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P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s✿ ❋❛st ❆❧❣♦r✐t❤♠s ❛♥❞ ◆❡✇ ❍❛r❞♥❡ss ❘❡s✉❧ts

❍❡♥♥✐♥❣ ❋❡r♥❛✉1 ❋❧♦r✐♥ ▼❛♥❡❛2 ❘♦❜❡rt ▼❡r❝❛➩2,3 ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞1

1❚r✐❡r ❯♥✐✈❡rs✐t②✱ ●❡r♠❛♥② 2❑✐❡❧ ❯♥✐✈❡rs✐t②✱ ●❡r♠❛♥② 3❑✐♥❣✬s ❈♦❧❧❡❣❡✱ ▲♦♥❞♦♥✱ ❯❑

❙❚❆❈❙ ✷✵✶✺

P❛tt❡r♥s ✇✐t❤ ❱❛r✐❛❜❧❡s

❋✐♥✐t❡ ❛❧♣❤❛❜❡t ♦❢ t❡r♠✐♥❛❧s Σ = {a, b, c, d} ❙❡t ♦❢ ✈❛r✐❛❜❧❡s X = {x1, x2, x3, . . .} P❛tt❡r♥s α ∈ (Σ ∪ X)+ ❲♦r❞s w ∈ Σ+ ❙✉❜st✐t✉t✐♦♥ h : X → Σ+ α = y1 . . . yn✱ h(α) = h(y1) . . . h(yn)✱ ✇✐t❤ h(a) = a✱ a ∈ Σ✳

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳ α = x1 x2 x1 x3 x2 w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳ α = a b b x2 a b b x3 x2 w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳ α = a b b b a a b b x3 b a w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳ α = a b b b a a b b a a a b a b a w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳ α = x1 a x2b x2x1 x2 w = b a c b a c b c b a c b c

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳ α = b a c b a x2b x2b a c b x2 w = b a c b a c b c b a c b c

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s

♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h(α) = w✳ α = b a c b a c b c b a c b c w = b a c b a c b c b a c b c

▼♦t✐✈❛t✐♦♥

▲❡❛r♥✐♥❣ t❤❡♦r② ✭✐♥❞✉❝t✐✈❡ ✐♥❢❡r❡♥❝❡✱ P❆❈ ❧❡❛r♥✐♥❣✮✱ ❧❛♥❣✉❛❣❡ t❤❡♦r② ✭♣❛tt❡r♥ ❧❛♥❣✉❛❣❡s✮✱ ❝♦♠❜✐♥❛t♦r✐❝s ♦♥ ✇♦r❞s ✭✇♦r❞ ❡q✉❛t✐♦♥s✱ ✉♥❛✈♦✐❞❛❜❧❡ ♣❛tt❡r♥s✱ ❛♠❜✐❣✉✐t② ♦❢ ♠♦r♣❤✐s♠s✱ ❡q✉❛❧✐t② s❡ts✮✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣ ✭♣❛r❛♠❡t❡r✐s❡❞ ♠❛t❝❤✐♥❣✱ ✭❣❡♥❡r❛❧✐s❡❞✮ ❢✉♥❝t✐♦♥ ♠❛t❝❤✐♥❣✮✱ ♠❛t❝❤t❡st ❢♦r r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ✇✐t❤ ❜❛❝❦r❡❢❡r❡♥❝❡s ✭t❡①t ❡❞✐t♦rs ✭❣r❡♣✱ ❡♠❛❝s✮✱ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ ✭P❡r❧✱ ❏❛✈❛✱ P②t❤♦♥✮✮✱ ❞❛t❛❜❛s❡ t❤❡♦r②✳

❈♦♠♣❧❡①✐t②

▼❛t❝❤✐♥❣ Pr♦❜❧❡♠ ✭▼❛t❝❤✮

• ✐✈❡♥ ❛ ♣❛tt❡r♥ α✱ ❛ ✇♦r❞ w✳ ❉♦❡s α ♠❛t❝❤ w ✭✐✳ ❡✳✱ ∃h : h(α) = w✮❄

▼❛t❝❤ ✐s ✭✐♥ ❣❡♥❡r❛❧✮ ◆P✲❝♦♠♣❧❡t❡✳ ❇❛❞ ♥❡✇s✿ ▼❛t❝❤ r❡♠❛✐♥s ❤❛r❞ ✐❢ ♥✉♠❡r✐❝❛❧ ♣❛r❛♠❡t❡rs ❛r❡ r❡str✐❝t❡❞ ✭❢❡✇ ❡①❝❡♣t✐♦♥s✮✿

▼❛t❝❤ ✐❢ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ♦r ✇♦r❞ ❧❡♥❣t❤ ❜♦✉♥❞❡❞ ✭tr✐✈✐❛❧✮✳ ▼❛t❝❤ st✐❧❧ ❤❛r❞ ✐❢

❛❧♣❤❛❜❡t s✐③❡ ✱ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❤❛s ❛t ♠♦st ♦❝❝✉rr❡♥❝❡s✱ ❢♦r ❡✈❡r② ✳

• ♦♦❞ ♥❡✇s✿ ❚r❛❝t❛❜❧❡ ✐❢ str✉❝t✉r❡ ♦❢ ♣❛tt❡r♥s ✐s r❡str✐❝t❡❞✳

❈♦♠♣❧❡①✐t②

▼❛t❝❤✐♥❣ Pr♦❜❧❡♠ ✭▼❛t❝❤✮

• ✐✈❡♥ ❛ ♣❛tt❡r♥ α✱ ❛ ✇♦r❞ w✳ ❉♦❡s α ♠❛t❝❤ w ✭✐✳ ❡✳✱ ∃h : h(α) = w✮❄

▼❛t❝❤ ✐s ✭✐♥ ❣❡♥❡r❛❧✮ ◆P✲❝♦♠♣❧❡t❡✳ ❇❛❞ ♥❡✇s✿ ▼❛t❝❤ r❡♠❛✐♥s ❤❛r❞ ✐❢ ♥✉♠❡r✐❝❛❧ ♣❛r❛♠❡t❡rs ❛r❡ r❡str✐❝t❡❞ ✭❢❡✇ ❡①❝❡♣t✐♦♥s✮✿

◮ ▼❛t❝❤ ∈ P ✐❢ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ♦r ✇♦r❞ ❧❡♥❣t❤ ❜♦✉♥❞❡❞ ✭tr✐✈✐❛❧✮✳ ◮ ▼❛t❝❤ st✐❧❧ ❤❛r❞ ✐❢ ⋆ ❛❧♣❤❛❜❡t s✐③❡ 2✱ ⋆ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❤❛s ❛t ♠♦st 2 ♦❝❝✉rr❡♥❝❡s✱ ⋆ |h(x)| ≤ 3 ❢♦r ❡✈❡r② x✳

• ♦♦❞ ♥❡✇s✿ ❚r❛❝t❛❜❧❡ ✐❢ str✉❝t✉r❡ ♦❢ ♣❛tt❡r♥s ✐s r❡str✐❝t❡❞✳

❈♦♠♣❧❡①✐t②

▼❛t❝❤✐♥❣ Pr♦❜❧❡♠ ✭▼❛t❝❤✮

• ✐✈❡♥ ❛ ♣❛tt❡r♥ α✱ ❛ ✇♦r❞ w✳ ❉♦❡s α ♠❛t❝❤ w ✭✐✳ ❡✳✱ ∃h : h(α) = w✮❄

▼❛t❝❤ ✐s ✭✐♥ ❣❡♥❡r❛❧✮ ◆P✲❝♦♠♣❧❡t❡✳ ❇❛❞ ♥❡✇s✿ ▼❛t❝❤ r❡♠❛✐♥s ❤❛r❞ ✐❢ ♥✉♠❡r✐❝❛❧ ♣❛r❛♠❡t❡rs ❛r❡ r❡str✐❝t❡❞ ✭❢❡✇ ❡①❝❡♣t✐♦♥s✮✿

◮ ▼❛t❝❤ ∈ P ✐❢ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ♦r ✇♦r❞ ❧❡♥❣t❤ ❜♦✉♥❞❡❞ ✭tr✐✈✐❛❧✮✳ ◮ ▼❛t❝❤ st✐❧❧ ❤❛r❞ ✐❢ ⋆ ❛❧♣❤❛❜❡t s✐③❡ 2✱ ⋆ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❤❛s ❛t ♠♦st 2 ♦❝❝✉rr❡♥❝❡s✱ ⋆ |h(x)| ≤ 3 ❢♦r ❡✈❡r② x✳

• ♦♦❞ ♥❡✇s✿ ❚r❛❝t❛❜❧❡ ✐❢ str✉❝t✉r❡ ♦❢ ♣❛tt❡r♥s ✐s r❡str✐❝t❡❞✳

◆♦t❛t✐♦♥

var(α) ❙❡t ♦❢ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ ♣❛tt❡r♥ α✳ |α|x ◆✉♠❜❡r ♦❢ ♦❝❝✉rr❡♥❝❡s ♦❢ ✈❛r✐❛❜❧❡ x ✐♥ ♣❛tt❡r♥ α✳

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s

❘❡❣✉❧❛r P❛tt❡r♥s✿ |α|x = 1✱ x ∈ var(α)✳ ❊✳ ❣✳✱ α = abx1x2bx3aaax4b✳ ◆♦♥✲❈r♦ss P❛tt❡r♥s✿ ✐s ♥♦t ♣♦ss✐❜❧❡✳ ❊✳ ❣✳✱

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s

❘❡❣✉❧❛r P❛tt❡r♥s✿ |α|x = 1✱ x ∈ var(α)✳ ❊✳ ❣✳✱ α = abx1x2bx3aaax4b✳ ◆♦♥✲❈r♦ss P❛tt❡r♥s✿ α = . . . x . . . y . . . x . . . ✐s ♥♦t ♣♦ss✐❜❧❡✳ ❊✳ ❣✳✱ α = x1abax1ax1x2x2bax2x3x3bbx3ax3

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s

k✲❘❡♣❡❛t❡❞✲❱❛r✐❛❜❧❡ P❛tt❡r♥s✿ |{x ∈ var(α) | |α|x ≥ 2}| ≤ k✳ ❊✳ ❣✳✱ α = x1abx2ax2ax3bax2bbx4x2x5 ✐s ❛ 1✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡ ♣❛tt❡r♥✳ P❛tt❡r♥ ✇✐t❤ ❇♦✉♥❞❡❞ ❙❝♦♣❡ ❈♦✐♥❝✐❞❡♥❝❡ ❉❡❣r❡❡✿ ❙❝♦♣❡ ✭♦❢ ✮✿ s❤♦rt❡st ❢❛❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ♦❝❝✳ ♦❢ ✱ ❙❝♦♣❡ ❝♦✐♥❝✐❞❡♥❝❡ ❞❡❣r❡❡✿ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦✐♥❝✐❞✐♥❣ s❝♦♣❡s✳

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s

k✲❘❡♣❡❛t❡❞✲❱❛r✐❛❜❧❡ P❛tt❡r♥s✿ |{x ∈ var(α) | |α|x ≥ 2}| ≤ k✳ ❊✳ ❣✳✱ α = x1abx2ax2ax3bax2bbx4x2x5 ✐s ❛ 1✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡ ♣❛tt❡r♥✳ P❛tt❡r♥ ✇✐t❤ ❇♦✉♥❞❡❞ ❙❝♦♣❡ ❈♦✐♥❝✐❞❡♥❝❡ ❉❡❣r❡❡✿ ❙❝♦♣❡ ✭♦❢ x✮✿ s❤♦rt❡st ❢❛❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ♦❝❝✳ ♦❢ x✱ ❙❝♦♣❡ ❝♦✐♥❝✐❞❡♥❝❡ ❞❡❣r❡❡✿ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦✐♥❝✐❞✐♥❣ s❝♦♣❡s✳ α1 = x1 x2 x1 x3 x2 x3 x1 x2 x3 scd(α1) = 3 α2 = x1 x2 x1 x1 x2 x3 x2 x3 x3 scd(α2) = 2

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s ✲ ❈♦♠♣❧❡①✐t②

❑♥♦✇♥ r❡s✉❧ts✿ ▼❛t❝❤ ✐s ✐♥ P ❢♦r r❡❣✉❧❛r ♣❛tt❡r♥s O(|α| + |w|)✱ ♥♦♥✲❝r♦ss ♣❛tt❡r♥s O(|α||w|4)✱ ♣❛tt❡r♥s ✇✐t❤ s❝❞ ≤ k O(|α||w|2(k+3)(k + 2)2)✳ ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❋✐♥❞ ✭❡✣❝✐❡♥t✮ ❛❧❣♦r✐t❤♠s ❢♦r t❤❡s❡ ❝❛s❡s✳ ❈❛♥ ✇❡ ❡①t❡♥❞ ♦✉r ❛❧❣♦r✐t❤♠s t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❝❛s❡ ✭✐✳ ❡✳✱ ❞✐✛❡r❡♥t ✈❛r✐❛❜❧❡s ❛r❡ r❡♣❧❛❝❡❞ ❜② ❞✐✛❡r❡♥t ✇♦r❞s✮❄

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s ✲ ❈♦♠♣❧❡①✐t②

❑♥♦✇♥ r❡s✉❧ts✿ ▼❛t❝❤ ✐s ✐♥ P ❢♦r r❡❣✉❧❛r ♣❛tt❡r♥s O(|α| + |w|)✱ ♥♦♥✲❝r♦ss ♣❛tt❡r♥s O(|α||w|4)✱ ♣❛tt❡r♥s ✇✐t❤ s❝❞ ≤ k O(|α||w|2(k+3)(k + 2)2)✳ ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❋✐♥❞ ✭❡✣❝✐❡♥t✮ ❛❧❣♦r✐t❤♠s ❢♦r t❤❡s❡ ❝❛s❡s✳ ❈❛♥ ✇❡ ❡①t❡♥❞ ♦✉r ❛❧❣♦r✐t❤♠s t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❝❛s❡ ✭✐✳ ❡✳✱ ❞✐✛❡r❡♥t ✈❛r✐❛❜❧❡s ❛r❡ r❡♣❧❛❝❡❞ ❜② ❞✐✛❡r❡♥t ✇♦r❞s✮❄

k✲❘❡♣❡❛t❡❞ ❱❛r✐❛❜❧❡ P❛tt❡r♥s

▲❡♠♠❛

▼❛t❝❤ ❢♦r 1✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡ ♣❛tt❡r♥s ✐s s♦❧✈❛❜❧❡ ✐♥ O(|w|2)✳

❚❤❡♦r❡♠

▼❛t❝❤ ❢♦r k✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡ ♣❛tt❡r♥s ✐s s♦❧✈❛❜❧❡ ✐♥ O

• |w|2k

((k−1)!)2

• ✳

◆♦♥✲❈r♦ss P❛tt❡r♥s

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✦ α ♥♦♥✲❝r♦ss ⇒ α = w0α1w1α2 . . . αℓwℓ✳ var(αi) = {xi}✱ wi ∈ Σ∗ ❈♦♠♣✉t❡ ❛❧❧ s✉❜✲♣r♦❜❧❡♠s✿ ❉♦❡s ♠❛t❝❤ ❄ ✱

◆♦♥✲❈r♦ss P❛tt❡r♥s

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✦ α ♥♦♥✲❝r♦ss ⇒ α = w0α1w1α2 . . . αℓwℓ✳ var(αi) = {xi}✱ wi ∈ Σ∗ ❈♦♠♣✉t❡ ❛❧❧ s✉❜✲♣r♦❜❧❡♠s✿ ❉♦❡s w0α1w1 . . . wi−1αi ♠❛t❝❤ w[1..j]❄ 1 ≤ i ≤ ℓ✱ 1 ≤ j ≤ |w|

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 1✿ αi = xi w0α1w1 . . . wi−1 αi ↓ w[1..j]

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 1✿ αi = xi w0α1w1 . . . wi−1 xi ↓ w[1..j]

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 1✿ αi = xi w0α1w1 . . . wi−1 xi ↓ w[1..j] ⇐ ⇒ w0α1w1 . . . wi−1 ↓ w[1..j′]

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 1✿ αi = xi w0α1w1 . . . wi−1 xi ↓ w[1..j] ⇐ ⇒ w0α1w1 . . . wi−1 xi ↓ ↓ w[1..j′] w[j′ + 1..j]

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2a✿ αi = (xi)k ✭xi ✐s ♠❛♣♣❡❞ t♦ ♣r✐♠✐t✐✈❡ ✇♦r❞ t✮ w0α1w1 . . . wi−1 αi ↓ w[1..j] ♣r✐♠✐t✐✈❡ ✇♦r❞ ✇✐t❤ s✉✣① ♦❢ ❛♥❞

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2a✿ αi = (xi)k ✭xi ✐s ♠❛♣♣❡❞ t♦ ♣r✐♠✐t✐✈❡ ✇♦r❞ t✮ w0α1w1 . . . wi−1 xixi . . . xi ↓ w[1..j] ♣r✐♠✐t✐✈❡ ✇♦r❞ ✇✐t❤ s✉✣① ♦❢ ❛♥❞

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2a✿ αi = (xi)k ✭xi ✐s ♠❛♣♣❡❞ t♦ ♣r✐♠✐t✐✈❡ ✇♦r❞ t✮ w0α1w1 . . . wi−1 xixi . . . xi ↓ w[1..j] ⇐ ⇒ ∃ ♣r✐♠✐t✐✈❡ ✇♦r❞ t ✇✐t❤ tk s✉✣① ♦❢ w[1..j] ❛♥❞ w0α1w1 . . . wi−1 ↓ w[1..j − (k|t|)]

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2a✿ αi = (xi)k ✭xi ✐s ♠❛♣♣❡❞ t♦ ♣r✐♠✐t✐✈❡ ✇♦r❞ t✮ w0α1w1 . . . wi−1 xixi . . . xi ↓ w[1..j] ⇐ ⇒ ∃ ♣r✐♠✐t✐✈❡ ✇♦r❞ t ✇✐t❤ tk s✉✣① ♦❢ w[1..j] ❛♥❞ w0α1w1 . . . wi−1 xixi . . . xi ↓ ↓ w[1..j − (k|t|)] tt . . . t

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2a✿ ❋✐♥❞ ❛❧❧ ♣r✐♠✐t✐✈❡ t s✉❝❤ t❤❛t w[1..j] ❤❛s t2 ❛s ❛ s✉✣①✦

▲❡♠♠❛ ✭❈r♦❝❤❡♠♦r❡✱ ✶✾✽✶✮

Pr✐♠✐t✐✈❡ u1, u2, u3✱ |u1| < |u2| < |u3|✱ w = wiuiui✱ 1 ≤ i ≤ 3 ⇒ 2|u1| < |u3|✳ ⇒ w ❤❛s ❛t ♠♦st 2 log |w| ♣r✐♠✐t✐✈❡❧② r♦♦t❡❞ sq✉❛r❡s ❛s s✉✣①✳

▲❡♠♠❛

❲❡ ❝❛♥ ❝♦♠♣✉t❡ ✐♥ t✐♠❡ ❛❧❧ t❤❡ s❡ts ♣r✐♠✐t✐✈❡ s✉✣① ♦❢ ✱ ✳ ❈❛s❡ ❝❛♥ ❜❡ ❞♦♥❡ ❡✣❝✐❡♥t❧②✳

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2a✿ ❋✐♥❞ ❛❧❧ ♣r✐♠✐t✐✈❡ t s✉❝❤ t❤❛t w[1..j] ❤❛s t2 ❛s ❛ s✉✣①✦

▲❡♠♠❛ ✭❈r♦❝❤❡♠♦r❡✱ ✶✾✽✶✮

Pr✐♠✐t✐✈❡ u1, u2, u3✱ |u1| < |u2| < |u3|✱ w = wiuiui✱ 1 ≤ i ≤ 3 ⇒ 2|u1| < |u3|✳ ⇒ w ❤❛s ❛t ♠♦st 2 log |w| ♣r✐♠✐t✐✈❡❧② r♦♦t❡❞ sq✉❛r❡s ❛s s✉✣①✳

▲❡♠♠❛

❲❡ ❝❛♥ ❝♦♠♣✉t❡ ✐♥ O(n log n) t✐♠❡ ❛❧❧ t❤❡ s❡ts Pi = {u | u ♣r✐♠✐t✐✈❡, u2 s✉✣① ♦❢ w[1..i]}✱ 1 ≤ i ≤ |w|✳ ⇒ ❈❛s❡ 2a ❝❛♥ ❜❡ ❞♦♥❡ ❡✣❝✐❡♥t❧②✳

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2b✿ αi = (xi)k ✭xi ✐s ♠❛♣♣❡❞ t♦ s♦♠❡ ✇♦r❞ t = vh+1✮ w0α1w1 . . . wi−1 xixi . . . xi ↓ w[1..j] ♣r✐♠✐t✐✈❡ ✇♦r❞ ✇✐t❤ s✉✣① ♦❢ ❛♥❞ ✇✐t❤

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 2b✿ αi = (xi)k ✭xi ✐s ♠❛♣♣❡❞ t♦ s♦♠❡ ✇♦r❞ t = vh+1✮ w0α1w1 . . . wi−1 xixi . . . xi ↓ w[1..j] ⇐ ⇒ ∃ ♣r✐♠✐t✐✈❡ ✇♦r❞ v ✇✐t❤ vk s✉✣① ♦❢ w[1..j] ❛♥❞ w0α1w1 . . . wi−1xixi . . . xi ✇✐t❤ h(xi) = vh ↓ w[1..j − k|v|)]

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 3✿ αi = xℓ0

i u1xℓ1 i u2 . . . xℓp−1 i

upxℓp

i

uk ∈ Σ+ w0α1w1 . . . wi−1 αi ↓ w[1..j] ✿ ♣r♦❝❡❡❞ s✐♠✐❧❛r t♦ ❈❛s❡ ✭♠♦r❡ ✐♥✈♦❧✈❡❞✱ ❞❡t❛✐❧s ♦♠✐tt❡❞✮✳ ✿ ✜♥❞ ❛❧❧ ♣r✐♠✐t✐✈❡ s✉❝❤ t❤❛t ✐s ❛ s✉✣① ♦❢ ✳

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 3✿ αi = xℓ0

i u1xℓ1 i u2 . . . xℓp−1 i

upxℓp

i

uk ∈ Σ+ w0α1w1 . . . wi−1 xℓ0

i u1xℓ1 i u2 . . . xℓp−1 i

upxℓp

i

↓ w[1..j] ✿ ♣r♦❝❡❡❞ s✐♠✐❧❛r t♦ ❈❛s❡ ✭♠♦r❡ ✐♥✈♦❧✈❡❞✱ ❞❡t❛✐❧s ♦♠✐tt❡❞✮✳ ✿ ✜♥❞ ❛❧❧ ♣r✐♠✐t✐✈❡ s✉❝❤ t❤❛t ✐s ❛ s✉✣① ♦❢ ✳

◆♦♥✲❈r♦ss P❛tt❡r♥s

❈❛s❡ 3✿ αi = xℓ0

i u1xℓ1 i u2 . . . xℓp−1 i

upxℓp

i

uk ∈ Σ+ w0α1w1 . . . wi−1 xℓ0

i u1xℓ1 i u2 . . . xℓp−1 i

upxℓp

i

↓ w[1..j] ℓp ≥ 2✿ ♣r♦❝❡❡❞ s✐♠✐❧❛r t♦ ❈❛s❡ 2 ✭♠♦r❡ ✐♥✈♦❧✈❡❞✱ ❞❡t❛✐❧s ♦♠✐tt❡❞✮✳ ℓp = 1✿ ✜♥❞ ❛❧❧ ♣r✐♠✐t✐✈❡ upt s✉❝❤ t❤❛t tupt ✐s ❛ s✉✣① ♦❢ w[1..j]✳

◆♦♥✲❈r♦ss P❛tt❡r♥s

• ❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ ❈r♦❝❤❡♠♦r❡✬s r❡s✉❧t✿

▲❡♠♠❛

❋♦r ❛ ✜①❡❞ v✱ w ❤❛s O(log |w|) ❢❛❝t♦rs uvu ✇✐t❤ uv ♣r✐♠✐t✐✈❡ ❛s s✉✣①❡s✳

▲❡♠♠❛

❋♦r ✜①❡❞ v✱ w✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ✐♥ O(n log n) t✐♠❡ ❛❧❧ t❤❡ s❡ts Rv

i = {u | uv ♣r✐♠✐t✐✈❡, uvu s✉✣① ♦❢ w[1..i]}✱ 1 ≤ i ≤ |w|✳

⇒ ❈❛s❡ 3 ❝❛♥ ❜❡ ❞♦♥❡ ❡✣❝✐❡♥t❧②✳

◆♦♥✲❈r♦ss P❛tt❡r♥s

❚❤❡♦r❡♠

▼❛t❝❤ ❢♦r ♥♦♥✲❝r♦ss ♣❛tt❡r♥s ✐s s♦❧✈❛❜❧❡ ✐♥ O(|w|m log |w|)✱ ✇❤❡r❡ m ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦♥❡✲✈❛r✐❛❜❧❡ ❜❧♦❝❦s ♦❢ t❤❡ ♣❛tt❡r♥✳

❚❤❡♦r❡♠

▼❛t❝❤ ❢♦r ♣❛tt❡r♥s ✇✐t❤ s❝♦♣❡ ❝♦✐♥❝✐❞❡♥❝❡ ❞❡❣r❡❡ ♦❢ ❛t ♠♦st k ✐s s♦❧✈❛❜❧❡ ✐♥ O

• |w|2km

((k−1)!)2

• ✱ ✇❤❡r❡ m ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦♥❡✲✈❛r✐❛❜❧❡ ❜❧♦❝❦s

♦❢ t❤❡ ♣❛tt❡r♥✳

■♥❥❡❝t✐✈❡ ▼❛t❝❤

■♥❥▼❛t❝❤✿ ▲✐❦❡ ▼❛t❝❤✱ ❜✉t ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❢♦r ❛♥ ✐♥❥❡❝t✐✈❡ s✉❜st✐t✉t✐♦♥ h✱ ✐✳ ❡✳✱ x = y ⇒ h(x) = h(y)✳ ❈❛♥ ✇❡ ✉s❡ ♦✉r ✭♦r ♦t❤❡r✮ ▼❛t❝❤✲❛❧❣♦r✐t❤♠s ❛❧s♦ ❢♦r ■♥❥▼❛t❝❤❄ ■♥❥▼❛t❝❤ r❡♠❛✐♥s ◆P✲❝♦♠♣❧❡t❡ ❢♦r ♣❛tt❡r♥s ❢♦r ✇❤✐❝❤ ▼❛t❝❤ ✐s ✭tr✐✈✐❛❧❧②✮ ✐♥ P✳

■♥❥❡❝t✐✈❡ ▼❛t❝❤

❚❤❡♦r❡♠

■♥❥▼❛t❝❤ ✐s ◆P✲❝♦♠♣❧❡t❡ ❡✈❡♥ ❢♦r ♣❛tt❡r♥s x1x2 . . . xn✱ n ≥ 1✳ ❲❡ ♣r♦✈❡ ◆P✲❝♦♠♣❧❡t❡♥❡ss ♦❢ t❤❡ ❡q✉✐✈❛❧❡♥t ♣r♦❜❧❡♠

❯♥❋❛❝t

■♥st❛♥❝❡✿ ❆ ✇♦r❞ w ❛♥❞ ❛♥ ✐♥t❡❣❡r k ≥ 1✳ ◗✉❡st✐♦♥✿ w = u1u2 . . . uk′ ✇✐t❤ k′ ≥ k ❛♥❞ ui = uj✱ 1 ≤ i < j ≤ k❄

❈♦r♦❧❧❛r②

■♥❥▼❛t❝❤ ✐s ◆P✲❝♦♠♣❧❡t❡ ❢♦r r❡❣✉❧❛r✱ ♥♦♥✲❝r♦ss✱ k✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡✱ ❜♦✉♥❞❡❞ s❝❞ ♣❛tt❡r♥s✳

❍❛r❞♥❡ss ♦❢ ■♥❥▼❛t❝❤ ✲ Pr♦♦❢ ■❞❡❛

✸❉✲▼❛t❝❤

■♥st❛♥❝❡✿ ❆♥ ✐♥t❡❣❡r ℓ ∈ N ❛♥❞ ❛ s❡t S ⊆ {(p, q, r) | 1 ≤ p < ℓ + 1 ≤ q < 2ℓ + 1 ≤ r ≤ 3ℓ}✳ ◗✉❡st✐♦♥✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ s✉❜s❡t S′ ♦❢ S ✇✐t❤ ❝❛r❞✐♥❛❧✐t② ℓ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ t✇♦ ❡❧❡♠❡♥ts (p, q, r), (p′, q′, r′) ∈ S′✱ p = p′✱ q = q′ ❛♥❞ r = r′❄

❍❛r❞♥❡ss ♦❢ ■♥❥▼❛t❝❤ ✲ Pr♦♦❢ ■❞❡❛

✸❉✲▼❛t❝❤ ✐♥st❛♥❝❡ (S, ℓ)✿ S = {s1, s2, . . . , sk} ❚r❛♥s❢♦r♠ ❡✈❡r② si = (pi, qi, ri)✱ 1 ≤ i ≤ k✱ ✐♥t♦ vi = ⋆i pi a bi,1 bi,2 qi a bi,3 bi,4 ri a ⋄i ⋆i✱ ⋄i✱ bi,j ❤❛✈❡ ♦♥❧② ♦♥❡ ♦❝❝✉rr❡♥❝❡✦ ▲❡t ✳ ✇✐t❤ ❛♥❞ ✱ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ ✳

❍❛r❞♥❡ss ♦❢ ■♥❥▼❛t❝❤ ✲ Pr♦♦❢ ■❞❡❛

✸❉✲▼❛t❝❤ ✐♥st❛♥❝❡ (S, ℓ)✿ S = {s1, s2, . . . , sk} ❚r❛♥s❢♦r♠ ❡✈❡r② si = (pi, qi, ri)✱ 1 ≤ i ≤ k✱ ✐♥t♦ vi = ⋆i pi a bi,1 bi,2 qi a bi,3 bi,4 ri a ⋄i ⋆i✱ ⋄i✱ bi,j ❤❛✈❡ ♦♥❧② ♦♥❡ ♦❝❝✉rr❡♥❝❡✦ ▲❡t S′ ⊆ S✳ (pi, qi, ri) / ∈ S′ ⇔ ⋆ipi abi,1 bi,2qi abi,3 bi,4ri a⋄i (pi, qi, ri) ∈ S′ ⇔ ⋆i pia bi,1bi,2 qia bi,3bi,4 ria ⋄i ✇✐t❤ ❛♥❞ ✱ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ ✳

❍❛r❞♥❡ss ♦❢ ■♥❥▼❛t❝❤ ✲ Pr♦♦❢ ■❞❡❛

✸❉✲▼❛t❝❤ ✐♥st❛♥❝❡ (S, ℓ)✿ S = {s1, s2, . . . , sk} ❚r❛♥s❢♦r♠ ❡✈❡r② si = (pi, qi, ri)✱ 1 ≤ i ≤ k✱ ✐♥t♦ vi = ⋆i pi a bi,1 bi,2 qi a bi,3 bi,4 ri a ⋄i ⋆i✱ ⋄i✱ bi,j ❤❛✈❡ ♦♥❧② ♦♥❡ ♦❝❝✉rr❡♥❝❡✦ ▲❡t S′ ⊆ S✳ (pi, qi, ri) / ∈ S′ ⇔ ⋆ipi abi,1 bi,2qi abi,3 bi,4ri a⋄i (pi, qi, ri) ∈ S′ ⇔ ⋆i pia bi,1bi,2 qia bi,3bi,4 ria ⋄i v = u1u2 . . . un ✇✐t❤ n = 7ℓ + 6(k − ℓ) ❛♥❞ ui = uj✱ 1 ≤ i < j ≤ n ⇐ ⇒ S′ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ (S, ℓ)✳

❆❧♣❤❛❜❡t ❙✐③❡

❖✉r ❘❡❞✉❝t✐♦♥ ♥❡❡❞s ❛♥ ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡t✦ ❍❛r❞♥❡ss ♦❢ ■♥❥▼❛t❝❤ ❢♦r ✜①❡❞ ❛❧♣❤❛❜❡ts ✐s ♦♣❡♥✱ ❜✉t✳✳✳

❚❤❡♦r❡♠

■♥❥▼❛t❝❤ ✭✇✐t❤ ❝♦♥st❛♥t ❛❧♣❤❛❜❡t✮ ✐s ◆P✲❝♦♠♣❧❡t❡ ❢♦r r❡❣✉❧❛r✱ ♥♦♥✲❝r♦ss✱ k✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡✱ ❜♦✉♥❞❡❞ s❝❞ ♣❛tt❡r♥s✳

❚❤❛♥❦ ②♦✉ ✈❡r② ♠✉❝❤ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳