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❆rt✉r ❏❡➺
▼❡❡t✐♥❣ ♦♥ ❙tr✐♥❣ ❈♦♥str❛✐♥ts ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✭▼❖❙❈❆✮ ✵✼✳✵✺✳✷✵✶✾
■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮ ❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ✳ ❯s✉❛❧❧②✿ ♥♦ ✱ ✐✳❡✳ ✳
■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮ aX bX Y bbb=X abaaY bY S(X) = aa, S(Y ) = bb ❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ✳ ❯s✉❛❧❧②✿ ♥♦ ✱ ✐✳❡✳ ✳
■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮ aX bX Y bbb=X abaaY bY S(X) = aa, S(Y ) = bb aaabaabbbbb=aaabaabbbbb ❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ✳ ❯s✉❛❧❧②✿ ♥♦ ✱ ✐✳❡✳ ✳
■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮ aX bX Y bbb=X abaaY bY S(X) = aa, S(Y ) = bb aaabaabbbbb=aaabaabbbbb ❲❡ ❡①t❡♥❞ S t♦ ❛ S : (Σ ∪ X)∗ → Σ∗❀ ✐❞❡♥t✐t② ♦♥ Σ✳ S(U) ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ S✿ ♠✐♥✐♠✐s❡s |S(U)|✳ ❯s✉❛❧❧②✿ ♥♦ S(X) = ǫ✱ ✐✳❡✳ S : X → Σ+✳
❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ ✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦ ✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪ ❚❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄
❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ ✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦ ✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪ ❚❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄
❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦
poly(log N)✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪ ❚❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄
❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦
poly(log N)✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪
◮ ❚❤❡ s✐③❡ N ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧②
❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄
❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦
poly(log N)✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪
◮ ❚❤❡ s✐③❡ N ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧②
❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪
◮ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪
❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄
❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦
poly(log N)✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪
◮ ❚❤❡ s✐③❡ N ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧②
❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪
◮ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ◮ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪
❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄
❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦
poly(log N)✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪
◮ ❚❤❡ s✐③❡ N ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧②
❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪
◮ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ◮ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪
❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄
❙✐♠♣❧❡ ✐s ❣♦♦❞ ♦♥ ✐ts ♦✇♥✳
❘❡❣✉❧❛r ❝♦♥str❛✐♥ts ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪ ■♥✈♦❧✉t✐♦♥ ✭ ✮ ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪ ❢r❡❡ ❣r♦✉♣s ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪ ❣❡♥❡r❛t✐♦♥ ♦❢ ❛❧❧ s♦❧✉t✐♦♥s ❬❏✳❪ ❢♦r ❢r❡❡ ❣r♦✉♣s ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪ ♣❛rt✐❛❧ ❝♦♠♠✉t❛t✐♦♥ ❬❉✐❡❦❡rt✱ ❏✳✱ ❑✉✢❡✐t♥❡r❪ ❛❧❧ s♦❧✉t✐♦♥s ❛r❡ ❊❉❚✵▲ ❧❛♥❣✉❛❣❡ ❬❈✐♦❜❛♥✉✱ ❉✐❡❦❡rt✱ ❊❧❞❡r❪ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❧✐♥❡❛r s♣❛❝❡ ❂ ❝♦♥t❡①t s❡♥s✐t✐✈❡ ❧❛♥❣✉❛❣❡ ❬❏✳❪ t✇✐st❡❞ ✇♦r❞ ❡q✉❛t✐♦♥s ✭♣❡r♠✉t❛t✐♦♥ ♦❢ ❧❡tt❡rs✮ ❬❉✐❡❦❡rt✱ ❊❧❞❡r❪ ❧✐♥❡❛r t✐♠❡ ❢♦r ♦♥❡ ✈❛r✐❛❜❧❡ ❬❏✳❪ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥ ✭t❡r♠s✮ ❬❏✳❪
❙✐♠♣❧❡ ✐s ❣♦♦❞ ♦♥ ✐ts ♦✇♥✳
◮ ❘❡❣✉❧❛r ❝♦♥str❛✐♥ts ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪ ◮ ■♥✈♦❧✉t✐♦♥ ✭aw = w a✮ ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪ ◮ ❢r❡❡ ❣r♦✉♣s ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪ ◮ ❣❡♥❡r❛t✐♦♥ ♦❢ ❛❧❧ s♦❧✉t✐♦♥s ❬❏✳❪
❢♦r ❢r❡❡ ❣r♦✉♣s ❬❉✐❡❦❡rt✱ ❏✳✱ P❧❛♥❞♦✇s❦✐❪
◮ ♣❛rt✐❛❧ ❝♦♠♠✉t❛t✐♦♥ ❬❉✐❡❦❡rt✱ ❏✳✱ ❑✉✢❡✐t♥❡r❪ ◮ ❛❧❧ s♦❧✉t✐♦♥s ❛r❡ ❊❉❚✵▲ ❧❛♥❣✉❛❣❡ ❬❈✐♦❜❛♥✉✱ ❉✐❡❦❡rt✱ ❊❧❞❡r❪ ◮ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❧✐♥❡❛r s♣❛❝❡ ❂ ❝♦♥t❡①t s❡♥s✐t✐✈❡ ❧❛♥❣✉❛❣❡ ❬❏✳❪ ◮ t✇✐st❡❞ ✇♦r❞ ❡q✉❛t✐♦♥s ✭♣❡r♠✉t❛t✐♦♥ ♦❢ ❧❡tt❡rs✮ ❬❉✐❡❦❡rt✱ ❊❧❞❡r❪ ◮ ❧✐♥❡❛r t✐♠❡ ❢♦r ♦♥❡ ✈❛r✐❛❜❧❡ ❬❏✳❪ ◮ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥ ✭t❡r♠s✮ ❬❏✳❪
❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
◮ ❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ◮ ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ◮ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
◮ ❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ♦❢ ❛ ❣r❛♠♠❛r ◮ ❲❡ ❜✉✐❧❞ ❛ ❣r❛♠♠❛r ❢♦r ❜♦t❤ str✐♥❣s✱ ❜♦tt♦♠✲✉♣✳ ◮ ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦
❚♦♣✲❞♦✇♥✱ ❝r❡❛t❡s ♠❛♥② ♣r♦❜❧❡♠s✳
❋♦r ❜♦t❤ s♦❧✉t✐♦♥ ✇♦r❞s ❝❤♦♦s❡ ❛ ♣❛✐r ✭♦r ❧❡tt❡r✮ ❛♥❞ ❝♦♠♣r❡ss ✐t✳ ✇❤✐❧❡ ❛♥❞ ❞♦ ▲ ❧❡tt❡rs ❢r♦♠ ❢♦r ❝❤♦♦s❡ ▲ ♦r ▲ ❞♦ r❡♣❧❛❝❡ ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ✐♥ ❛♥❞ ✭♦r r❡♣❧❛❝❡ ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ❜❧♦❝❦s ♦❢ ✮ ❍♦✇ t♦ ❞♦ t❤✐s ❢♦r ❡q✉❛t✐♦♥s❄
❋♦r ❜♦t❤ s♦❧✉t✐♦♥ ✇♦r❞s ❝❤♦♦s❡ ❛ ♣❛✐r ✭♦r ❧❡tt❡r✮ ❛♥❞ ❝♦♠♣r❡ss ✐t✳ ✇❤✐❧❡ U / ∈ Σ ❛♥❞ V / ∈ Σ ❞♦ ▲ ← ❧❡tt❡rs ❢r♦♠ S(U) = S(V ) ❢♦r ❝❤♦♦s❡ ab ∈ ▲2 ♦r a ∈ ▲ ❞♦ r❡♣❧❛❝❡ ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ab ✐♥ S(U) ❛♥❞ S(V ) ✭♦r r❡♣❧❛❝❡ ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ❜❧♦❝❦s ♦❢ a✮ ❍♦✇ t♦ ❞♦ t❤✐s ❢♦r ❡q✉❛t✐♦♥s❄
❋♦r ❜♦t❤ s♦❧✉t✐♦♥ ✇♦r❞s ❝❤♦♦s❡ ❛ ♣❛✐r ✭♦r ❧❡tt❡r✮ ❛♥❞ ❝♦♠♣r❡ss ✐t✳ ✇❤✐❧❡ U / ∈ Σ ❛♥❞ V / ∈ Σ ❞♦ ▲ ← ❧❡tt❡rs ❢r♦♠ S(U) = S(V ) ❢♦r ❝❤♦♦s❡ ab ∈ ▲2 ♦r a ∈ ▲ ❞♦ r❡♣❧❛❝❡ ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ab ✐♥ S(U) ❛♥❞ S(V ) ✭♦r r❡♣❧❛❝❡ ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ❜❧♦❝❦s ♦❢ a✮ ❍♦✇ t♦ ❞♦ t❤✐s ❢♦r ❡q✉❛t✐♦♥s❄
XbaY b = baaababbab ❤❛s ❛ s♦❧✉t✐♦♥ S(X) = baaa✱ S(Y ) = bba ❲❡ ✇❛♥t t♦ r❡♣❧❛❝❡ ♣❛✐r ❜② ❛ ♥❡✇ ❧❡tt❡r ✳ ❚❤❡♥ ❢♦r ❢♦r ❆♥❞ ✇❤❛t ❛❜♦✉t r❡♣❧❛❝✐♥❣ ❜② ❄ ❢♦r ❚❤❡r❡ ✐s ❛ ♣r♦❜❧❡♠ ✇✐t❤ ❵❝r♦ss✐♥❣ ♣❛✐rs✬✳ ❲❡ ✇✐❧❧ ✜①✦
XbaY b = baaababbab ❤❛s ❛ s♦❧✉t✐♦♥ S(X) = baaa✱ S(Y ) = bba ❲❡ ✇❛♥t t♦ r❡♣❧❛❝❡ ♣❛✐r ba ❜② ❛ ♥❡✇ ❧❡tt❡r c✳ ❚❤❡♥ XbaY b=baaababbab ❢♦r S(X) = baaa S(Y ) = bba ❢♦r ❆♥❞ ✇❤❛t ❛❜♦✉t r❡♣❧❛❝✐♥❣ ❜② ❄ ❢♦r ❚❤❡r❡ ✐s ❛ ♣r♦❜❧❡♠ ✇✐t❤ ❵❝r♦ss✐♥❣ ♣❛✐rs✬✳ ❲❡ ✇✐❧❧ ✜①✦
XbaY b = baaababbab ❤❛s ❛ s♦❧✉t✐♦♥ S(X) = baaa✱ S(Y ) = bba ❲❡ ✇❛♥t t♦ r❡♣❧❛❝❡ ♣❛✐r ba ❜② ❛ ♥❡✇ ❧❡tt❡r c✳ ❚❤❡♥ XbaY b=baaababbab ❢♦r S(X) = baaa S(Y ) = bba X c Y b= c aa c b c b ❢♦r S′(X) = caa S′(Y ) = bc ❆♥❞ ✇❤❛t ❛❜♦✉t r❡♣❧❛❝✐♥❣ ❜② ❄ ❢♦r ❚❤❡r❡ ✐s ❛ ♣r♦❜❧❡♠ ✇✐t❤ ❵❝r♦ss✐♥❣ ♣❛✐rs✬✳ ❲❡ ✇✐❧❧ ✜①✦
XbaY b = baaababbab ❤❛s ❛ s♦❧✉t✐♦♥ S(X) = baaa✱ S(Y ) = bba ❲❡ ✇❛♥t t♦ r❡♣❧❛❝❡ ♣❛✐r ba ❜② ❛ ♥❡✇ ❧❡tt❡r c✳ ❚❤❡♥ XbaY b=baaababbab ❢♦r S(X) = baaa S(Y ) = bba X c Y b= c aa c b c b ❢♦r S′(X) = caa S′(Y ) = bc ❆♥❞ ✇❤❛t ❛❜♦✉t r❡♣❧❛❝✐♥❣ ab ❜② d❄ XbaY b = baaababbab ❢♦r S(X) = baaa S(Y ) = bba ❚❤❡r❡ ✐s ❛ ♣r♦❜❧❡♠ ✇✐t❤ ❵❝r♦ss✐♥❣ ♣❛✐rs✬✳ ❲❡ ✇✐❧❧ ✜①✦
XbaY b = baaababbab ❤❛s ❛ s♦❧✉t✐♦♥ S(X) = baaa✱ S(Y ) = bba ❲❡ ✇❛♥t t♦ r❡♣❧❛❝❡ ♣❛✐r ba ❜② ❛ ♥❡✇ ❧❡tt❡r c✳ ❚❤❡♥ XbaY b=baaababbab ❢♦r S(X) = baaa S(Y ) = bba X c Y b= c aa c b c b ❢♦r S′(X) = caa S′(Y ) = bc ❆♥❞ ✇❤❛t ❛❜♦✉t r❡♣❧❛❝✐♥❣ ab ❜② d❄ XbaY b = baaababbab ❢♦r S(X) = baaa S(Y ) = bba ❚❤❡r❡ ✐s ❛ ♣r♦❜❧❡♠ ✇✐t❤ ❵❝r♦ss✐♥❣ ♣❛✐rs✬✳ ❲❡ ✇✐❧❧ ✜①✦
❖❝❝✉rr❡♥❝❡ ♦❢ ab ✐♥ ❛ s♦❧✉t✐♦♥ ✇♦r❞ ✭s♦ ❢♦r ❛ ✜①❡❞ s♦❧✉t✐♦♥✮ ✐s ❡①♣❧✐❝✐t ✐t ❝♦♠❡s ❢r♦♠ U ♦r V ❀ ✐♠♣❧✐❝✐t ❝♦♠❡s s♦❧❡❧② ❢r♦♠ S(X)❀ ❝r♦ss✐♥❣ ✐♥ ♦t❤❡r ❝❛s❡✳ ab ✐s ❝r♦ss✐♥❣ ✐❢ ✐t ❤❛s ❛ ❝r♦ss✐♥❣ ♦❝❝✉rr❡♥❝❡✱ ♥♦♥✲❝r♦ss✐♥❣ ♦t❤❡r✇✐s❡✳ ❡①♣❧✐❝✐t ✐♠♣❧✐❝✐t ❝r♦ss✐♥❣
❖❝❝✉rr❡♥❝❡ ♦❢ ab ✐♥ ❛ s♦❧✉t✐♦♥ ✇♦r❞ ✭s♦ ❢♦r ❛ ✜①❡❞ s♦❧✉t✐♦♥✮ ✐s ❡①♣❧✐❝✐t ✐t ❝♦♠❡s ❢r♦♠ U ♦r V ❀ ✐♠♣❧✐❝✐t ❝♦♠❡s s♦❧❡❧② ❢r♦♠ S(X)❀ ❝r♦ss✐♥❣ ✐♥ ♦t❤❡r ❝❛s❡✳ ab ✐s ❝r♦ss✐♥❣ ✐❢ ✐t ❤❛s ❛ ❝r♦ss✐♥❣ ♦❝❝✉rr❡♥❝❡✱ ♥♦♥✲❝r♦ss✐♥❣ ♦t❤❡r✇✐s❡✳ X baa Y b = baaabaabbab S(X) = baaa S(Y ) = bba ❡①♣❧✐❝✐t ✐♠♣❧✐❝✐t ❝r♦ss✐♥❣
❖❝❝✉rr❡♥❝❡ ♦❢ ab ✐♥ ❛ s♦❧✉t✐♦♥ ✇♦r❞ ✭s♦ ❢♦r ❛ ✜①❡❞ s♦❧✉t✐♦♥✮ ✐s ❡①♣❧✐❝✐t ✐t ❝♦♠❡s ❢r♦♠ U ♦r V ❀ ✐♠♣❧✐❝✐t ❝♦♠❡s s♦❧❡❧② ❢r♦♠ S(X)❀ ❝r♦ss✐♥❣ ✐♥ ♦t❤❡r ❝❛s❡✳ ab ✐s ❝r♦ss✐♥❣ ✐❢ ✐t ❤❛s ❛ ❝r♦ss✐♥❣ ♦❝❝✉rr❡♥❝❡✱ ♥♦♥✲❝r♦ss✐♥❣ ♦t❤❡r✇✐s❡✳ X baa Y b = baaabaabbab S(X) = baaa S(Y ) = bba baaa baa bba b = baaabaabbab ❡①♣❧✐❝✐t baaa baa bba b = baaabaabbab ✐♠♣❧✐❝✐t baaa baa bba b = baaabaabbab ❝r♦ss✐♥❣
✶✿ ❧❡t c ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r ✷✿ r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ab ✐♥ U ❛♥❞ V ❜② c
❚❤❡ P❛✐r❈♦♠♣ ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ♣❛✐rs✳ ❝♦♠♣❧❡t❡ ✐❢ t❤❡ ♦❧❞ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ t❤❡ ♥❡✇ ♦♥❡ ❤❛s s♦✉♥❞ ✐❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ t❤❡ ♦❧❞ ♦♥❡ ❤❛s
✶✿ ❧❡t c ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r ✷✿ r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ab ✐♥ U ❛♥❞ V ❜② c
❚❤❡ P❛✐r❈♦♠♣(a, b) ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ♣❛✐rs✳ ❝♦♠♣❧❡t❡ ✐❢ t❤❡ ♦❧❞ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ t❤❡ ♥❡✇ ♦♥❡ ❤❛s s♦✉♥❞ ✐❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ t❤❡ ♦❧❞ ♦♥❡ ❤❛s
✶✿ ❧❡t c ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r ✷✿ r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ab ✐♥ U ❛♥❞ V ❜② c
❚❤❡ P❛✐r❈♦♠♣(a, b) ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ♣❛✐rs✳ ❝♦♠♣❧❡t❡ ✐❢ t❤❡ ♦❧❞ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ t❤❡ ♥❡✇ ♦♥❡ ❤❛s s♦✉♥❞ ✐❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ t❤❡ ♦❧❞ ♦♥❡ ❤❛s
S′(U ′) ✐s S(U) ✇✐t❤ ❡✈❡r② ab r❡♣❧❛❝❡❞❀ s✐♠✐❧❛r❧② S′(V ′)✿ ❡①♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ❡①♣❧✐❝✐t❧② ✐♠♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ✐♠♣❧✐❝✐t❧② ✭✐♥ t❤❡ s♦❧✉t✐♦♥✮ ❝r♦ss✐♥❣ t❤❡r❡ ❛r❡ ♥♦♥❡
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❛❜❧❡✿ r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳
S′(U ′) ✐s S(U) ✇✐t❤ ❡✈❡r② ab r❡♣❧❛❝❡❞❀ s✐♠✐❧❛r❧② S′(V ′)✿ ❡①♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ❡①♣❧✐❝✐t❧② ✐♠♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ✐♠♣❧✐❝✐t❧② ✭✐♥ t❤❡ s♦❧✉t✐♦♥✮ ❝r♦ss✐♥❣ t❤❡r❡ ❛r❡ ♥♦♥❡
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❛❜❧❡✿ r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba baaabaabbab=baaabaabbab
S′(U ′) ✐s S(U) ✇✐t❤ ❡✈❡r② ab r❡♣❧❛❝❡❞❀ s✐♠✐❧❛r❧② S′(V ′)✿ ❡①♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ❡①♣❧✐❝✐t❧② ✐♠♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ✐♠♣❧✐❝✐t❧② ✭✐♥ t❤❡ s♦❧✉t✐♦♥✮ ❝r♦ss✐♥❣ t❤❡r❡ ❛r❡ ♥♦♥❡
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❛❜❧❡✿ r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba baaabaabbab=baaabaabbab c aa c ab c b= c aa c ab c b X c a Y b= c aa c ab c b S′(X) = caa S′(Y ) = bc
S′(U ′) ✐s S(U) ✇✐t❤ ❡✈❡r② ab r❡♣❧❛❝❡❞❀ s✐♠✐❧❛r❧② S′(V ′)✿ ❡①♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ❡①♣❧✐❝✐t❧② ✐♠♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ✐♠♣❧✐❝✐t❧② ✭✐♥ t❤❡ s♦❧✉t✐♦♥✮ ❝r♦ss✐♥❣ t❤❡r❡ ❛r❡ ♥♦♥❡
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❛❜❧❡✿ r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳
S′(U ′) ✐s S(U) ✇✐t❤ ❡✈❡r② ab r❡♣❧❛❝❡❞❀ s✐♠✐❧❛r❧② S′(V ′)✿ ❡①♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ❡①♣❧✐❝✐t❧② ✐♠♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ✐♠♣❧✐❝✐t❧② ✭✐♥ t❤❡ s♦❧✉t✐♦♥✮ ❝r♦ss✐♥❣ t❤❡r❡ ❛r❡ ♥♦♥❡
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❛❜❧❡✿ r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab c aa c ab c b= c aa c ab c b X c a Y b= c aa c ab c b S′(X) = caa S′(Y ) = bc
S′(U ′) ✐s S(U) ✇✐t❤ ❡✈❡r② ab r❡♣❧❛❝❡❞❀ s✐♠✐❧❛r❧② S′(V ′)✿ ❡①♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ❡①♣❧✐❝✐t❧② ✐♠♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ✐♠♣❧✐❝✐t❧② ✭✐♥ t❤❡ s♦❧✉t✐♦♥✮ ❝r♦ss✐♥❣ t❤❡r❡ ❛r❡ ♥♦♥❡
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❛❜❧❡✿ r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba c aa c ab c b= c aa c ab c b X c a Y b= c aa c ab c b S′(X) = caa S′(Y ) = bc
S′(U ′) ✐s S(U) ✇✐t❤ ❡✈❡r② ab r❡♣❧❛❝❡❞❀ s✐♠✐❧❛r❧② S′(V ′)✿ ❡①♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ❡①♣❧✐❝✐t❧② ✐♠♣❧✐❝✐t ♣❛✐rs r❡♣❧❛❝❡❞ ✐♠♣❧✐❝✐t❧② ✭✐♥ t❤❡ s♦❧✉t✐♦♥✮ ❝r♦ss✐♥❣ t❤❡r❡ ❛r❡ ♥♦♥❡
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ✐s s❛t✐s✜❛❜❧❡✿ r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba baaabaabbab=baaabaabbab c aa c ab c b= c aa c ab c b X c a Y b= c aa c ab c b S′(X) = caa S′(Y ) = bc
❚❤❡r❡ ✐s X s✉❝❤ t❤❛t S(X) = bw ❛♥❞ aX ♦❝❝✉rs ✐♥ U = V ✭♦r s②♠♠❡tr✐❝✮✳
✶✿ ❢♦r
❞♦
✷✿
✐❢ ✜rst ❧❡tt❡r ♦❢ ✐s t❤❡♥
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ ❜② P♦♣ ❈❤❛♥❣❡ ❛❝❝♦r❞✐♥❣❧②
✹✿
✐❢ t❤❡♥ r❡♠♦✈❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
✺✿
♣❡r❢♦r♠ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞
❆❢t❡r ✉♥❝r♦ss✐♥❣ ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳ ❲❡ ❝❛♥ ❝♦♠♣r❡ss ✐t✳
❚❤❡r❡ ✐s X s✉❝❤ t❤❛t S(X) = bw ❛♥❞ aX ♦❝❝✉rs ✐♥ U = V ✭♦r s②♠♠❡tr✐❝✮✳
✶✿ ❢♦r X ∈ X ❞♦ ✷✿
✐❢ ✜rst ❧❡tt❡r ♦❢ S(X) ✐s b t❤❡♥
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ X ❜② bX ⊲ P♦♣ ⊲ ❈❤❛♥❣❡ S ❛❝❝♦r❞✐♥❣❧②
✹✿
✐❢ S(X) = ǫ t❤❡♥ r❡♠♦✈❡ X ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
✺✿
⊲ ♣❡r❢♦r♠ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞ a
❆❢t❡r ✉♥❝r♦ss✐♥❣ ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳ ❲❡ ❝❛♥ ❝♦♠♣r❡ss ✐t✳
❚❤❡r❡ ✐s X s✉❝❤ t❤❛t S(X) = bw ❛♥❞ aX ♦❝❝✉rs ✐♥ U = V ✭♦r s②♠♠❡tr✐❝✮✳
✶✿ ❢♦r X ∈ X ❞♦ ✷✿
✐❢ ✜rst ❧❡tt❡r ♦❢ S(X) ✐s b t❤❡♥
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ X ❜② bX ⊲ P♦♣ ⊲ ❈❤❛♥❣❡ S ❛❝❝♦r❞✐♥❣❧②
✹✿
✐❢ S(X) = ǫ t❤❡♥ r❡♠♦✈❡ X ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
✺✿
⊲ ♣❡r❢♦r♠ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞ a
❆❢t❡r ✉♥❝r♦ss✐♥❣ ab ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳ ❲❡ ❝❛♥ ❝♦♠♣r❡ss ✐t✳
❚❤❡r❡ ✐s X s✉❝❤ t❤❛t S(X) = bw ❛♥❞ aX ♦❝❝✉rs ✐♥ U = V ✭♦r s②♠♠❡tr✐❝✮✳
✶✿ ❢♦r X ∈ X ❞♦ ✷✿
✐❢ ✜rst ❧❡tt❡r ♦❢ S(X) ✐s b t❤❡♥
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ X ❜② bX ⊲ P♦♣ ⊲ ❈❤❛♥❣❡ S ❛❝❝♦r❞✐♥❣❧②
✹✿
✐❢ S(X) = ǫ t❤❡♥ r❡♠♦✈❡ X ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
✺✿
⊲ ♣❡r❢♦r♠ s②♠♠❡tr✐❝❛❧❧② ❢♦r t❤❡ ❧❛st ❧❡tt❡r ❛♥❞ a
❆❢t❡r ✉♥❝r♦ss✐♥❣ ab ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳ ❲❡ ❝❛♥ ❝♦♠♣r❡ss ✐t✳
X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba
X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba baaabaa bba b=baaabaabbab
X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba baaabaa bba b=baaabaabbab bXabaabY ab=baaabaabbab S′(X) = aa S′(Y ) = b
X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba baaabaa bba b=baaabaabbab baaabaab bab=baaabaabbab bXabaabY ab=baaabaabbab S′(X) = aa S′(Y ) = b
❲❤❡♥ aℓ ♦❝❝✉rs ✐♥ S(U) = S(V ) ❛♥❞ ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✳ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳ ❇❧♦❝❦ ♦❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ❡①♣❧✐❝✐t✱ ✐♠♣❧✐❝✐t ♦r ❝r♦ss✐♥❣✳ ▲❡tt❡r ✐s ❝r♦ss✐♥❣ ✭❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✮ ✐❢ t❤❡r❡ ✐s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦ ♦❢ ✳
■❢ ✐s ❛ ♠❛①✐♠❛❧ ❜❧♦❝❦ ✐♥ ❛ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡♥ ✳
❲❤❡♥ aℓ ♦❝❝✉rs ✐♥ S(U) = S(V ) ❛♥❞ ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✳ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳ ❇❧♦❝❦ ♦❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ❡①♣❧✐❝✐t✱ ✐♠♣❧✐❝✐t ♦r ❝r♦ss✐♥❣✳ ▲❡tt❡r ✐s ❝r♦ss✐♥❣ ✭❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✮ ✐❢ t❤❡r❡ ✐s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦ ♦❢ ✳
■❢ ✐s ❛ ♠❛①✐♠❛❧ ❜❧♦❝❦ ✐♥ ❛ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡♥ ✳
❲❤❡♥ aℓ ♦❝❝✉rs ✐♥ S(U) = S(V ) ❛♥❞ ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✳ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳
◮ ❇❧♦❝❦ ♦❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ❡①♣❧✐❝✐t✱ ✐♠♣❧✐❝✐t ♦r ❝r♦ss✐♥❣✳ ◮ ▲❡tt❡r a ✐s ❝r♦ss✐♥❣ ✭❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✮ ✐❢ t❤❡r❡ ✐s ❛ ❝r♦ss✐♥❣
❜❧♦❝❦ ♦❢ a✳
■❢ ✐s ❛ ♠❛①✐♠❛❧ ❜❧♦❝❦ ✐♥ ❛ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡♥ ✳
❲❤❡♥ aℓ ♦❝❝✉rs ✐♥ S(U) = S(V ) ❛♥❞ ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✳ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳
◮ ❇❧♦❝❦ ♦❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ❡①♣❧✐❝✐t✱ ✐♠♣❧✐❝✐t ♦r ❝r♦ss✐♥❣✳ ◮ ▲❡tt❡r a ✐s ❝r♦ss✐♥❣ ✭❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✮ ✐❢ t❤❡r❡ ✐s ❛ ❝r♦ss✐♥❣
❜❧♦❝❦ ♦❢ a✳ X baa Y b = baabbaabbb S(X) = baab S(Y ) = bb baab baa bb b = baabbaabbb
■❢ ✐s ❛ ♠❛①✐♠❛❧ ❜❧♦❝❦ ✐♥ ❛ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡♥ ✳
❲❤❡♥ aℓ ♦❝❝✉rs ✐♥ S(U) = S(V ) ❛♥❞ ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞✳ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳
◮ ❇❧♦❝❦ ♦❝❝✉rr❡♥❝❡ ❝❛♥ ❜❡ ❡①♣❧✐❝✐t✱ ✐♠♣❧✐❝✐t ♦r ❝r♦ss✐♥❣✳ ◮ ▲❡tt❡r a ✐s ❝r♦ss✐♥❣ ✭❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✮ ✐❢ t❤❡r❡ ✐s ❛ ❝r♦ss✐♥❣
❜❧♦❝❦ ♦❢ a✳ X baa Y b = baabbaabbb S(X) = baab S(Y ) = bb baab baa bb b = baabbaabbb
■❢ aℓ ✐s ❛ ♠❛①✐♠❛❧ ❜❧♦❝❦ ✐♥ ❛ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ U = V t❤❡♥ ℓ ≤ 2c|UV |✳
✶✿ ❢♦r ❛❧❧ ♠❛①✐♠❛❧ ❜❧♦❝❦s aℓ ♦❢ a ❛♥❞ ℓ > 1 ❞♦ ✷✿
❧❡t aℓ ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ♠❛①✐♠❛❧ aℓ ✐♥ U = V ❜② aℓ
❚❤❡ ❇❧♦❝❦❈♦♠♣ ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ❜❧♦❝❦s ♦❢ ✳
✶✿ ❢♦r ❛❧❧ ♠❛①✐♠❛❧ ❜❧♦❝❦s aℓ ♦❢ a ❛♥❞ ℓ > 1 ❞♦ ✷✿
❧❡t aℓ ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ♠❛①✐♠❛❧ aℓ ✐♥ U = V ❜② aℓ
❚❤❡ ❇❧♦❝❦❈♦♠♣(a) ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ❜❧♦❝❦s ♦❢ a✳
✶✿ ❢♦r ❛❧❧ ♠❛①✐♠❛❧ ❜❧♦❝❦s aℓ ♦❢ a ❛♥❞ ℓ > 1 ❞♦ ✷✿
❧❡t aℓ ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ♠❛①✐♠❛❧ aℓ ✐♥ U = V ❜② aℓ
❚❤❡ ❇❧♦❝❦❈♦♠♣(a) ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ❜❧♦❝❦s ♦❢ a✳ X baaY baaa=baabbaabbbaaa S(X) = baab S(Y ) = bb
✶✿ ❢♦r ❛❧❧ ♠❛①✐♠❛❧ ❜❧♦❝❦s aℓ ♦❢ a ❛♥❞ ℓ > 1 ❞♦ ✷✿
❧❡t aℓ ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ♠❛①✐♠❛❧ aℓ ✐♥ U = V ❜② aℓ
❚❤❡ ❇❧♦❝❦❈♦♠♣(a) ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ❜❧♦❝❦s ♦❢ a✳ X baaY baaa=baabbaabbbaaa S(X) = baab S(Y ) = bb baabbaabbbaaa=baabbaabbbaaa
✶✿ ❢♦r ❛❧❧ ♠❛①✐♠❛❧ ❜❧♦❝❦s aℓ ♦❢ a ❛♥❞ ℓ > 1 ❞♦ ✷✿
❧❡t aℓ ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ♠❛①✐♠❛❧ aℓ ✐♥ U = V ❜② aℓ
❚❤❡ ❇❧♦❝❦❈♦♠♣(a) ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ❜❧♦❝❦s ♦❢ a✳ X baaY baaa=baabbaabbbaaa S(X) = baab S(Y ) = bb baabbaabbbaaa=baabbaabbbaaa X ba2Y b a3 =ba2bba2bbb a3 S′(X) = ba2b S′(Y ) = bb
✶✿ ❢♦r ❛❧❧ ♠❛①✐♠❛❧ ❜❧♦❝❦s aℓ ♦❢ a ❛♥❞ ℓ > 1 ❞♦ ✷✿
❧❡t aℓ ∈ Σ ❜❡ ❛♥ ✉♥✉s❡❞ ❧❡tt❡r
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ❡①♣❧✐❝✐t ♠❛①✐♠❛❧ aℓ ✐♥ U = V ❜② aℓ
❚❤❡ ❇❧♦❝❦❈♦♠♣(a) ♣r♦♣❡r❧② ❝♦♠♣r❡ss❡s ♥♦♥❝r♦ss✐♥❣ ❜❧♦❝❦s ♦❢ a✳ X baaY baaa=baabbaabbbaaa S(X) = baab S(Y ) = bb baabbaabbbaaa=baabbaabbbaaa ba2bba2bbb a3 =ba2bba2bbb a3 X ba2Y b a3 =ba2bba2bbb a3 S′(X) = ba2b S′(Y ) = bb
◮ ❈r♦ss✐♥❣ a✲❝❤❛✐♥✿ s✐♠✐❧❛r t♦ ❝r♦ss✐♥❣ ab✳
♣♦♣ ✇❤♦❧❡ ✲♣r❡✜① ❛♥❞ ✲s✉✣①✿ ✿ ❝❤❛♥❣❡ ✐t t♦
✶✿ ❢♦r
❞♦
✷✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ ❜②
✸✿
❛♥❞ ❛r❡ t❤❡ ✲♣r❡✜① ❛♥❞ s✉✣① ♦❢
✹✿
✐❢ t❤❡♥
✺✿
r❡♠♦✈❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
❆❢t❡r ✉♥❝r♦ss✐♥❣ ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳
◮ ❈r♦ss✐♥❣ a✲❝❤❛✐♥✿ s✐♠✐❧❛r t♦ ❝r♦ss✐♥❣ ab✳ ◮ ♣♦♣ ✇❤♦❧❡ a✲♣r❡✜① ❛♥❞ a✲s✉✣①✿
S(X) = aℓXwarX✿ ❝❤❛♥❣❡ ✐t t♦ S(X) = w
✶✿ ❢♦r
❞♦
✷✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ ❜②
✸✿
❛♥❞ ❛r❡ t❤❡ ✲♣r❡✜① ❛♥❞ s✉✣① ♦❢
✹✿
✐❢ t❤❡♥
✺✿
r❡♠♦✈❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
❆❢t❡r ✉♥❝r♦ss✐♥❣ ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳
◮ ❈r♦ss✐♥❣ a✲❝❤❛✐♥✿ s✐♠✐❧❛r t♦ ❝r♦ss✐♥❣ ab✳ ◮ ♣♦♣ ✇❤♦❧❡ a✲♣r❡✜① ❛♥❞ a✲s✉✣①✿
S(X) = aℓXwarX✿ ❝❤❛♥❣❡ ✐t t♦ S(X) = w
✶✿ ❢♦r X ∈ X ❞♦ ✷✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ X ❜② aℓXXarX ⊲ ℓX, rX ≥ 0
✸✿
⊲ aℓX ❛♥❞ arX ❛r❡ t❤❡ a✲♣r❡✜① ❛♥❞ s✉✣① ♦❢ S(X)
✹✿
✐❢ S(X) = ǫ t❤❡♥
✺✿
r❡♠♦✈❡ X ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
❆❢t❡r ✉♥❝r♦ss✐♥❣ ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳
◮ ❈r♦ss✐♥❣ a✲❝❤❛✐♥✿ s✐♠✐❧❛r t♦ ❝r♦ss✐♥❣ ab✳ ◮ ♣♦♣ ✇❤♦❧❡ a✲♣r❡✜① ❛♥❞ a✲s✉✣①✿
S(X) = aℓXwarX✿ ❝❤❛♥❣❡ ✐t t♦ S(X) = w
✶✿ ❢♦r X ∈ X ❞♦ ✷✿
r❡♣❧❛❝❡ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ♦❢ X ❜② aℓXXarX ⊲ ℓX, rX ≥ 0
✸✿
⊲ aℓX ❛♥❞ arX ❛r❡ t❤❡ a✲♣r❡✜① ❛♥❞ s✉✣① ♦❢ S(X)
✹✿
✐❢ S(X) = ǫ t❤❡♥
✺✿
r❡♠♦✈❡ X ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥
❆❢t❡r ✉♥❝r♦ss✐♥❣ a ✐s ♥♦ ❧♦♥❣❡r ❝r♦ss✐♥❣✳
✇❤✐❧❡ U / ∈ Σ ❛♥❞ V / ∈ Σ ❞♦ ▲ ← ❧❡tt❡rs ❢r♦♠ U = V ❝❤♦♦s❡ ❛ ♣❛✐r ♦❢ ❧❡tt❡rs ♦r ❛ ❜❧♦❝❦ ❢r♦♠ ▲ ✐❢ ✐t ✐s ❝r♦ss✐♥❣ t❤❡♥ ❯♥❝r♦ss ✐t ❈♦♠♣r❡ss ✐t
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ❛❧s♦ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡ ❤❛❞✳ ❏✉st r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ❛❧s♦ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡ ❤❛❞✳ ❏✉st r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ❛❧s♦ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡ ❤❛❞✳ ❏✉st r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab X c a Y b= c aa c ab c b S(X) = caa S(Y ) = bc
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ❛❧s♦ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡ ❤❛❞✳ ❏✉st r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab c aa c ab c b= c aa c ab c b X c a Y b= c aa c ab c b S(X) = caa S(Y ) = bc
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ❛❧s♦ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡ ❤❛❞✳ ❏✉st r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba c aa c ab c b= c aa c ab c b X c a Y b= c aa c ab c b S(X) = caa S(Y ) = bc
■❢ t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ s♦❧✉t✐♦♥✱ t❤❡♥ ❛❧s♦ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡ ❤❛❞✳ ❏✉st r♦❧❧ ❜❛❝❦ t❤❡ ❝❤❛♥❣❡s✳ X baa Y b=baaabaabbab S(X) = baaa S(Y ) = bba baaabaabbab=baaabaabbab c aa c ab c b= c aa c ab c b X c a Y b= c aa c ab c b S(X) = caa S(Y ) = bc
■❢ t❤❡ ❡q✉❛t✐♦♥ ❤❛s t❤❡ s♦❧✉t✐♦♥✱ t❤❡♥ ❢♦r s♦♠❡ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♥❡✳ ▼❛❦❡ t❤❡ ❝❤♦✐❝❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s♦❧✉t✐♦♥✳
■❢ t❤❡ ❡q✉❛t✐♦♥ ❤❛s t❤❡ s♦❧✉t✐♦♥✱ t❤❡♥ ❢♦r s♦♠❡ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♥❡✳ ▼❛❦❡ t❤❡ ❝❤♦✐❝❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s♦❧✉t✐♦♥✳
■❢ t❤❡ ❡q✉❛t✐♦♥ ❤❛s t❤❡ s♦❧✉t✐♦♥✱ t❤❡♥ ❢♦r s♦♠❡ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s t❤❡ ♥❡✇ ❡q✉❛t✐♦♥ ❤❛s ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♥❡✳ ▼❛❦❡ t❤❡ ❝❤♦✐❝❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s♦❧✉t✐♦♥✳
❲❡ s❤♦✇ t❤❛t
◮ ✇❡ st❛② ✐♥ O(n2) s♣❛❝❡✳ ◮ ❆❢t❡r ❡❛❝❤ ♦♣❡r❛t✐♦♥ t❤❡ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ s❤♦rt❡♥s✳
❙♦ ✇❡ t❡r♠✐♥❛t❡ ♦♥ ♣♦s✐t✐✈❡ ✐♥st❛♥❝❡s✳
❊❛❝❤ ❝♦♠♣r❡ss✐♦♥ ❞❡❝r❡❛s❡s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥✳
❲❡ ♣❡r❢♦r♠ t❤❡ ❝♦♠♣r❡ss✐♦♥ ♦♥ t❤❡ s♦❧✉t✐♦♥ ✇♦r❞✳
❲❡ s❤♦✇ t❤❛t
◮ ✇❡ st❛② ✐♥ O(n2) s♣❛❝❡✳ ◮ ❆❢t❡r ❡❛❝❤ ♦♣❡r❛t✐♦♥ t❤❡ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ s❤♦rt❡♥s✳
❙♦ ✇❡ t❡r♠✐♥❛t❡ ♦♥ ♣♦s✐t✐✈❡ ✐♥st❛♥❝❡s✳
❊❛❝❤ ❝♦♠♣r❡ss✐♦♥ ❞❡❝r❡❛s❡s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥✳
❲❡ ♣❡r❢♦r♠ t❤❡ ❝♦♠♣r❡ss✐♦♥ ♦♥ t❤❡ s♦❧✉t✐♦♥ ✇♦r❞✳
❲❡ s❤♦✇ t❤❛t
◮ ✇❡ st❛② ✐♥ O(n2) s♣❛❝❡✳ ◮ ❆❢t❡r ❡❛❝❤ ♦♣❡r❛t✐♦♥ t❤❡ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ s❤♦rt❡♥s✳
❙♦ ✇❡ t❡r♠✐♥❛t❡ ♦♥ ♣♦s✐t✐✈❡ ✐♥st❛♥❝❡s✳
❊❛❝❤ ❝♦♠♣r❡ss✐♦♥ ❞❡❝r❡❛s❡s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧❡♥❣t❤✲♠✐♥✐♠❛❧ s♦❧✉t✐♦♥✳
❲❡ ♣❡r❢♦r♠ t❤❡ ❝♦♠♣r❡ss✐♦♥ ♦♥ t❤❡ s♦❧✉t✐♦♥ ✇♦r❞✳
❈♦♠♣r❡ss✐♦♥ ♦❢ ❛ ♥♦♥✲❝r♦ss✐♥❣ ♣❛✐r✴❜❧♦❝❦ ❞❡❝r❡❛s❡s ❡q✉❛t✐♦♥✬s s✐③❡✳
❙♦♠❡t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ ❡q✉❛t✐♦♥✳
■❢ t❤❡r❡ ✐s s♦♠❡t❤✐♥❣ ♥♦♥✲❝r♦ss✐♥❣✿ ❝♦♠♣r❡ss ✐t✳ ■❢ t❤❡r❡ ❛r❡ ♦♥❧② ❝r♦ss✐♥❣✿ ❝❤♦♦s❡ ♦♥❡ t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ❡q✉❛t✐♦♥✳
❈♦♠♣r❡ss✐♦♥ ♦❢ ❛ ♥♦♥✲❝r♦ss✐♥❣ ♣❛✐r✴❜❧♦❝❦ ❞❡❝r❡❛s❡s ❡q✉❛t✐♦♥✬s s✐③❡✳
❙♦♠❡t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ ❡q✉❛t✐♦♥✳
◮ ■❢ t❤❡r❡ ✐s s♦♠❡t❤✐♥❣ ♥♦♥✲❝r♦ss✐♥❣✿ ❝♦♠♣r❡ss ✐t✳ ◮ ■❢ t❤❡r❡ ❛r❡ ♦♥❧② ❝r♦ss✐♥❣✿ ❝❤♦♦s❡ ♦♥❡ t❤❛t ♠✐♥✐♠✐s❡s t❤❡
❡q✉❛t✐♦♥✳
❚❤❡r❡ ❛r❡ ❛t ♠♦st 2n ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s✳ ❊❛❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐❞❡ ♦❢ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ✈❛r✐❛❜❧❡✳
❯♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ❛t ♠♦st ❧❡tt❡rs t♦ t❤❡ ❡q✉❛t✐♦♥✳ ❊❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s ❧❡❢t ❛♥❞ r✐❣❤t ♦♥❡ ❧❡tt❡r ❢♦r ✲❝❤❛✐♥s✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r✇❛r❞s✳
❚❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ❝❤♦✐❝❡ t♦ ❜❡ ✳ ❚❤❡r❡ ❛r❡ ❧❡tt❡rs ❛♥❞ ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ❜❧♦❝❦s✴♣❛✐rs✳ ❙♦♠❡ ❝♦✈❡rs ❧❡tt❡rs✳ ■ts ❝♦♠♣r❡ss✐♦♥ r❡♠♦✈❡s ❧❡tt❡rs ❛♥❞ ✐♥tr♦❞✉❝❡s ❧❡tt❡rs✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛t ♠♦st
❚❤❡r❡ ❛r❡ ❛t ♠♦st 2n ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s✳ ❊❛❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐❞❡ ♦❢ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ✈❛r✐❛❜❧❡✳
❯♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ❛t ♠♦st ❧❡tt❡rs t♦ t❤❡ ❡q✉❛t✐♦♥✳ ❊❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s ❧❡❢t ❛♥❞ r✐❣❤t ♦♥❡ ❧❡tt❡r ❢♦r ✲❝❤❛✐♥s✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r✇❛r❞s✳
❚❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ❝❤♦✐❝❡ t♦ ❜❡ ✳ ❚❤❡r❡ ❛r❡ ❧❡tt❡rs ❛♥❞ ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ❜❧♦❝❦s✴♣❛✐rs✳ ❙♦♠❡ ❝♦✈❡rs ❧❡tt❡rs✳ ■ts ❝♦♠♣r❡ss✐♦♥ r❡♠♦✈❡s ❧❡tt❡rs ❛♥❞ ✐♥tr♦❞✉❝❡s ❧❡tt❡rs✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛t ♠♦st
❚❤❡r❡ ❛r❡ ❛t ♠♦st 2n ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s✳ ❊❛❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐❞❡ ♦❢ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ✈❛r✐❛❜❧❡✳
❯♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ❛t ♠♦st 2n ❧❡tt❡rs t♦ t❤❡ ❡q✉❛t✐♦♥✳ ❊❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s ❧❡❢t ❛♥❞ r✐❣❤t ♦♥❡ ❧❡tt❡r ❢♦r ✲❝❤❛✐♥s✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r✇❛r❞s✳
❚❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ❝❤♦✐❝❡ t♦ ❜❡ ✳ ❚❤❡r❡ ❛r❡ ❧❡tt❡rs ❛♥❞ ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ❜❧♦❝❦s✴♣❛✐rs✳ ❙♦♠❡ ❝♦✈❡rs ❧❡tt❡rs✳ ■ts ❝♦♠♣r❡ss✐♦♥ r❡♠♦✈❡s ❧❡tt❡rs ❛♥❞ ✐♥tr♦❞✉❝❡s ❧❡tt❡rs✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛t ♠♦st
❚❤❡r❡ ❛r❡ ❛t ♠♦st 2n ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s✳ ❊❛❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐❞❡ ♦❢ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ✈❛r✐❛❜❧❡✳
❯♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ❛t ♠♦st 2n ❧❡tt❡rs t♦ t❤❡ ❡q✉❛t✐♦♥✳ ❊❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s ❧❡❢t ❛♥❞ r✐❣❤t ♦♥❡ ❧❡tt❡r ❢♦r a✲❝❤❛✐♥s✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r✇❛r❞s✳
❚❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ❝❤♦✐❝❡ t♦ ❜❡ ✳ ❚❤❡r❡ ❛r❡ ❧❡tt❡rs ❛♥❞ ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ❜❧♦❝❦s✴♣❛✐rs✳ ❙♦♠❡ ❝♦✈❡rs ❧❡tt❡rs✳ ■ts ❝♦♠♣r❡ss✐♦♥ r❡♠♦✈❡s ❧❡tt❡rs ❛♥❞ ✐♥tr♦❞✉❝❡s ❧❡tt❡rs✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛t ♠♦st
❚❤❡r❡ ❛r❡ ❛t ♠♦st 2n ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s✳ ❊❛❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐❞❡ ♦❢ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ✈❛r✐❛❜❧❡✳
❯♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ❛t ♠♦st 2n ❧❡tt❡rs t♦ t❤❡ ❡q✉❛t✐♦♥✳ ❊❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s ❧❡❢t ❛♥❞ r✐❣❤t ♦♥❡ ❧❡tt❡r ❢♦r a✲❝❤❛✐♥s✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r✇❛r❞s✳
❚❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ❝❤♦✐❝❡ t♦ ❜❡ ≤ 8n2✳ ❚❤❡r❡ ❛r❡ ❧❡tt❡rs ❛♥❞ ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ❜❧♦❝❦s✴♣❛✐rs✳ ❙♦♠❡ ❝♦✈❡rs ❧❡tt❡rs✳ ■ts ❝♦♠♣r❡ss✐♦♥ r❡♠♦✈❡s ❧❡tt❡rs ❛♥❞ ✐♥tr♦❞✉❝❡s ❧❡tt❡rs✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛t ♠♦st
❚❤❡r❡ ❛r❡ ❛t ♠♦st 2n ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s✳ ❊❛❝❤ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ s✐❞❡ ♦❢ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ✈❛r✐❛❜❧❡✳
❯♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ❛t ♠♦st 2n ❧❡tt❡rs t♦ t❤❡ ❡q✉❛t✐♦♥✳ ❊❛❝❤ ✈❛r✐❛❜❧❡ ♣♦♣s ❧❡❢t ❛♥❞ r✐❣❤t ♦♥❡ ❧❡tt❡r ❢♦r a✲❝❤❛✐♥s✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐♠♠❡❞✐❛t❡❧② ❛❢t❡r✇❛r❞s✳
❚❤❡r❡ ✐s ❛❧✇❛②s s♦♠❡ ❝❤♦✐❝❡ t♦ ❜❡ ≤ 8n2✳ ❚❤❡r❡ ❛r❡ m ≤ 8n2 ❧❡tt❡rs ❛♥❞ k ≤ 2n ❞✐✛❡r❡♥t ❝r♦ss✐♥❣ ❜❧♦❝❦s✴♣❛✐rs✳ ❙♦♠❡ ❝♦✈❡rs ≥ m/k ❧❡tt❡rs✳ ■ts ❝♦♠♣r❡ss✐♦♥ r❡♠♦✈❡s ≥ m/2k ❧❡tt❡rs ❛♥❞ ✐♥tr♦❞✉❝❡s 2n ❧❡tt❡rs✳ ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ ❛t ♠♦st (1 − 1/2k) · m + 2n ≤ (1 − 1/4n) · 8n2 + 2n = 8n2 .
◮ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛♥ t❤❡
❝♦♠❜✐♥❛t♦r✐❝s✳
❆r❡ ✇♦r❞ ❡q✉❛t✐♦♥s ✐♥ ◆P❄ ✭❆r❡ s♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄✮ ❚♦ ✇❤✐❝❤ ♣r♦❜❧❡♠s ❝❛♥ ✇❡ ❣❡♥❡r❛❧✐s❡ t❤✐s ❛♣♣r♦❛❝❤❄
◮ ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ♠♦r❡ ✐♠♣♦rt❛♥t t❤❛♥ t❤❡
❝♦♠❜✐♥❛t♦r✐❝s✳
◮ ❆r❡ ✇♦r❞ ❡q✉❛t✐♦♥s ✐♥ ◆P❄ ✭❆r❡ s♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄✮ ◮ ❚♦ ✇❤✐❝❤ ♣r♦❜❧❡♠s ❝❛♥ ✇❡ ❣❡♥❡r❛❧✐s❡ t❤✐s ❛♣♣r♦❛❝❤❄
❋♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✿ ❝♦♥str❛✐♥ts ♦❢ t❤❡ ❢♦r♠ X ∈ R, X / ∈ R′ ✿ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❧❡tt❡rs t♦ tr❛♥s✐t✐♦♥ ♠❛tr✐❝❡s ♦❢ ◆❋❆s ❡①t❡♥❞ ❛❧s♦ t♦ ✈❛r✐❛❜❧❡s✿ ✱ r❡q✉✐r❡ ✇❤❡♥ ✐s r❡♣❧❛❝❡❞ ❜② ✿ ✇❤❡♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ✿ s✉❝❤ t❤❛t ✇❤❡♥ ✐ r❡♠♦✈❡❞✿ ❝❤❡❝❦ ✭s♦♠❡ ❡①tr❛ tr✐❝❦s ✐♥ t❤❡ ❛♥❛❧②s✐s✮
❋♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✿ ❝♦♥str❛✐♥ts ♦❢ t❤❡ ❢♦r♠ X ∈ R, X / ∈ R′ ρ✿ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❧❡tt❡rs t♦ tr❛♥s✐t✐♦♥ ♠❛tr✐❝❡s ♦❢ ◆❋❆s ❡①t❡♥❞ ❛❧s♦ t♦ ✈❛r✐❛❜❧❡s✿ ρX✱ r❡q✉✐r❡ ρ(S(X)) = ρX ✇❤❡♥ ✐s r❡♣❧❛❝❡❞ ❜② ✿ ✇❤❡♥ ✐s r❡♣❧❛❝❡❞ ✇✐t❤ ✿ s✉❝❤ t❤❛t ✇❤❡♥ ✐ r❡♠♦✈❡❞✿ ❝❤❡❝❦ ✭s♦♠❡ ❡①tr❛ tr✐❝❦s ✐♥ t❤❡ ❛♥❛❧②s✐s✮
❋♦r ❡❛❝❤ ✈❛r✐❛❜❧❡✿ ❝♦♥str❛✐♥ts ♦❢ t❤❡ ❢♦r♠ X ∈ R, X / ∈ R′ ρ✿ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❧❡tt❡rs t♦ tr❛♥s✐t✐♦♥ ♠❛tr✐❝❡s ♦❢ ◆❋❆s ❡①t❡♥❞ ❛❧s♦ t♦ ✈❛r✐❛❜❧❡s✿ ρX✱ r❡q✉✐r❡ ρ(S(X)) = ρX ✇❤❡♥ w ✐s r❡♣❧❛❝❡❞ ❜② c✿ ρ(c) ← ρ(w) ✇❤❡♥ X ✐s r❡♣❧❛❝❡❞ ✇✐t❤ wX✿ ρX ← ρ′
X s✉❝❤ t❤❛t ρX = ρ(w)ρ′ X
✇❤❡♥ X ✐ r❡♠♦✈❡❞✿ ❝❤❡❝❦ ρX = ρ(ǫ) ✭s♦♠❡ ❡①tr❛ tr✐❝❦s ✐♥ t❤❡ ❛♥❛❧②s✐s✮
❯s✐♥❣ ♣❛r❛❧❧❡❧ ❝♦♠♣r❡ss✐♦♥✿ ❧❡♥❣t❤ O(n) = ⇒ O(n log n) ❜✐ts ❯s✐♥❣ ❍✉✛♠❛♥ ❝♦❞✐♥❣✿ ❧✐♥❡❛r✲s✐③❡ ✭✐♥ t❡r♠s ♦❢ ❜✐ts✮ ❊✈❡♥ ✐❢ ✐♥♣✉t ✐s ❍✉✛♠❛♥✲❝♦❞❡❞✳
❯s✐♥❣ ♣❛r❛❧❧❡❧ ❝♦♠♣r❡ss✐♦♥✿ ❧❡♥❣t❤ O(n) = ⇒ O(n log n) ❜✐ts ❯s✐♥❣ ❍✉✛♠❛♥ ❝♦❞✐♥❣✿ ❧✐♥❡❛r✲s✐③❡ ✭✐♥ t❡r♠s ♦❢ ❜✐ts✮ ❊✈❡♥ ✐❢ ✐♥♣✉t ✐s ❍✉✛♠❛♥✲❝♦❞❡❞✳