ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠ ❛❝❝♦r❞✐♥❣ t♦ ❙▼✲✇✐♥s ♦✉t♣✉ts ❚❣ = {❢ , ❛, ❜, ❞, ❡}✳ ❚❤❡ t✇♦ ♦t❤❡r s♣❛♥♥✐♥❣ tr❡❡s ❛r❡ ❚✶ = {❜, ❝, ❞, ❡, ❢ } ❛♥❞ ❚✷ = {❛, ❝, ❞, ❡, ❢ }✳ ❚✶ \ ❚❣ = {❝} ❛♥❞ ❚❣ \ ❚✶ = {❛} ⇒ ❚✶ ⊲❙ ❚❣ ✳ ❚✷ \ ❚❣ = {❝} ❛♥❞ ❚❣ \ ❚✷ = {❜} ⇒ ❚✷ ⊲❙ ❚❣ ✳ ■✳❡✳ ❛❝❝♦r❞✐♥❣ t♦ ❙ ❡✈❡r② ♦t❤❡r s♣❛♥♥✐♥❣ tr❡❡ ♦❢ t❤❡ ❣r❛♣❤ ✐s str✐❝t❧② ♣r❡❢❡rr❡❞ t♦ ❚❣ ✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

Pr♦♣♦s✐t✐♦♥ ▲❡t =s♠✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❤♦❧❞✿

✶ ❚❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ● = (❱ , ❊) ❛♥❞ ❛ ✈♦t❡r ♣r❡❢❡r❡♥❝❡ ♣r♦✜❧❡ ♦♥ ❊

s✉❝❤ t❤❛t ❛ ❜❡st tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ ❙ ❞♦❡s ♥♦t ❡①✐st✳

✷ ❚❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ● = (❱ , ❊) ❛♥❞ ❛ ✈♦t❡r ♣r❡❢❡r❡♥❝❡ ♣r♦✜❧❡ ♦♥ ❊

s✉❝❤ t❤❛t ❛ ❜❡st tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ ❙ ❡①✐sts ❜✉t t❤❡ ❣❡♥❡r❛❧✐③❡❞ ✈❡rs✐♦♥ ♦❢ ❑r✉s❦❛❧✬s ❛❧❣♦r✐t❤♠ ❢❛✐❧s t♦ ❞❡t❡r♠✐♥❡ s✉❝❤ ❛ ❜❡st tr❡❡✳ ■♥ ❢❛❝t✱ t❤❡ tr❡❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❛❧❣♦r✐t❤♠ ♠❛② ❡✈❡♥ ❜❡ t❤❡ ✇♦rst tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ ❙✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❈♦♠♣❛r✐♥❣ tr❡❡s

❚❤r❡❡ ♠♦r❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ

■❞❡❛✿ ❈♦♠♣❛r❡ tr❡❡s ♦♥ ❜❛s✐s ♦❢ ❛ ❣✐✈❡♥ ✇❡❛❦ ♦r❞❡r ♦♥ ❊✳ ❲❤❡♥ ❝♦♠♣❛r✐♥❣ t✇♦ tr❡❡s✱ t❤♦s❡ ❡❞❣❡s t❤❛t ❛r❡ s✐♠✉❧t❛♥❡♦✉s❧② ❝♦♥t❛✐♥❡❞ ✐♥ ❜♦t❤ tr❡❡s s❤♦✉❧❞ ♥♦t ♣❧❛② ❛ r♦❧❡✳ ◆♦t❛t✐♦♥✿

• ✐✈❡♥ ❚✶, ❚✷ ∈ τ✱ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ ˜

❚✶ := ❚✶ \ ❚✷✱ ˜ ❚✷ := ❚✷ \ ❚✶ ❛♥❞ r := | ˜ ❚✶|

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❚❤r❡❡ ♠♦r❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ

❈♦♥❝❡♣t ✶✿ ❉❡r✐✈❡❞ ❢r♦♠ t❤❡ ♠❛①♠✐♥✲♦r❞❡r ♦♥ s❡ts ♣r❡s❡♥t❡❞ ✐♥ ❬❇❇P✵✹❪ ❉❡✜♥✐t✐♦♥ ▲❡t ❚✶, ❚✷ ∈ τ✳ ❚❤❡♥ ✇❡ ❞❡✜♥❡ t❤❡ ♠❛①♠✐♥✲♦r❞❡r ♠①♥ ♦♥ τ ❜② ❚✶ ♠①♥ ❚✷ :⇐ ⇒ [ ˜ ❚✶ = ∅ ♦r ♠❛① ˜ ❚✶ ≻ ♠❛① ˜ ❚✷ ♦r (♠❛① ˜ ❚✶ ∼ ♠❛① ˜ ❚✷ ❛♥❞ ♠✐♥ ˜ ❚✶ ♠✐♥ ˜ ❚✷)]

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❈♦♥❝❡♣t ✷✿ ❉❡r✐✈❡❞ ❢r♦♠ t❤❡ ❧❡①✐♠❛① ♦r❞❡r ♦♥ s❡ts ♣r❡s❡♥t❡❞ ✐♥ ❬❇❇P✵✹❪ ❉❡✜♥✐t✐♦♥ ▲❡t ❚✶, ❚✷ ∈ τ✳ ▲❡t ˜ ❚✶ := {❡✶, ❡✷, ..., ❡r}✱ ˜ ❚✷ := {❢✶, ❢✷, ..., ❢r} s✉❝❤ t❤❛t ❡✐ ❡✐+✶ ❛♥❞ ❢✐ ❢✐+✶ ❤♦❧❞s ❢♦r ✶ ≤ ✐ ≤ r − ✶✳ ❚❤❡♥ t❤❡ ❧❡①✐♠❛① ♦r❞❡r ❧❡① ♦♥ τ ✐s ❞❡✜♥❡❞ ❜② ❚✶ ❧❡① ❚✷ :⇐ ⇒ [ ˜ ❚✶ = ∅ ♦r ❡✐ ∼ ❢✐ ❢♦r ❛❧❧ ✶ ≤ ✐ ≤ r ♦r (∃❥ ∈ {✶, ..., r} s✉❝❤ t❤❛t ❡✐ ∼ ❢✐ ❢♦r ❛❧❧ ✐ < ❥ ❛♥❞ ❡❥ ≻ ❢❥)]

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❈♦♥❝❡♣t ✸✿ ❘❛♥❦ t❤❡ ❡❞❣❡s ♦❢ t❤❡ ❞✐s❥♦✐♥t ✉♥✐♦♥ ♦❢ ❚✶, ❚✷ ∈ τ ❛❝❝♦r❞✐♥❣ t♦ ✳ ❋♦r t❤❡ r❡s✉❧t✐♥❣ r❛♥❦✐♥❣ ✉s❡ ❛ ♣♦s✐t✐♦♥❛❧ s❝♦r✐♥❣ ❝♦♥❝❡♣t t♦ ❝♦♠♣❛r❡ t❤❡ tr❡❡s✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❚✶, ❚✷ ∈ τ✳ ▲❡t ˜ ❚✶ ∪ ˜ ❚✷ := {❞✶, ❞✷, ..., ❞✷r} s✉❝❤ t❤❛t ❞✐ ❞✐+✶ ❤♦❧❞s ❢♦r ✶ ≤ ✐ ≤ ✷r − ✶. ▲❡t ❜ : ❊ → R ❜❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ✱ t❤❛t ✐s✱ ❢♦r ✶ ≤ ✐ ≤ ✷r − ✶✱ ❜(❞✐) = ❜(❞✐+✶) ✐❢ ❞✐ ∼ ❞✐+✶ ❜(❞✐) > ❜(❞✐+✶) ✐❢ ❞✐ ≻ ❞✐+✶ ▲❡t ❜( ˜ ❚✶) :=

❡∈ ˜ ❚✶ ❜(❡) ❛♥❞ ❜( ˜

❚✷) :=

❢ ∈ ˜ ❚✷ ❜(❢ )✳

❚❤❡♥ ✇❡ ❞❡✜♥❡ ❚✶ ♣s ❚✷ :⇐ ⇒ ❜( ˜ ❚✶) ≥ ❜( ˜ ❚✷) . ❘❡♠❛r❦✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❛❞❛♣ts t❤❡ ❝♦♥❝❡♣t ♦❢ t❤❡ ♣♦s✐t✐♦♥❛❧ s❝♦r✐♥❣ ♣r♦❝❡❞✉r❡s ♣r❡s❡♥t❡❞ ✐♥ ❬❇❋✵✷❪

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❘❡♠❛r❦s✳ ❚❤❡ ♦r❞❡rs ❙✱ ❧❡①✱ ♠①♥ ❛♥❞ ♣s ❛r❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ✳ ■♥ t❤❡ ❛❜♦✈❡ ❝♦♥❝❡♣ts t❤❡ ♦r❞❡r ♦♥ ❊ ❞♦❡s ♥♦t ♥❡❡❞ t♦ ❜❡ ♦❢ ♥✉♠❡r✐❝❛❧ ♥❛t✉r❡✱ ✐✳❡✳ ❞♦❡s ♥♦t ❤❛✈❡ t♦ ❛❧❧♦❝❛t❡ ♥✉♠❜❡rs t♦ t❤❡ ❡❞❣❡s✳ ❆✐♠✿ ❋✐♥❞ ❛ ❜❡st tr❡❡ ✇✳r✳t✳ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦r❞❡r

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❘❡s✉❧ts ♦♥ ♠❛①✲s♣❛♥♥✐♥❣ tr❡❡s ❛♥❞ ❜❡st tr❡❡s

❘❡❝❛❧❧✿ ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛ ❣✐✈❡♥ ✇❡❛❦ ♦r❞❡r ♦♥ ❊✱ ✐✳❡✳ ✐s ❝♦♠♣❧❡t❡ ❛♥❞ tr❛♥s✐t✐✈❡ ❚❤❡♦r❡♠ ❆ ♠❛①✲s♣❛♥♥✐♥❣ tr❡❡ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O(|❊| + |❱ | ❧♦❣ |❱ |) t✐♠❡✳ Pr♦♦❢✳ ■♠♠❡❞✐❛t❡❧② ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ♠❛①✐♠✉♠ s♣❛♥♥✐♥❣ tr❡❡ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ O(|❊| + |❱ | ❧♦❣ |❱ |) t✐♠❡✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

▼❛✐♥ ❚❤❡♦r❡♠

❚❤❡♦r❡♠ ▲❡t ▼ ∈ τ ❛♥❞ ❧❡t ❜❡ ❛ ✇❡❛❦ ♦r❞❡r ♦♥ ❊✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✿

✶ ▼ ✐s ❛ ♠❛①✲s♣❛♥♥✐♥❣ tr❡❡ ✷ ∄❇ ∈ τ : ❇ ⊲❧❡① ▼ ✸ ∄❇ ∈ τ : ❇ ⊲❙ ▼ ✹ ∄❇ ∈ τ : ❇ ⊲♠①♥ ▼ ✺ ∄❇ ∈ τ : ❇ ⊲♣s ▼

❈♦r♦❧❧❛r② ❊✈❡r② ♣♦s✐t✐♦♥❛❧ s❝♦r✐♥❣ ♠❡t❤♦❞ t❤❛t ②✐❡❧❞s t❤❡ s❛♠❡ r❛♥❦✐♥❣ ♦♥ ❊ ②✐❡❧❞s t❤❡ s❛♠❡ s❡t ♦❢ ❜❡st tr❡❡s ✇✳r✳t✳ ♣s✱ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s ❛ss✐❣♥❡❞ t♦ t❤❡ ❡❞❣❡s✳

■♥tr♦❞✉❝t✐♦♥ ❋♦r♠❛❧ ❋r❛♠❡✇♦r❦ ❙♦♠❡ ❝♦♠♣❧❡t❡ ♦r❞❡rs ♦♥ τ ❈♦♠♣❛r✐♥❣ tr❡❡s

❈♦♥s❡q✉❡♥❝❡s

▲❡t ❜❡ ❛ ✇❡❛❦ ♦r❞❡r ♦♥ ❊ ❛♥❞ ❧❡t ∈ {❧❡①, ❙, ♠①♥, ♣s} ❈♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ ❚❤❡♦r❡♠✿

✶ ❆ ❜❡st tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ ❛❧✇❛②s ❡①✐sts ✷ ❆ ❜❡st tr❡❡ ✇✐t❤ r❡s♣❡❝t t♦ ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❡✣❝✐❡♥t❧② ✸ ❋♦r t❤❡ ♦r❞❡rs ❧❡①, ❙, ♠①♥, ♣s✱ t❤❡ s❡ts ♦❢ ❜❡st tr❡❡s ❝♦✐♥❝✐❞❡

❈♦♥❝❧✉❞✐♥❣ ❘❡♠❛r❦✿ ❘❡s✉❧ts ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ t♦ ❜❛s❡s ♦❢ ♠❛tr♦✐❞s

❘❡❢❡r❡♥❝❡s

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❘✳❑✳ ❆❤✉❥❛✱ ❚✳▲✳ ▼❛❣♥❛♥t✐✱ ❛♥❞ ❏✳❇✳ ❖r❧✐♥✳ ◆❡t✇♦r❦ ✢♦✇s✿ t❤❡♦r②✱ ❛❧❣♦r✐t❤♠s✱ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ Pr❡♥t✐❝❡ ❍❛❧❧✱ ✶✾✾✸✳ ❙✳ ❇❛r❜❡r❛✱ ❲✳ ❇♦ss❡rt✱ ❛♥❞ P✳❑✳ P❛tt❛♥❛✐❦✳ ❘❛♥❦✐♥❣ s❡ts ♦❢ ♦❜❥❡❝ts✳ ■♥ ❍❛♥❞❜♦♦❦ ♦❢ ❯t✐❧✐t② ❚❤❡♦r②✱ ✈♦❧✉♠❡ ✷✱ ♣❛❣❡s ✽✾✸✕✾✼✼✳ ✷✵✵✹✳ ❙✳❏✳ ❇r❛♠s ❛♥❞ P✳❈✳ ❋✐s❤❜✉r♥✳ ❆♣♣r♦✈❛❧ ❱♦t✐♥❣✳ ❇✐r❦❤ä✉s❡r✱ ❇♦st♦♥✱ ✶✾✽✸✳ ❙✳❏✳ ❇r❛♠s ❛♥❞ P✳❈✳ ❋✐s❤❜✉r♥✳ ❱♦t✐♥❣ ♣r♦❝❡❞✉r❡s✳ ■♥ ❑✳❏✳ ❆rr♦✇✱ ❆✳❑✳ ❙❡♥✱ ❛♥❞ ❑✳ ❙✉③✉♠✉r❛✱ ❡❞✐t♦rs✱ ❍❛♥❞❜♦♦❦ ♦❢ ❙♦❝✐❛❧ ❈❤♦✐❝❡ ❛♥❞ ❲❡❧❢❛r❡✱ ✈♦❧✉♠❡ ✶✱ ♣❛❣❡s ✶✼✸✕✷✸✻✳ ❊❧s❡✈✐❡r✱ ✷✵✵✷✳ ❋✳❙✳ ❘♦❜❡rts✳ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ t❤❡ ♣❧✉r❛❧✐t② ❢✉♥❝t✐♦♥✳ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❛❧ ❙❝✐❡♥❝❡s✱ ✷✶✭✷✮✿✶✵✶✕✶✷✼✱ ❆♣r✐❧ ✶✾✾✶✳