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■♥t✉✐t✐♦♥✿ r❡❝♦♠♣r❡ss✐♦♥ ❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ❲❡ ❜✉✐❧❞ ❙▲Ps ❢♦r ❜♦t❤ str✐♥❣s ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦ ■♥t✉✐t✐♦♥ a 3 b d c d a b 2 d c b a a 3 b d c d a b 2 d c b a ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✻ ✴ ✸✼

■♥t✉✐t✐♦♥✿ r❡❝♦♠♣r❡ss✐♦♥ ❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ❲❡ ❜✉✐❧❞ ❙▲Ps ❢♦r ❜♦t❤ str✐♥❣s ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦ ■♥t✉✐t✐♦♥ a 3 b d c d a b 2 d c e a 3 b d c d a b 2 d c e ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✻ ✴ ✸✼

■♥t✉✐t✐♦♥✿ r❡❝♦♠♣r❡ss✐♦♥ ❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ❲❡ ❜✉✐❧❞ ❙▲Ps ❢♦r ❜♦t❤ str✐♥❣s ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦ ■♥t✉✐t✐♦♥ a 3 b d c d a b 2 d c e a 3 b d c d a b 2 d c e ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✻ ✴ ✸✼

■♥t✉✐t✐♦♥ a 3 b d c d a b 2 d c e a 3 b d c d a b 2 d c e ■♥t✉✐t✐♦♥✿ r❡❝♦♠♣r❡ss✐♦♥ ❚❤✐♥❦ ♦❢ ♥❡✇ ❧❡tt❡rs ❛s ♥♦♥t❡r♠✐♥❛❧s ❲❡ ❜✉✐❧❞ ❙▲Ps ❢♦r ❜♦t❤ str✐♥❣s ❊✈❡r②t❤✐♥❣ ✐s ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛②✦ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✻ ✴ ✸✼

❚❖❉❖ ❢♦r♠❛❧✐s❡ ❢♦r ❙▲Ps t✐♠❡✴s♣❛❝❡ ❛♥❛❧②s✐s ❊ss❡♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞ ■❞❡❛ ❋♦r ❜♦t❤ str✐♥❣s r❡♣❧❛❝❡ ♣❛✐rs ♦❢ ❧❡tt❡rs r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ♦❢ t❤❡ s❛♠❡ ❧❡tt❡r ❲❤❡♥ ❡✈❡r② ❧❡tt❡r ✐s ❝♦♠♣r❡ss❡❞✱ ❧❡♥❣t❤ ❤❛❧✈❡s ✐♥ ✐t❡r❛t✐♦♥✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✼ ✴ ✸✼

❊ss❡♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞ ■❞❡❛ ❋♦r ❜♦t❤ str✐♥❣s r❡♣❧❛❝❡ ♣❛✐rs ♦❢ ❧❡tt❡rs r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ♦❢ t❤❡ s❛♠❡ ❧❡tt❡r ❲❤❡♥ ❡✈❡r② ❧❡tt❡r ✐s ❝♦♠♣r❡ss❡❞✱ ❧❡♥❣t❤ ❤❛❧✈❡s ✐♥ ✐t❡r❛t✐♦♥✳ ❚❖❉❖ ❢♦r♠❛❧✐s❡ ❢♦r ❙▲Ps t✐♠❡✴s♣❛❝❡ ❛♥❛❧②s✐s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✼ ✴ ✸✼

❢♦r ❡✈❡r② ❧❡tt❡r ❛ ▲ ❞♦ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ❧❡tt❡r ❛❜ P ❞♦ ❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ ❧♦❣ ◆ ♣❤❛s❡s✳ ❋♦r♠❛❧✐s❛t✐♦♥ ■♥ ♦♥❡ ♣❤❛s❡ ▲ ← ❧✐st ♦❢ ❧❡tt❡rs✱ P ← ❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼

❢♦r ❡✈❡r② ♣❛✐r ♦❢ ❧❡tt❡r ❛❜ P ❞♦ ❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ ❧♦❣ ◆ ♣❤❛s❡s✳ ❋♦r♠❛❧✐s❛t✐♦♥ ■♥ ♦♥❡ ♣❤❛s❡ ▲ ← ❧✐st ♦❢ ❧❡tt❡rs✱ P ← ❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ❛ ℓ ✇✐t❤ ❛ ℓ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼

❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ ❧♦❣ ◆ ♣❤❛s❡s✳ ❋♦r♠❛❧✐s❛t✐♦♥ ■♥ ♦♥❡ ♣❤❛s❡ ▲ ← ❧✐st ♦❢ ❧❡tt❡rs✱ P ← ❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ❛ ℓ ✇✐t❤ ❛ ℓ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ❧❡tt❡r ❛❜ ∈ P ❞♦ r❡♣❧❛❝❡ ♣❛✐rs ❛❜ ✇✐t❤ ❝ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼

❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ ❧♦❣ ◆ ♣❤❛s❡s✳ ❋♦r♠❛❧✐s❛t✐♦♥ ■♥ ♦♥❡ ♣❤❛s❡ ▲ ← ❧✐st ♦❢ ❧❡tt❡rs✱ P ← ❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ❛ ℓ ✇✐t❤ ❛ ℓ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ❧❡tt❡r ❛❜ ∈ P ❞♦ r❡♣❧❛❝❡ ♣❛✐rs ❛❜ ✇✐t❤ ❝ ❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼

❋♦r♠❛❧✐s❛t✐♦♥ ■♥ ♦♥❡ ♣❤❛s❡ ▲ ← ❧✐st ♦❢ ❧❡tt❡rs✱ P ← ❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ❛ ℓ ✇✐t❤ ❛ ℓ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ❧❡tt❡r ❛❜ ∈ P ❞♦ r❡♣❧❛❝❡ ♣❛✐rs ❛❜ ✇✐t❤ ❝ ❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ O ( ❧♦❣ ◆ ) ♣❤❛s❡s✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼

❋♦r♠❛❧✐s❛t✐♦♥ ■♥ ♦♥❡ ♣❤❛s❡ ▲ ← ❧✐st ♦❢ ❧❡tt❡rs✱ P ← ❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ❛ ℓ ✇✐t❤ ❛ ℓ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ❧❡tt❡r ❛❜ ∈ P ❞♦ r❡♣❧❛❝❡ ♣❛✐rs ❛❜ ✇✐t❤ ❝ ❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ O ( ❧♦❣ ◆ ) ♣❤❛s❡s✳ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼

❋♦r♠❛❧✐s❛t✐♦♥ ■♥ ♦♥❡ ♣❤❛s❡ ▲ ← ❧✐st ♦❢ ❧❡tt❡rs✱ P ← ❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ r❡♣❧❛❝❡ ✭♠❛①✐♠❛❧✮ ❜❧♦❝❦s ❛ ℓ ✇✐t❤ ❛ ℓ ❢♦r ❡✈❡r② ♣❛✐r ♦❢ ❧❡tt❡r ❛❜ ∈ P ❞♦ r❡♣❧❛❝❡ ♣❛✐rs ❛❜ ✇✐t❤ ❝ ❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ O ( ❧♦❣ ◆ ) ♣❤❛s❡s✳ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐❢ ❛ � = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼

▲❡♠♠❛ ❚❤❡r❡ ❛r❡ ● ✹ ♥ ❞✐✛❡r❡♥t ♠❛①✐♠❛❧ ❧❡♥❣t❤s ♦❢ ❜❧♦❝❦s ✐♥ ●✳ Pr♦♦❢✳ ❜❧♦❝❦s t❤❛t ❛r❡ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛ss✐❣♥ t♦ ❡①♣❧✐❝✐t ❧❡tt❡rs ❜❧♦❝❦s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛t ♠♦st ✹ ♣❡r r✉❧❡ ▲❡♠♠❛ ✹ ♥ ❞✐✛❡r❡♥t ♣❛✐rs ♦❢ ❧❡tt❡rs ✐♥ ●✳ ❚❤❡r❡ ❛r❡ ● ❙▲Ps ●r❛♠♠❛r ❢♦r♠ ▼♦r❡ ❣❡♥❡r❛❧ r✉❧❡s✿ ❳ ✐ → ✉❳ ❥ ✈❳ ❦ ✇ , ❥ , ❦ < ✐ ✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✾ ✴ ✸✼

▲❡♠♠❛ ✹ ♥ ❞✐✛❡r❡♥t ♠❛①✐♠❛❧ ❧❡♥❣t❤s ♦❢ ❜❧♦❝❦s ✐♥ ●✳ ❚❤❡r❡ ❛r❡ ● Pr♦♦❢✳ ❜❧♦❝❦s t❤❛t ❛r❡ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛ss✐❣♥ t♦ ❡①♣❧✐❝✐t ❧❡tt❡rs ❜❧♦❝❦s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛t ♠♦st ✹ ♣❡r r✉❧❡ ▲❡♠♠❛ ❚❤❡r❡ ❛r❡ ● ✹ ♥ ❞✐✛❡r❡♥t ♣❛✐rs ♦❢ ❧❡tt❡rs ✐♥ ●✳ ❙▲Ps ●r❛♠♠❛r ❢♦r♠ ▼♦r❡ ❣❡♥❡r❛❧ r✉❧❡s✿ ❳ ✐ → ✉❳ ❥ ✈❳ ❦ ✇ , ❥ , ❦ < ✐ ✳ ❡①♣❧✐❝✐t ❧❡tt❡rs ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✾ ✴ ✸✼

▲❡♠♠❛ ✹ ♥ ❞✐✛❡r❡♥t ♣❛✐rs ♦❢ ❧❡tt❡rs ✐♥ ●✳ ❚❤❡r❡ ❛r❡ ● ❙▲Ps ●r❛♠♠❛r ❢♦r♠ ▼♦r❡ ❣❡♥❡r❛❧ r✉❧❡s✿ ❳ ✐ → ✉❳ ❥ ✈❳ ❦ ✇ , ❥ , ❦ < ✐ ✳ ▲❡♠♠❛ ❚❤❡r❡ ❛r❡ | ● | + ✹ ♥ ❞✐✛❡r❡♥t ♠❛①✐♠❛❧ ❧❡♥❣t❤s ♦❢ ❜❧♦❝❦s ✐♥ ●✳ Pr♦♦❢✳ ❜❧♦❝❦s t❤❛t ❛r❡ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛ss✐❣♥ t♦ ❡①♣❧✐❝✐t ❧❡tt❡rs ❜❧♦❝❦s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛t ♠♦st ✹ ♣❡r r✉❧❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✾ ✴ ✸✼

❙▲Ps ●r❛♠♠❛r ❢♦r♠ ▼♦r❡ ❣❡♥❡r❛❧ r✉❧❡s✿ ❳ ✐ → ✉❳ ❥ ✈❳ ❦ ✇ , ❥ , ❦ < ✐ ✳ ▲❡♠♠❛ ❚❤❡r❡ ❛r❡ | ● | + ✹ ♥ ❞✐✛❡r❡♥t ♠❛①✐♠❛❧ ❧❡♥❣t❤s ♦❢ ❜❧♦❝❦s ✐♥ ●✳ Pr♦♦❢✳ ❜❧♦❝❦s t❤❛t ❛r❡ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛ss✐❣♥ t♦ ❡①♣❧✐❝✐t ❧❡tt❡rs ❜❧♦❝❦s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❡①♣❧✐❝✐t ✇♦r❞s✿ ❛t ♠♦st ✹ ♣❡r r✉❧❡ ▲❡♠♠❛ ❚❤❡r❡ ❛r❡ | ● | + ✹ ♥ ❞✐✛❡r❡♥t ♣❛✐rs ♦❢ ❧❡tt❡rs ✐♥ ●✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✾ ✴ ✸✼

❝♦♠♣r❡ss✐♦♥ ♦❢ ❛❜ ✿ ❡❛s② ❝♦♠♣r❡ss✐♦♥ ♦❢ ❜❛ ✿ ♣r♦❜❧❡♠ ❉❡✜♥✐t✐♦♥ ✭❈r♦ss✐♥❣ ♣❛✐r✮ ❛❜ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r ✐✛ ❛❳ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❜❡❣✐♥s ✇✐t❤ ❜ ❳❜ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❡♥❞s ✇✐t❤ ❛ ❈♦♠♣r❡ss✐♦♥ ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ✐s ❡❛s②✿ r❡♣❧❛❝❡ t❤❡♠ ✐♥ ❙▲P✳ P❛✐r ❝♦♠♣r❡ss✐♦♥ ❳ ✶ → ❛❜❛❜❝❛❜ ✱ ❳ ✷ → ❛❜❝❜❳ ✶ ❛❜❳ ✶ ❛ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✵ ✴ ✸✼

❝♦♠♣r❡ss✐♦♥ ♦❢ ❜❛ ✿ ♣r♦❜❧❡♠ ❉❡✜♥✐t✐♦♥ ✭❈r♦ss✐♥❣ ♣❛✐r✮ ❛❜ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r ✐✛ ❛❳ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❜❡❣✐♥s ✇✐t❤ ❜ ❳❜ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❡♥❞s ✇✐t❤ ❛ ❈♦♠♣r❡ss✐♦♥ ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ✐s ❡❛s②✿ r❡♣❧❛❝❡ t❤❡♠ ✐♥ ❙▲P✳ P❛✐r ❝♦♠♣r❡ss✐♦♥ ❳ ✶ → ❛❜❛❜❝❛❜ ✱ ❳ ✷ → ❛❜❝❜❳ ✶ ❛❜❳ ✶ ❛ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❛❜ ✿ ❡❛s② ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✵ ✴ ✸✼

❉❡✜♥✐t✐♦♥ ✭❈r♦ss✐♥❣ ♣❛✐r✮ ❛❜ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r ✐✛ ❛❳ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❜❡❣✐♥s ✇✐t❤ ❜ ❳❜ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❡♥❞s ✇✐t❤ ❛ ❈♦♠♣r❡ss✐♦♥ ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ✐s ❡❛s②✿ r❡♣❧❛❝❡ t❤❡♠ ✐♥ ❙▲P✳ P❛✐r ❝♦♠♣r❡ss✐♦♥ ❳ ✶ → ❛❜❛❜❝❛❜ ✱ ❳ ✷ → ❛❜❝❜❳ ✶ ❛❜❳ ✶ ❛ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❛❜ ✿ ❡❛s② ❝♦♠♣r❡ss✐♦♥ ♦❢ ❜❛ ✿ ♣r♦❜❧❡♠ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✵ ✴ ✸✼

❈♦♠♣r❡ss✐♦♥ ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ✐s ❡❛s②✿ r❡♣❧❛❝❡ t❤❡♠ ✐♥ ❙▲P✳ P❛✐r ❝♦♠♣r❡ss✐♦♥ ❳ ✶ → ❛❜❛❜❝❛❜ ✱ ❳ ✷ → ❛❜❝❜❳ ✶ ❛❜❳ ✶ ❛ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❛❜ ✿ ❡❛s② ❝♦♠♣r❡ss✐♦♥ ♦❢ ❜❛ ✿ ♣r♦❜❧❡♠ ❉❡✜♥✐t✐♦♥ ✭❈r♦ss✐♥❣ ♣❛✐r✮ ❛❜ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r ✐✛ ❛❳ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❜❡❣✐♥s ✇✐t❤ ❜ ❳❜ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❡♥❞s ✇✐t❤ ❛ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✵ ✴ ✸✼

P❛✐r ❝♦♠♣r❡ss✐♦♥ ❳ ✶ → ❛❜❛❜❝❛❜ ✱ ❳ ✷ → ❛❜❝❜❳ ✶ ❛❜❳ ✶ ❛ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❛❜ ✿ ❡❛s② ❝♦♠♣r❡ss✐♦♥ ♦❢ ❜❛ ✿ ♣r♦❜❧❡♠ ❉❡✜♥✐t✐♦♥ ✭❈r♦ss✐♥❣ ♣❛✐r✮ ❛❜ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r ✐✛ ❛❳ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❜❡❣✐♥s ✇✐t❤ ❜ ❳❜ ❛♣♣❡❛rs ✐♥ ❛ r✉❧❡✱ ❳ ❡♥❞s ✇✐t❤ ❛ ❈♦♠♣r❡ss✐♦♥ ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ✐s ❡❛s②✿ r❡♣❧❛❝❡ t❤❡♠ ✐♥ ❙▲P✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✵ ✴ ✸✼

❯♥❝r♦ss✐♥❣ ❛❜ ✶✿ ❢♦r ✐ ✶ ♥ ❞♦ ✐❢ t❤❡ ✜rst s②♠❜♦❧ ✐♥ ❳ ✐ ✐s ❜ t❤❡♥ ✷✿ r❡♠♦✈❡ t❤✐s ❜ ✸✿ r❡♣❧❛❝❡ ❳ ✐ ✐♥ ♣r♦❞✉❝t✐♦♥s ❜② ❜❳ ✐ ✹✿ s✐♠✐❧❛r❧② ❢♦r ❛ ▲❡♠♠❛ ❆❢t❡r ✉♥❝r♦ss✐♥❣ t❤❡ ❛❜ ✐s ♥♦♥❝r♦ss✐♥❣✳ ❈r♦ss✐♥❣ ♣❛✐rs ❲❤❡♥ ❛❜ ❤❛s ❛ ❵❝r♦ss✐♥❣✬ ❛♣♣❡❛r❛♥❝❡✿ ❛❳ ✐ ♦r ❳ ✐ ❜ ❳ ✐ ❞❡✜♥❡s ❜✇ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ ✱ r❡♣❧❛❝❡ ❳ ✐ ❜② ❜❳ ✐ s②♠♠❡tr✐❝❛❧❧② ❢♦r ❡♥❞✐♥❣ ❛ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✶ ✴ ✸✼

❈r♦ss✐♥❣ ♣❛✐rs ❲❤❡♥ ❛❜ ❤❛s ❛ ❵❝r♦ss✐♥❣✬ ❛♣♣❡❛r❛♥❝❡✿ ❛❳ ✐ ♦r ❳ ✐ ❜ ❳ ✐ ❞❡✜♥❡s ❜✇ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ ✱ r❡♣❧❛❝❡ ❳ ✐ ❜② ❜❳ ✐ s②♠♠❡tr✐❝❛❧❧② ❢♦r ❡♥❞✐♥❣ ❛ ❯♥❝r♦ss✐♥❣ ❛❜ ✶✿ ❢♦r ✐ ← ✶ . . ♥ ❞♦ ✐❢ t❤❡ ✜rst s②♠❜♦❧ ✐♥ ❳ ✐ → α ✐s ❜ t❤❡♥ ✷✿ r❡♠♦✈❡ t❤✐s ❜ ✸✿ r❡♣❧❛❝❡ ❳ ✐ ✐♥ ♣r♦❞✉❝t✐♦♥s ❜② ❜❳ ✐ ✹✿ s✐♠✐❧❛r❧② ❢♦r ❛ ▲❡♠♠❛ ❆❢t❡r ✉♥❝r♦ss✐♥❣ t❤❡ ❛❜ ✐s ♥♦♥❝r♦ss✐♥❣✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✶ ✴ ✸✼

❈r♦ss✐♥❣ ❜❧♦❝❦s ❛ ❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✱ ✐❢ ❛❛ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r✳ ◆♦♥❝r♦sss✐♥❣ ❜❧♦❝❦s ❝♦♠♣r❡ss✐♦♥ ❲❤❡♥ ❛ ❤❛s ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦s✿ r❡♣❧❛❝❡ ❡①♣❧✐❝✐t ❜❧♦❝❦s ✐♥ r✉❧❡s ❲❤❛t ❛❜♦✉t ❜❧♦❝❦s❄ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳ ❈♦♠♣r❡ss ❛♥❞ ✉♥❝r♦ss s✐♠✐❧❛r❧②✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✷ ✴ ✸✼

◆♦♥❝r♦sss✐♥❣ ❜❧♦❝❦s ❝♦♠♣r❡ss✐♦♥ ❲❤❡♥ ❛ ❤❛s ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦s✿ r❡♣❧❛❝❡ ❡①♣❧✐❝✐t ❜❧♦❝❦s ✐♥ r✉❧❡s ❲❤❛t ❛❜♦✉t ❜❧♦❝❦s❄ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳ ❈♦♠♣r❡ss ❛♥❞ ✉♥❝r♦ss s✐♠✐❧❛r❧②✳ ❈r♦ss✐♥❣ ❜❧♦❝❦s ❛ ❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✱ ✐❢ ❛❛ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✷ ✴ ✸✼

❲❤❛t ❛❜♦✉t ❜❧♦❝❦s❄ ❊q✉✐✈❛❧❡♥ts ♦❢ ♣❛✐rs✳ ❈♦♠♣r❡ss ❛♥❞ ✉♥❝r♦ss s✐♠✐❧❛r❧②✳ ❈r♦ss✐♥❣ ❜❧♦❝❦s ❛ ❤❛s ❛ ❝r♦ss✐♥❣ ❜❧♦❝❦✱ ✐❢ ❛❛ ✐s ❛ ❝r♦ss✐♥❣ ♣❛✐r✳ ◆♦♥❝r♦sss✐♥❣ ❜❧♦❝❦s ❝♦♠♣r❡ss✐♦♥ ❲❤❡♥ ❛ ❤❛s ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦s✿ r❡♣❧❛❝❡ ❡①♣❧✐❝✐t ❜❧♦❝❦s ✐♥ r✉❧❡s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✷ ✴ ✸✼

❯♥❝r♦ss ❛ ✶✿ ❢♦r ✐ ✶ t♦ ♥ ❞♦ ❝❛❧❝✉❧❛t❡ ❛♥❞ r❡♠♦✈❡ ❛ ✲♣r❡✜① ❛ ✐ ❛♥❞ ❛ ✲s✉✣① ❛ r ✐ ♦❢ ❳ ✐ ✷✿ r❡♣❧❛❝❡ ❡❛❝❤ ❳ ✐ ✐♥ r✉❧❡s ❜♦❞✐❡s ❜② ❛ ✐ ❳ ✐ ❛ r ✐ ✸✿ ▲❡♠♠❛ ❆❢t❡r ✉♥❝r♦ss✐♥❣ ❛ ✐t ❤❛s ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦✳ ❯♥❝r♦ss✐♥❣ ❜❧♦❝❦s ❯♥❝r♦ss✐♥❣ P♦♣ ✇❤♦❧❡ ♣r❡✜①❡s✴s✉✣①❡s✱ ♥♦t s✐♥❣❧❡ ❧❡tt❡rs ❳ ✐ ❞❡✜♥❡s ❛ ℓ ✐ ✇❛ r ✐ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ r❡♣❧❛❝❡ ❳ ✐ ✐♥ r✉❧❡s ❜② ❛ ℓ ✐ ❳ ✐ ❛ r ✐ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✸ ✴ ✸✼

▲❡♠♠❛ ❆❢t❡r ✉♥❝r♦ss✐♥❣ ❛ ✐t ❤❛s ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦✳ ❯♥❝r♦ss✐♥❣ ❜❧♦❝❦s ❯♥❝r♦ss✐♥❣ P♦♣ ✇❤♦❧❡ ♣r❡✜①❡s✴s✉✣①❡s✱ ♥♦t s✐♥❣❧❡ ❧❡tt❡rs ❳ ✐ ❞❡✜♥❡s ❛ ℓ ✐ ✇❛ r ✐ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ r❡♣❧❛❝❡ ❳ ✐ ✐♥ r✉❧❡s ❜② ❛ ℓ ✐ ❳ ✐ ❛ r ✐ ❯♥❝r♦ss ❛ ✶✿ ❢♦r ✐ ← ✶ t♦ ♥ ❞♦ ❝❛❧❝✉❧❛t❡ ❛♥❞ r❡♠♦✈❡ ❛ ✲♣r❡✜① ❛ ℓ ✐ ❛♥❞ ❛ ✲s✉✣① ❛ r ✐ ♦❢ ❳ ✐ ✷✿ r❡♣❧❛❝❡ ❡❛❝❤ ❳ ✐ ✐♥ r✉❧❡s ❜♦❞✐❡s ❜② ❛ ℓ ✐ ❳ ✐ ❛ r ✐ ✸✿ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✸ ✴ ✸✼

❯♥❝r♦ss✐♥❣ ❜❧♦❝❦s ❯♥❝r♦ss✐♥❣ P♦♣ ✇❤♦❧❡ ♣r❡✜①❡s✴s✉✣①❡s✱ ♥♦t s✐♥❣❧❡ ❧❡tt❡rs ❳ ✐ ❞❡✜♥❡s ❛ ℓ ✐ ✇❛ r ✐ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ r❡♣❧❛❝❡ ❳ ✐ ✐♥ r✉❧❡s ❜② ❛ ℓ ✐ ❳ ✐ ❛ r ✐ ❯♥❝r♦ss ❛ ✶✿ ❢♦r ✐ ← ✶ t♦ ♥ ❞♦ ❝❛❧❝✉❧❛t❡ ❛♥❞ r❡♠♦✈❡ ❛ ✲♣r❡✜① ❛ ℓ ✐ ❛♥❞ ❛ ✲s✉✣① ❛ r ✐ ♦❢ ❳ ✐ ✷✿ r❡♣❧❛❝❡ ❡❛❝❤ ❳ ✐ ✐♥ r✉❧❡s ❜♦❞✐❡s ❜② ❛ ℓ ✐ ❳ ✐ ❛ r ✐ ✸✿ ▲❡♠♠❛ ❆❢t❡r ✉♥❝r♦ss✐♥❣ ❛ ✐t ❤❛s ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✸ ✴ ✸✼

❢♦r ❡✈❡r② ❧❡tt❡r ❛ ▲ ❞♦ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ▲ ❞♦ ✉♥❝r♦ss ❛ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ P ♥♦♥❝r♦ss✐♥❣ ♣❛✐rs P ❝r♦ss✐♥❣ ♣❛✐rs ❢♦r ❛❜ P ❞♦ ❝♦♠♣r❡ss ❛❜ ❢♦r ❛❜ P ❞♦ ✉♥❝r♦ss ❛❜ ❝♦♠♣r❡ss ❛❜ ❆❧❣♦r✐t❤♠ ✇❤✐❧❡ ♥♦♥tr✐✈✐❛❧ ❞♦ ▲ ← ❧❡tt❡rs ✇✐t❤ ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦s ▲ ′ ← ❧❡tt❡rs ✇✐t❤ ❝r♦ss✐♥❣ ❜❧♦❝❦s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼

❢♦r ❡✈❡r② ❧❡tt❡r ❛ ▲ ❞♦ ✉♥❝r♦ss ❛ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ P ♥♦♥❝r♦ss✐♥❣ ♣❛✐rs P ❝r♦ss✐♥❣ ♣❛✐rs ❢♦r ❛❜ P ❞♦ ❝♦♠♣r❡ss ❛❜ ❢♦r ❛❜ P ❞♦ ✉♥❝r♦ss ❛❜ ❝♦♠♣r❡ss ❛❜ ❆❧❣♦r✐t❤♠ ✇❤✐❧❡ ♥♦♥tr✐✈✐❛❧ ❞♦ ▲ ← ❧❡tt❡rs ✇✐t❤ ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦s ▲ ′ ← ❧❡tt❡rs ✇✐t❤ ❝r♦ss✐♥❣ ❜❧♦❝❦s ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼

P ♥♦♥❝r♦ss✐♥❣ ♣❛✐rs P ❝r♦ss✐♥❣ ♣❛✐rs ❢♦r ❛❜ P ❞♦ ❝♦♠♣r❡ss ❛❜ ❢♦r ❛❜ P ❞♦ ✉♥❝r♦ss ❛❜ ❝♦♠♣r❡ss ❛❜ ❆❧❣♦r✐t❤♠ ✇❤✐❧❡ ♥♦♥tr✐✈✐❛❧ ❞♦ ▲ ← ❧❡tt❡rs ✇✐t❤ ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦s ▲ ′ ← ❧❡tt❡rs ✇✐t❤ ❝r♦ss✐♥❣ ❜❧♦❝❦s ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ′ ❞♦ ✉♥❝r♦ss ❛ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼

❆❧❣♦r✐t❤♠ ✇❤✐❧❡ ♥♦♥tr✐✈✐❛❧ ❞♦ ▲ ← ❧❡tt❡rs ✇✐t❤ ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦s ▲ ′ ← ❧❡tt❡rs ✇✐t❤ ❝r♦ss✐♥❣ ❜❧♦❝❦s ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ❞♦ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ ❢♦r ❡✈❡r② ❧❡tt❡r ❛ ∈ ▲ ′ ❞♦ ✉♥❝r♦ss ❛ ❝♦♠♣r❡ss ❜❧♦❝❦s ♦❢ ❛ P ← ♥♦♥❝r♦ss✐♥❣ ♣❛✐rs P ′ ← ❝r♦ss✐♥❣ ♣❛✐rs ❢♦r ❛❜ ∈ P ❞♦ ❝♦♠♣r❡ss ❛❜ ❢♦r ❛❜ ∈ P ′ ❞♦ ✉♥❝r♦ss ❛❜ ❝♦♠♣r❡ss ❛❜ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼

✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ✶ ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ♥ ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ✸ ❝♥ ✷ ● ✹ ● ♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳ ♥ ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ✸ ❝♥ ✷ ● ✹ ● ♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳ ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ✸ ❝♥ ✷ ● ✹ ● ♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳ ✸ ❝♥ ✷ ● ✹ ● ♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

✐❢ ❛ ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳ ✸ ❝♥ ✷ ● ✹ ● ♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ◮ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳ ✸ ❝♥ ✷ ● ✹ ● ♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ◮ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ◮ ✐❢ ❛ � = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

✸ ❝♥ ✷ ● ✹ ● ♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ◮ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ◮ ✐❢ ❛ � = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ O ( ❧♦❣ ◆ ) ♣❤❛s❡s✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

♥ ✷ ● ✐s ♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ◮ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ◮ ✐❢ ❛ � = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ O ( ❧♦❣ ◆ ) ♣❤❛s❡s✳ | ● ′ | ≤ ✸ ✹ | ● | + ❝♥ ✷ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

♥ ✷ ❧♦❣ ◆ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ◮ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ◮ ✐❢ ❛ � = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ O ( ❧♦❣ ◆ ) ♣❤❛s❡s✳ | ● ′ | ≤ ✸ ✹ | ● | + ❝♥ ✷ ● ✐s O ( ♥ ✷ ) ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

●r❛♠♠❛r s✐③❡ ●r❛♠♠❛r s✐③❡ ■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O ( ♥ ✷ ) ◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O ( ✶ ) ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ◮ O ( ♥ ) ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ● s❤♦rt❡♥s ❜② ❛ ❝♦♥st❛♥t ❢❛❝t♦r ✐♥ ❛ ♣❤❛s❡✿ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ◮ ✐❢ ❛ = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ◮ ✐❢ ❛ � = ❜ ✿ ✐t ✐s ❝♦♠♣r❡ss❡❞ ✉♥❧❡ss ❛ ♦r ❜ ✇❛s ❝♦♠♣r❡ss❡❞ ❛❧r❡❛❞② ❙♦ O ( ❧♦❣ ◆ ) ♣❤❛s❡s✳ | ● ′ | ≤ ✸ ✹ | ● | + ❝♥ ✷ ● ✐s O ( ♥ ✷ ) ❘✉♥♥✐♥❣ t✐♠❡ ✐s O ( ♥ ✷ ❧♦❣ ◆ ) ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼

■♠♣r♦✈❡♠❡♥ts ❛♥❞ ♠♦❞✐✜❝❛t✐♦♥s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✻ ✴ ✸✼

❈❛♥ ❜❡ ❞♦♥❡ st❡♣✲❜②✲st❡♣ ❜② ♣❛✐r✐♥❣ ❛❛ ❛❧♦♥❡ ♠❛♥❛❣❡❛❜❧❡ ✭♥❡✇ ♥♦♥t❡r♠✐♥❛❧s✮ ❛❧❣♦r✐t❤♠ ♠♦r❡ ❞✐✣❝✉❧t✱ ❛♥❛❧②s✐s s✐♠✐❧❛r ❇❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛r❛❧❧❡❧ ❛♥❞ ♦♥❝❡ ♦♥❧②✳ ❳ ✐ ❞❡✜♥❡s ❛ ℓ ✐ ✇❜ r ✐ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ r❡♣❧❛❝❡ ❳ ✐ ✐♥ r✉❧❡s ❜② ❛ ℓ ✐ ❳ ✐ ❜ r ✐ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✼ ✴ ✸✼

♠❛♥❛❣❡❛❜❧❡ ✭♥❡✇ ♥♦♥t❡r♠✐♥❛❧s✮ ❛❧❣♦r✐t❤♠ ♠♦r❡ ❞✐✣❝✉❧t✱ ❛♥❛❧②s✐s s✐♠✐❧❛r ❇❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛r❛❧❧❡❧ ❛♥❞ ♦♥❝❡ ♦♥❧②✳ ❳ ✐ ❞❡✜♥❡s ❛ ℓ ✐ ✇❜ r ✐ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ r❡♣❧❛❝❡ ❳ ✐ ✐♥ r✉❧❡s ❜② ❛ ℓ ✐ ❳ ✐ ❜ r ✐ ❈❛♥ ❜❡ ❞♦♥❡ st❡♣✲❜②✲st❡♣ ❜② ♣❛✐r✐♥❣ ❛❛ ❛❧♦♥❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✼ ✴ ✸✼

❇❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ ❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛r❛❧❧❡❧ ❛♥❞ ♦♥❝❡ ♦♥❧②✳ ❳ ✐ ❞❡✜♥❡s ❛ ℓ ✐ ✇❜ r ✐ ✿ ❝❤❛♥❣❡ ✐t t♦ ✇ r❡♣❧❛❝❡ ❳ ✐ ✐♥ r✉❧❡s ❜② ❛ ℓ ✐ ❳ ✐ ❜ r ✐ ❈❛♥ ❜❡ ❞♦♥❡ st❡♣✲❜②✲st❡♣ ❜② ♣❛✐r✐♥❣ ❛❛ ❛❧♦♥❡ ♠❛♥❛❣❡❛❜❧❡ ✭♥❡✇ ♥♦♥t❡r♠✐♥❛❧s✮ ❛❧❣♦r✐t❤♠ ♠♦r❡ ❞✐✣❝✉❧t✱ ❛♥❛❧②s✐s s✐♠✐❧❛r ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✼ ✴ ✸✼

❡❛s② t♦ ❞❡r❛♥❞♦♠✐s❡ ❲❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❖♥❡ ♣❛rt✐t✐♦♥ ✇✐t❤ ♠❛♥② ♦❝❝✉rr❡♥❝❡s ♥ ❧♦❣ ◆ t✐♠❡ ❧♦❣ ♥ ♣❛rt✐t✐♦♥s ♦❢ ❛❧❧ ❧❡tt❡rs ♥ ❧♦❣ ◆ ❧♦❣ ♥ t✐♠❡ P❛✐rs P❛r❛❧❧❡❧ ❝♦♠♣r❡ss✐♦♥ P❛rt✐t✐♦♥ Σ ✐♥t♦ Σ ✶ ❛♥❞ Σ ✷ ✭r❛♥❞♦♠ ✇✐❧❧ ❞♦✮✳ P❛✐rs ❛❜ ✿ ❛ ∈ Σ ✶ ✱ ❜ ∈ Σ ✷ ✿ ◮ ❝❛♥♥♦t ♦✈❡r❧❛♣ ◮ ❝♦♠♣r❡ss ✐♥ ♣❛r❛❧❧❡❧ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✽ ✴ ✸✼

❲❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❖♥❡ ♣❛rt✐t✐♦♥ ✇✐t❤ ♠❛♥② ♦❝❝✉rr❡♥❝❡s ♥ ❧♦❣ ◆ t✐♠❡ ❧♦❣ ♥ ♣❛rt✐t✐♦♥s ♦❢ ❛❧❧ ❧❡tt❡rs ♥ ❧♦❣ ◆ ❧♦❣ ♥ t✐♠❡ P❛✐rs P❛r❛❧❧❡❧ ❝♦♠♣r❡ss✐♦♥ P❛rt✐t✐♦♥ Σ ✐♥t♦ Σ ✶ ❛♥❞ Σ ✷ ✭r❛♥❞♦♠ ✇✐❧❧ ❞♦✮✳ P❛✐rs ❛❜ ✿ ❛ ∈ Σ ✶ ✱ ❜ ∈ Σ ✷ ✿ ◮ ❝❛♥♥♦t ♦✈❡r❧❛♣ ◮ ❝♦♠♣r❡ss ✐♥ ♣❛r❛❧❧❡❧ ❡❛s② t♦ ❞❡r❛♥❞♦♠✐s❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✽ ✴ ✸✼

P❛✐rs P❛r❛❧❧❡❧ ❝♦♠♣r❡ss✐♦♥ P❛rt✐t✐♦♥ Σ ✐♥t♦ Σ ✶ ❛♥❞ Σ ✷ ✭r❛♥❞♦♠ ✇✐❧❧ ❞♦✮✳ P❛✐rs ❛❜ ✿ ❛ ∈ Σ ✶ ✱ ❜ ∈ Σ ✷ ✿ ◮ ❝❛♥♥♦t ♦✈❡r❧❛♣ ◮ ❝♦♠♣r❡ss ✐♥ ♣❛r❛❧❧❡❧ ❡❛s② t♦ ❞❡r❛♥❞♦♠✐s❡ ❲❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ❖♥❡ ♣❛rt✐t✐♦♥ ✇✐t❤ ♠❛♥② ♦❝❝✉rr❡♥❝❡s O ( ♥ ❧♦❣ ◆ ) t✐♠❡ O ( ❧♦❣ ♥ ) ♣❛rt✐t✐♦♥s ♦❢ ❛❧❧ ❧❡tt❡rs O ( ♥ ❧♦❣ ◆ ❧♦❣ ♥ ) t✐♠❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✽ ✴ ✸✼

❙✳ ❆❧str✉♣✱ ●✳ ❇r♦❞❛❧ ❛♥❞ ❚❤✳ ❘❛✉❤❡ P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✐♥ ❉②♥❛♠✐❝ ❚❡①ts ✱ ✬✵✵ ♥♦t ♣❛✐rs ❜✉t ❧♦♥❣❡r ❢r❛❣♠❡♥ts ♦❢ ❧❡♥❣t❤ ✶ ♣❛r❛❧❧❡❧ ❢r❛❣♠❡♥ts ❞❡♣❡♥❞ ♦♥❧② ♦♥ ❧♦❝❛❧ ♥❡✐❣❤❜♦✉r❤♦♦❞ ✭ ❧♦❣ ♥ ✮ ✇ s ✐♥ ①✇②✇③ ❝♦♠♣r❡ss❡❞ t❤❡ s❛♠❡✱ ❡①❝❡♣t ❧❡❛❞✐♥❣✴❡♥❞✐♥❣ ❧♦❣ ♥ ♠✉❝❤ ♦❢ ❧♦❝❛❧ str✉❝t✉r❡ ✐s ♣r❡s❡r✈❡❞ s❧✐❣❤t❧② s❧♦✇❡r ✭❧♦❣ ♥ ❧♦❣ ♥ ❢❛❝t♦r✮ ▼♦r❡ r♦❜✉st ♣❛✐r✐♥❣ ❑✳ ▼❡❤❧❤♦r♥✱ ❘✳ ❙✉♥❞❛r ❛♥❞ ❈❤✳ ❯❤r✐❣✱ ▼❛✐♥t❛✐♥✐♥❣ ❉②♥❛♠✐❝ ❙❡q✉❡♥❝❡s ✉♥❞❡r ❊q✉❛❧✐t② ❚❡sts ✐♥ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❚✐♠❡ ✱ ❵✾✼ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✾ ✴ ✸✼

♥♦t ♣❛✐rs ❜✉t ❧♦♥❣❡r ❢r❛❣♠❡♥ts ♦❢ ❧❡♥❣t❤ ✶ ♣❛r❛❧❧❡❧ ❢r❛❣♠❡♥ts ❞❡♣❡♥❞ ♦♥❧② ♦♥ ❧♦❝❛❧ ♥❡✐❣❤❜♦✉r❤♦♦❞ ✭ ❧♦❣ ♥ ✮ ✇ s ✐♥ ①✇②✇③ ❝♦♠♣r❡ss❡❞ t❤❡ s❛♠❡✱ ❡①❝❡♣t ❧❡❛❞✐♥❣✴❡♥❞✐♥❣ ❧♦❣ ♥ ♠✉❝❤ ♦❢ ❧♦❝❛❧ str✉❝t✉r❡ ✐s ♣r❡s❡r✈❡❞ s❧✐❣❤t❧② s❧♦✇❡r ✭❧♦❣ ♥ ❧♦❣ ♥ ❢❛❝t♦r✮ ▼♦r❡ r♦❜✉st ♣❛✐r✐♥❣ ❑✳ ▼❡❤❧❤♦r♥✱ ❘✳ ❙✉♥❞❛r ❛♥❞ ❈❤✳ ❯❤r✐❣✱ ▼❛✐♥t❛✐♥✐♥❣ ❉②♥❛♠✐❝ ❙❡q✉❡♥❝❡s ✉♥❞❡r ❊q✉❛❧✐t② ❚❡sts ✐♥ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❚✐♠❡ ✱ ❵✾✼ ❙✳ ❆❧str✉♣✱ ●✳ ❇r♦❞❛❧ ❛♥❞ ❚❤✳ ❘❛✉❤❡ P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✐♥ ❉②♥❛♠✐❝ ❚❡①ts ✱ ✬✵✵ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✾ ✴ ✸✼

▼♦r❡ r♦❜✉st ♣❛✐r✐♥❣ ❑✳ ▼❡❤❧❤♦r♥✱ ❘✳ ❙✉♥❞❛r ❛♥❞ ❈❤✳ ❯❤r✐❣✱ ▼❛✐♥t❛✐♥✐♥❣ ❉②♥❛♠✐❝ ❙❡q✉❡♥❝❡s ✉♥❞❡r ❊q✉❛❧✐t② ❚❡sts ✐♥ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❚✐♠❡ ✱ ❵✾✼ ❙✳ ❆❧str✉♣✱ ●✳ ❇r♦❞❛❧ ❛♥❞ ❚❤✳ ❘❛✉❤❡ P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✐♥ ❉②♥❛♠✐❝ ❚❡①ts ✱ ✬✵✵ ♥♦t ♣❛✐rs ❜✉t ❧♦♥❣❡r ❢r❛❣♠❡♥ts ♦❢ ❧❡♥❣t❤ O ( ✶ ) ♣❛r❛❧❧❡❧ ❢r❛❣♠❡♥ts ❞❡♣❡♥❞ ♦♥❧② ♦♥ ❧♦❝❛❧ ♥❡✐❣❤❜♦✉r❤♦♦❞ ✭ O ( ❧♦❣ ∗ ♥ ) ✮ ✇ s ✐♥ ①✇②✇③ ❝♦♠♣r❡ss❡❞ t❤❡ s❛♠❡✱ ❡①❝❡♣t ❧❡❛❞✐♥❣✴❡♥❞✐♥❣ O ( ❧♦❣ ∗ ♥ ) ♠✉❝❤ ♦❢ ❧♦❝❛❧ str✉❝t✉r❡ ✐s ♣r❡s❡r✈❡❞ s❧✐❣❤t❧② s❧♦✇❡r ✭❧♦❣ ♥ ❧♦❣ ∗ ♥ ❢❛❝t♦r✮ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✾ ✴ ✸✼

❛♥❛❧②s✐s ✐s s✐♠✐❧❛r✿ str✐♥❣s ❜❡t✇❡❡♥ ❨ s ❛r❡ t❤❡ s❛♠❡✿ ✶ ❨✈ ❳ ✉ ❨✇ t❤❡r❡ ❛r❡ ♦♥❧② ✸ str✐♥❣s t♦ ❝♦♥s✐❞❡r ✐♥ ❛ r✉❧❡ s♣❛❝❡ ❛♥❞ t✐♠❡ t❤❡ s❛♠❡ ❊①t❡♥❞ ❙▲Ps ❜② ❡①♣♦♥❡♥❡t✐❛t✐♦♥ ❛❧❧♦✇ ❡①♣♦♥❡♥t✐❛t✐♦♥✿ r✉❧❡s ❳ → ❨ ℓ ✭s✐③❡ O ( ✶ ) ✮ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✵ ✴ ✸✼

s♣❛❝❡ ❛♥❞ t✐♠❡ t❤❡ s❛♠❡ ❊①t❡♥❞ ❙▲Ps ❜② ❡①♣♦♥❡♥❡t✐❛t✐♦♥ ❛❧❧♦✇ ❡①♣♦♥❡♥t✐❛t✐♦♥✿ r✉❧❡s ❳ → ❨ ℓ ✭s✐③❡ O ( ✶ ) ✮ ❛♥❛❧②s✐s ✐s s✐♠✐❧❛r✿ ◮ str✐♥❣s ❜❡t✇❡❡♥ ❨ s ❛r❡ t❤❡ s❛♠❡✿ ❳ → ✉ ( ❨✇ ) ℓ − ✶ ❨✈ ◮ t❤❡r❡ ❛r❡ ♦♥❧② ✸ str✐♥❣s t♦ ❝♦♥s✐❞❡r ✐♥ ❛ r✉❧❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✵ ✴ ✸✼

❊①t❡♥❞ ❙▲Ps ❜② ❡①♣♦♥❡♥❡t✐❛t✐♦♥ ❛❧❧♦✇ ❡①♣♦♥❡♥t✐❛t✐♦♥✿ r✉❧❡s ❳ → ❨ ℓ ✭s✐③❡ O ( ✶ ) ✮ ❛♥❛❧②s✐s ✐s s✐♠✐❧❛r✿ ◮ str✐♥❣s ❜❡t✇❡❡♥ ❨ s ❛r❡ t❤❡ s❛♠❡✿ ❳ → ✉ ( ❨✇ ) ℓ − ✶ ❨✈ ◮ t❤❡r❡ ❛r❡ ♦♥❧② ✸ str✐♥❣s t♦ ❝♦♥s✐❞❡r ✐♥ ❛ r✉❧❡ s♣❛❝❡ ❛♥❞ t✐♠❡ t❤❡ s❛♠❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✵ ✴ ✸✼

❡❛s② t♦ ❣❡♥❡r❛❧✐s❡ ❝♦♥str❛✐♥ts ♣❛rt✐❛❧ ❝♦♠♠✉t❛t✐♦♥ ✐♥✈❡rs✐♦♥ ❛❧❧ s♦❧✉t✐♦♥s ❱❡r② ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✳ ❣❡♥❡r❛❧✐s❡s ❙▲Ps ❡✈❡r②t❤✐♥❣ ✇♦r❦s t❤❡ s❛♠❡ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❣✉❡ss❡s✮ P❙P❆❈❊ ❛❧❣♦r✐t❤♠ ✭✇❛s ❦♥♦✇♥✮ ▼♦r❡ ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✿ ✇♦r❞ ❡q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥s ❯ ✐ = ❱ ✐ ✱ ✇❤❡r❡ ❯ ✐ , ❱ ✐ ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛♥ ❛ss✐❣♥♠❡♥t ❙ : X �→ Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥s❄ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼

❡❛s② t♦ ❣❡♥❡r❛❧✐s❡ ❝♦♥str❛✐♥ts ♣❛rt✐❛❧ ❝♦♠♠✉t❛t✐♦♥ ✐♥✈❡rs✐♦♥ ❛❧❧ s♦❧✉t✐♦♥s ❣❡♥❡r❛❧✐s❡s ❙▲Ps ❡✈❡r②t❤✐♥❣ ✇♦r❦s t❤❡ s❛♠❡ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❣✉❡ss❡s✮ P❙P❆❈❊ ❛❧❣♦r✐t❤♠ ✭✇❛s ❦♥♦✇♥✮ ▼♦r❡ ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✿ ✇♦r❞ ❡q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥s ❯ ✐ = ❱ ✐ ✱ ✇❤❡r❡ ❯ ✐ , ❱ ✐ ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛♥ ❛ss✐❣♥♠❡♥t ❙ : X �→ Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥s❄ ❱❡r② ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼

❡❛s② t♦ ❣❡♥❡r❛❧✐s❡ ❝♦♥str❛✐♥ts ♣❛rt✐❛❧ ❝♦♠♠✉t❛t✐♦♥ ✐♥✈❡rs✐♦♥ ❛❧❧ s♦❧✉t✐♦♥s ▼♦r❡ ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✿ ✇♦r❞ ❡q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥s ❯ ✐ = ❱ ✐ ✱ ✇❤❡r❡ ❯ ✐ , ❱ ✐ ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛♥ ❛ss✐❣♥♠❡♥t ❙ : X �→ Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥s❄ ❱❡r② ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✳ ❣❡♥❡r❛❧✐s❡s ❙▲Ps ❡✈❡r②t❤✐♥❣ ✇♦r❦s t❤❡ s❛♠❡ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❣✉❡ss❡s✮ P❙P❆❈❊ ❛❧❣♦r✐t❤♠ ✭✇❛s ❦♥♦✇♥✮ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼

▼♦r❡ ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✿ ✇♦r❞ ❡q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥s ❯ ✐ = ❱ ✐ ✱ ✇❤❡r❡ ❯ ✐ , ❱ ✐ ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛♥ ❛ss✐❣♥♠❡♥t ❙ : X �→ Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥s❄ ❱❡r② ✐♠♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥✳ ❣❡♥❡r❛❧✐s❡s ❙▲Ps ❡✈❡r②t❤✐♥❣ ✇♦r❦s t❤❡ s❛♠❡ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❣✉❡ss❡s✮ P❙P❆❈❊ ❛❧❣♦r✐t❤♠ ✭✇❛s ❦♥♦✇♥✮ ❡❛s② t♦ ❣❡♥❡r❛❧✐s❡ ◮ ❝♦♥str❛✐♥ts ◮ ♣❛rt✐❛❧ ❝♦♠♠✉t❛t✐♦♥ ◮ ✐♥✈❡rs✐♦♥ ◮ ❛❧❧ s♦❧✉t✐♦♥s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼

✉s❡ t❤❡ s❛♠❡ ❛❧❣♦r✐t❤♠ ✭♦♥ t❤❡ ❡①♣❧✐❝✐t str✐♥❣✮ ✉s❡ t❤❡ ♠❡t❤♦❞ ♦♥❧② ❢♦r ❛♥❛❧②s✐s✿ ❣r❛♠♠❛r ✐s ✐♠♣❧✐❝✐t t❤❡ ❝♦♠♣r❡ss✐♦♥s ❛r❡ ♦❜❧✐✈✐♦✉s ♦❢ t❤❡ ❣r❛♠♠❛r ❧♦❣ ♥ ❣ ❛♣♣r♦①✐♠❛t✐♦♥ ❑✳ ▼❡❤❧❤♦r♥✱ ❘✳ ❙✉♥❞❛r ❛♥❞ ❈❤✳ ❯❤r✐❣ ▼❛✐♥t❛✐♥✐♥❣ ❉②♥❛♠✐❝ ❙❡q✉❡♥❝❡s ✉♥❞❡r ❊q✉❛❧✐t② ❚❡sts ✐♥ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❚✐♠❡ ✱ ❵✾✼ ❧♦❣ ♥ ❣ ❧♦❣ ♥ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❙▲P ♦♥❧② ✐♥ ❛♥❛❧②s✐s Pr♦❜❧❡♠ ❙♠❛❧❧❡st ❙▲P ❢♦r ❛ str✐♥❣✳ ✭◆P✲❤❛r❞✮ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✷ ✴ ✸✼

❑✳ ▼❡❤❧❤♦r♥✱ ❘✳ ❙✉♥❞❛r ❛♥❞ ❈❤✳ ❯❤r✐❣ ▼❛✐♥t❛✐♥✐♥❣ ❉②♥❛♠✐❝ ❙❡q✉❡♥❝❡s ✉♥❞❡r ❊q✉❛❧✐t② ❚❡sts ✐♥ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❚✐♠❡ ✱ ❵✾✼ ❧♦❣ ♥ ❣ ❧♦❣ ♥ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❙▲P ♦♥❧② ✐♥ ❛♥❛❧②s✐s Pr♦❜❧❡♠ ❙♠❛❧❧❡st ❙▲P ❢♦r ❛ str✐♥❣✳ ✭◆P✲❤❛r❞✮ ✉s❡ t❤❡ s❛♠❡ ❛❧❣♦r✐t❤♠ ✭♦♥ t❤❡ ❡①♣❧✐❝✐t str✐♥❣✮ ✉s❡ t❤❡ ♠❡t❤♦❞ ♦♥❧② ❢♦r ❛♥❛❧②s✐s✿ ❣r❛♠♠❛r ✐s ✐♠♣❧✐❝✐t t❤❡ ❝♦♠♣r❡ss✐♦♥s ❛r❡ ♦❜❧✐✈✐♦✉s ♦❢ t❤❡ ❣r❛♠♠❛r O ( ❧♦❣ ( ♥ / ❣ )) ❛♣♣r♦①✐♠❛t✐♦♥ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✷ ✴ ✸✼

❙▲P ♦♥❧② ✐♥ ❛♥❛❧②s✐s Pr♦❜❧❡♠ ❙♠❛❧❧❡st ❙▲P ❢♦r ❛ str✐♥❣✳ ✭◆P✲❤❛r❞✮ ✉s❡ t❤❡ s❛♠❡ ❛❧❣♦r✐t❤♠ ✭♦♥ t❤❡ ❡①♣❧✐❝✐t str✐♥❣✮ ✉s❡ t❤❡ ♠❡t❤♦❞ ♦♥❧② ❢♦r ❛♥❛❧②s✐s✿ ❣r❛♠♠❛r ✐s ✐♠♣❧✐❝✐t t❤❡ ❝♦♠♣r❡ss✐♦♥s ❛r❡ ♦❜❧✐✈✐♦✉s ♦❢ t❤❡ ❣r❛♠♠❛r O ( ❧♦❣ ( ♥ / ❣ )) ❛♣♣r♦①✐♠❛t✐♦♥ ❑✳ ▼❡❤❧❤♦r♥✱ ❘✳ ❙✉♥❞❛r ❛♥❞ ❈❤✳ ❯❤r✐❣ ▼❛✐♥t❛✐♥✐♥❣ ❉②♥❛♠✐❝ ❙❡q✉❡♥❝❡s ✉♥❞❡r ❊q✉❛❧✐t② ❚❡sts ✐♥ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❚✐♠❡ ✱ ❵✾✼ O ( ❧♦❣ ( ♥ / ❣ ) ❧♦❣ ∗ ♥ ) ❛♣♣r♦①✐♠❛t✐♦♥✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✷ ✴ ✸✼

Pr♦❜❧❡♠✿ ❲♦r❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♦♥❡ ✈❛r✐❛❜❧❡ ❆ ✵ ❳❆ ✶ ❆ ❦ ✶ ❳❆ ❦ ❳❇ ✶ ❇ ✶ ❳❇ ✱ ♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠s✿ ♥ ❧♦❣ ♥ r❡❝♦♠♣r❡ss✐♦♥ ❜❛s❡❞✿ ♥ ❧♦❣ ♥ ✭s✐♠♣❧❡ ❛♥❛❧②s✐s✮ r❡❝♦♠♣r❡ss ❛♥❞ t❡st ❧♦st s♦❧✉t✐♦♥s✱ ❜② s✉❜st✐t✉t✐♦♥ s♣❡❡❞ ✉♣ t❡st✐♥❣✿ ♣❛rts ♦❢ ❡q✉❛t✐♦♥ ❛r❡ ♣❡r✐♦❞✐❝ ✇❤❛t ✐s ❧❡❢t✿ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✉✈ ❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮ ❈♦♠❜✐♥✐♥❣ ✇✐t❤ str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼

♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠s✿ ♥ ❧♦❣ ♥ r❡❝♦♠♣r❡ss✐♦♥ ❜❛s❡❞✿ ♥ ❧♦❣ ♥ ✭s✐♠♣❧❡ ❛♥❛❧②s✐s✮ r❡❝♦♠♣r❡ss ❛♥❞ t❡st ❧♦st s♦❧✉t✐♦♥s✱ ❜② s✉❜st✐t✉t✐♦♥ s♣❡❡❞ ✉♣ t❡st✐♥❣✿ ♣❛rts ♦❢ ❡q✉❛t✐♦♥ ❛r❡ ♣❡r✐♦❞✐❝ ✇❤❛t ✐s ❧❡❢t✿ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✉✈ ❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮ ❈♦♠❜✐♥✐♥❣ ✇✐t❤ str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s Pr♦❜❧❡♠✿ ❲♦r❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♦♥❡ ✈❛r✐❛❜❧❡ ❆ ✵ ❳❆ ✶ . . . ❆ ❦ − ✶ ❳❆ ❦ = ❳❇ ✶ . . . ❇ ℓ − ✶ ❳❇ ℓ ✱ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼

r❡❝♦♠♣r❡ss ❛♥❞ t❡st ❧♦st s♦❧✉t✐♦♥s✱ ❜② s✉❜st✐t✉t✐♦♥ s♣❡❡❞ ✉♣ t❡st✐♥❣✿ ♣❛rts ♦❢ ❡q✉❛t✐♦♥ ❛r❡ ♣❡r✐♦❞✐❝ ✇❤❛t ✐s ❧❡❢t✿ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✉✈ ❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮ ❈♦♠❜✐♥✐♥❣ ✇✐t❤ str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s Pr♦❜❧❡♠✿ ❲♦r❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♦♥❡ ✈❛r✐❛❜❧❡ ❆ ✵ ❳❆ ✶ . . . ❆ ❦ − ✶ ❳❆ ❦ = ❳❇ ✶ . . . ❇ ℓ − ✶ ❳❇ ℓ ✱ ♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠s✿ O ( ♥ ❧♦❣ ♥ ) r❡❝♦♠♣r❡ss✐♦♥ ❜❛s❡❞✿ O ( ♥ ❧♦❣ ♥ ) ✭s✐♠♣❧❡ ❛♥❛❧②s✐s✮ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼

s♣❡❡❞ ✉♣ t❡st✐♥❣✿ ♣❛rts ♦❢ ❡q✉❛t✐♦♥ ❛r❡ ♣❡r✐♦❞✐❝ ✇❤❛t ✐s ❧❡❢t✿ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✉✈ ❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮ ❈♦♠❜✐♥✐♥❣ ✇✐t❤ str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s Pr♦❜❧❡♠✿ ❲♦r❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♦♥❡ ✈❛r✐❛❜❧❡ ❆ ✵ ❳❆ ✶ . . . ❆ ❦ − ✶ ❳❆ ❦ = ❳❇ ✶ . . . ❇ ℓ − ✶ ❳❇ ℓ ✱ ♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠s✿ O ( ♥ ❧♦❣ ♥ ) r❡❝♦♠♣r❡ss✐♦♥ ❜❛s❡❞✿ O ( ♥ ❧♦❣ ♥ ) ✭s✐♠♣❧❡ ❛♥❛❧②s✐s✮ r❡❝♦♠♣r❡ss ❛♥❞ t❡st ❧♦st s♦❧✉t✐♦♥s✱ ❜② s✉❜st✐t✉t✐♦♥ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼

✇❤❛t ✐s ❧❡❢t✿ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ✉✈ ❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮ ❈♦♠❜✐♥✐♥❣ ✇✐t❤ str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s Pr♦❜❧❡♠✿ ❲♦r❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♦♥❡ ✈❛r✐❛❜❧❡ ❆ ✵ ❳❆ ✶ . . . ❆ ❦ − ✶ ❳❆ ❦ = ❳❇ ✶ . . . ❇ ℓ − ✶ ❳❇ ℓ ✱ ♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠s✿ O ( ♥ ❧♦❣ ♥ ) r❡❝♦♠♣r❡ss✐♦♥ ❜❛s❡❞✿ O ( ♥ ❧♦❣ ♥ ) ✭s✐♠♣❧❡ ❛♥❛❧②s✐s✮ r❡❝♦♠♣r❡ss ❛♥❞ t❡st ❧♦st s♦❧✉t✐♦♥s✱ ❜② s✉❜st✐t✉t✐♦♥ s♣❡❡❞ ✉♣ t❡st✐♥❣✿ ♣❛rts ♦❢ ❡q✉❛t✐♦♥ ❛r❡ ♣❡r✐♦❞✐❝ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼

❈♦♠❜✐♥✐♥❣ ✇✐t❤ str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s Pr♦❜❧❡♠✿ ❲♦r❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♦♥❡ ✈❛r✐❛❜❧❡ ❆ ✵ ❳❆ ✶ . . . ❆ ❦ − ✶ ❳❆ ❦ = ❳❇ ✶ . . . ❇ ℓ − ✶ ❳❇ ℓ ✱ ♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠s✿ O ( ♥ ❧♦❣ ♥ ) r❡❝♦♠♣r❡ss✐♦♥ ❜❛s❡❞✿ O ( ♥ ❧♦❣ ♥ ) ✭s✐♠♣❧❡ ❛♥❛❧②s✐s✮ r❡❝♦♠♣r❡ss ❛♥❞ t❡st ❧♦st s♦❧✉t✐♦♥s✱ ❜② s✉❜st✐t✉t✐♦♥ s♣❡❡❞ ✉♣ t❡st✐♥❣✿ ♣❛rts ♦❢ ❡q✉❛t✐♦♥ ❛r❡ ♣❡r✐♦❞✐❝ ✇❤❛t ✐s ❧❡❢t✿ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ( ✉✈ ❦ ) ℓ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼

❝r♦ss✐♥❣✴♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ❣❡♥❡r❛❧✐s❡ t❤❡ r❡❝♦♠♣r❡ss✐♦♥ ❛♣♣r♦❛❝❤✿ ❡①♣❧✐❝✐t ✇♦r❞✿ ❡❛s✐❧② ✭❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s♠❛❧❧❡st ❙▲P✮ ✐♠♣❧✐❝✐t ✇♦r❞ ✭♦♥❧② ▲❩✼✼✮✿ ❝❛♥ ❜❡ ❞♦♥❡ ▲❩✼✼ ❘❡♣r❡s❡♥t ✇ ❛s ✇ = ❢ ✶ ❢ ✷ ❢ ✸ . . . ❢ ℓ ✳ ❊❛❝❤ ❢ = ✇ [ ✐ . . ✐ + | ❢ | ) ✐s ◮ ❛ ❧❡tt❡r ♦r ◮ ❡q✉❛❧ t♦ ✇ [ ❥ . . ❥ + | ❢ | ) ❢♦r s♦♠❡ ❥ < ✐ ✳ ❝♦♠♣r❡ss❡❞ r❡♣r❡s❡♥t❛t✐♦♥✿ ( ✵ , ❛ ) ♦r ( ❞✐st , ❧❡♥ ) ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✹ ✴ ✸✼

❝r♦ss✐♥❣✴♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ❣❡♥❡r❛❧✐s❡ t❤❡ r❡❝♦♠♣r❡ss✐♦♥ ❛♣♣r♦❛❝❤✿ ❡①♣❧✐❝✐t ✇♦r❞✿ ❡❛s✐❧② ✭❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s♠❛❧❧❡st ❙▲P✮ ✐♠♣❧✐❝✐t ✇♦r❞ ✭♦♥❧② ▲❩✼✼✮✿ ❝❛♥ ❜❡ ❞♦♥❡ ▲❩✼✼ ❘❡♣r❡s❡♥t ✇ ❛s ✇ = ❢ ✶ ❢ ✷ ❢ ✸ . . . ❢ ℓ ✳ ❊❛❝❤ ❢ = ✇ [ ✐ . . ✐ + | ❢ | ) ✐s ◮ ❛ ❧❡tt❡r ♦r ◮ ❡q✉❛❧ t♦ ✇ [ ❥ . . ❥ + | ❢ | ) ❢♦r s♦♠❡ ❥ < ✐ ✳ ❝♦♠♣r❡ss❡❞ r❡♣r❡s❡♥t❛t✐♦♥✿ ( ✵ , ❛ ) ♦r ( ❞✐st , ❧❡♥ ) a b a a c c d a a c c b a a c c d a b d ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✹ ✴ ✸✼

▲❩✼✼ ❘❡♣r❡s❡♥t ✇ ❛s ✇ = ❢ ✶ ❢ ✷ ❢ ✸ . . . ❢ ℓ ✳ ❊❛❝❤ ❢ = ✇ [ ✐ . . ✐ + | ❢ | ) ✐s ◮ ❛ ❧❡tt❡r ♦r ◮ ❡q✉❛❧ t♦ ✇ [ ❥ . . ❥ + | ❢ | ) ❢♦r s♦♠❡ ❥ < ✐ ✳ ❝♦♠♣r❡ss❡❞ r❡♣r❡s❡♥t❛t✐♦♥✿ ( ✵ , ❛ ) ♦r ( ❞✐st , ❧❡♥ ) a b a a c c d a a c c b a a c c d a b d ❝r♦ss✐♥❣✴♥♦♥✲❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦s ❣❡♥❡r❛❧✐s❡ t❤❡ r❡❝♦♠♣r❡ss✐♦♥ ❛♣♣r♦❛❝❤✿ ◮ ❡①♣❧✐❝✐t ✇♦r❞✿ ❡❛s✐❧② ✭❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s♠❛❧❧❡st ❙▲P✮ ◮ ✐♠♣❧✐❝✐t ✇♦r❞ ✭♦♥❧② ▲❩✼✼✮✿ ❝❛♥ ❜❡ ❞♦♥❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✹ ✴ ✸✼

❉✐✛❡r❡♥t ❢r♦♠ str✐♥❣s✿ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❡r♠ r❡✇r✐t✐♥❣ ❚r❡❡s ▼♦❞❡❧ f r♦♦t❡❞ b a ♦r❞❡r❡❞ c ❧❛❜❡❧❧❡❞ ❛❧♣❤❛❜❡t Σ c c a r❛♥❦❡❞ r❛♥❦ : Σ �→ N ♥♦❞❡ ❛ ❤❛s r❛♥❦ ( ❛ ) ❝❤✐❧❞r❡♥ c ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✺ ✴ ✸✼

❚r❡❡s ▼♦❞❡❧ f r♦♦t❡❞ b a ♦r❞❡r❡❞ c ❧❛❜❡❧❧❡❞ ❛❧♣❤❛❜❡t Σ c c a r❛♥❦❡❞ r❛♥❦ : Σ �→ N ♥♦❞❡ ❛ ❤❛s r❛♥❦ ( ❛ ) ❝❤✐❧❞r❡♥ c ❉✐✛❡r❡♥t ❢r♦♠ str✐♥❣s✿ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❡r♠ r❡✇r✐t✐♥❣ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✺ ✴ ✸✼

✐❢ ② ✐ ♠❛② ❜❡ ♠✉❧t✐♣❧✐❡❞✿ t✉r♥s ✈❡r② ❞✐✣❝✉❧t✳ ❚r❡❡ ❙▲P ❚❙▲P ❣r❛♠♠❛r ♦r❞❡r❡❞ r❛♥❦❡❞ ♥♦♥t❡r♠✐♥❛❧s ❳ ✶ , ❳ ✷ , . . . r✉❧❡s ❳ ✐ ( ② ✶ , ② ✷ , . . . , ② ♠ ) → t ✱ ✇❤❡r❡ ◮ ♠ ✐s t❤❡ r❛♥❦ ♦❢ ❳ ✐ ◮ t ❝♦♥t❛✐♥s ❛ s✐♥❣❧❡ ❧❡❛❢ ② ✶ ✱ ✳ ✳ ✳ ✱ ② ♠ ◮ ♠❛② ❝♦♥t❛✐♥ ❳ ✶ ✱ ✳ ✳ ✳ ✱ ❳ ✐ − ✶ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✻ ✴ ✸✼

✐❢ ② ✐ ♠❛② ❜❡ ♠✉❧t✐♣❧✐❡❞✿ t✉r♥s ✈❡r② ❞✐✣❝✉❧t✳ ❚r❡❡ ❙▲P ❚❙▲P ❣r❛♠♠❛r ♦r❞❡r❡❞ r❛♥❦❡❞ ♥♦♥t❡r♠✐♥❛❧s ❳ ✶ , ❳ ✷ , . . . r✉❧❡s ❳ ✐ ( ② ✶ , ② ✷ , . . . , ② ♠ ) → t ✱ ✇❤❡r❡ ◮ ♠ ✐s t❤❡ r❛♥❦ ♦❢ ❳ ✐ ◮ t ❝♦♥t❛✐♥s ❛ s✐♥❣❧❡ ❧❡❛❢ ② ✶ ✱ ✳ ✳ ✳ ✱ ② ♠ ◮ ♠❛② ❝♦♥t❛✐♥ ❳ ✶ ✱ ✳ ✳ ✳ ✱ ❳ ✐ − ✶ f f c A c f A → f ( f ( y 1 , c ) , y 2 ) a f f f f a c a c a c c c c f c c c ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✻ ✴ ✸✼

❚r❡❡ ❙▲P ❚❙▲P ❣r❛♠♠❛r ♦r❞❡r❡❞ r❛♥❦❡❞ ♥♦♥t❡r♠✐♥❛❧s ❳ ✶ , ❳ ✷ , . . . r✉❧❡s ❳ ✐ ( ② ✶ , ② ✷ , . . . , ② ♠ ) → t ✱ ✇❤❡r❡ ◮ ♠ ✐s t❤❡ r❛♥❦ ♦❢ ❳ ✐ ◮ t ❝♦♥t❛✐♥s ❛ s✐♥❣❧❡ ❧❡❛❢ ② ✶ ✱ ✳ ✳ ✳ ✱ ② ♠ ◮ ♠❛② ❝♦♥t❛✐♥ ❳ ✶ ✱ ✳ ✳ ✳ ✱ ❳ ✐ − ✶ ✐❢ ② ✐ ♠❛② ❜❡ ♠✉❧t✐♣❧✐❡❞✿ t✉r♥s ✈❡r② ❞✐✣❝✉❧t✳ f f c A c f A → f ( f ( y 1 , c ) , y 2 ) a f f f f a c a c a c c c c f c c c ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✻ ✴ ✸✼

♥♦♥t❡r♠✐♥❛❧s ♦❢ r❛♥❦ ✵✿ tr❡❡s ♥♦♥t❡r♠✐♥❛❧s ♦❢ r❛♥❦ ✶✿ tr❡❡s ✇✐t❤ ♦♥❡ ♣❛r❛♠❡t❡r✴❤♦❧❡ ❙✐♠♣❧✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ✭▲♦❤r❡②✱ ▼❛♥❡t❤✱ ❙❝❤♠✐❞t✲❙❝❤❛✉ÿ✮ ❆♥② ❚❙▲P ❣r❛♠♠❛r ❝❛♥ ❜❡ ❝♦♥✈❡rt❡❞ t♦ ❛♥ ❡q✉✐✈❛❧❡♥t ♦♥❡✱ ✇❤✐❝❤ ✉s❡s ♦♥❧② ♥♦♥t❡r♠✐♥❛❧s ♦❢ r❛♥❦ ✵ ❛♥❞ ✶ ❤❛s s✐③❡ r t✐♠❡s ❧❛r❣❡r ✇❤❡r❡ r ✐s t❤❡ ♠❛①✐♠❛❧ ❛r✐t② ♦❢ ❧❡tt❡rs ✐♥ Σ ✳ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✼ ✴ ✸✼

❙✐♠♣❧✐✜❝❛t✐♦♥ ❚❤❡♦r❡♠ ✭▲♦❤r❡②✱ ▼❛♥❡t❤✱ ❙❝❤♠✐❞t✲❙❝❤❛✉ÿ✮ ❆♥② ❚❙▲P ❣r❛♠♠❛r ❝❛♥ ❜❡ ❝♦♥✈❡rt❡❞ t♦ ❛♥ ❡q✉✐✈❛❧❡♥t ♦♥❡✱ ✇❤✐❝❤ ✉s❡s ♦♥❧② ♥♦♥t❡r♠✐♥❛❧s ♦❢ r❛♥❦ ✵ ❛♥❞ ✶ ❤❛s s✐③❡ r t✐♠❡s ❧❛r❣❡r ✇❤❡r❡ r ✐s t❤❡ ♠❛①✐♠❛❧ ❛r✐t② ♦❢ ❧❡tt❡rs ✐♥ Σ ✳ ♥♦♥t❡r♠✐♥❛❧s ♦❢ r❛♥❦ ✵✿ tr❡❡s ♥♦♥t❡r♠✐♥❛❧s ♦❢ r❛♥❦ ✶✿ tr❡❡s ✇✐t❤ ♦♥❡ ♣❛r❛♠❡t❡r✴❤♦❧❡ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✼ ✴ ✸✼

▲❡❛✈❡s ❝♦♠♣r❡ss✐♦♥ ❵❆❜s♦r❜✬ ❛ ❧❡❛❢ ❜② ✐ts ❢❛t❤❡r ✭❛♥❞ ❝❤❛♥❣❡ ❢❛t❤❡r✬s ❧❛❜❡❧✮✳ ❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛r❛❧❧❡❧ ❍♦✇ t♦ ❣❡♥❡r❛❧✐s❡ ❑❡❡♣ ♣r❡✈✐♦✉s ❝♦♠♣r❡ss✐♦♥ ❛s t❤❡② ✇❡r❡ ✭♥♦❞❡s ♦❢ ❛r✐t② ✶✮✳ ❆♣♣❧② t♦ ❝❤❛✐♥s ♦❢ ✉♥❛r② ♥♦❞❡s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✽ ✴ ✸✼

❍♦✇ t♦ ❣❡♥❡r❛❧✐s❡ ❑❡❡♣ ♣r❡✈✐♦✉s ❝♦♠♣r❡ss✐♦♥ ❛s t❤❡② ✇❡r❡ ✭♥♦❞❡s ♦❢ ❛r✐t② ✶✮✳ ❆♣♣❧② t♦ ❝❤❛✐♥s ♦❢ ✉♥❛r② ♥♦❞❡s ▲❡❛✈❡s ❝♦♠♣r❡ss✐♦♥ ❵❆❜s♦r❜✬ ❛ ❧❡❛❢ ❜② ✐ts ❢❛t❤❡r ✭❛♥❞ ❝❤❛♥❣❡ ❢❛t❤❡r✬s ❧❛❜❡❧✮✳ ❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛r❛❧❧❡❧ ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✽ ✴ ✸✼

❍♦✇ t♦ ❣❡♥❡r❛❧✐s❡ ❑❡❡♣ ♣r❡✈✐♦✉s ❝♦♠♣r❡ss✐♦♥ ❛s t❤❡② ✇❡r❡ ✭♥♦❞❡s ♦❢ ❛r✐t② ✶✮✳ ❆♣♣❧② t♦ ❝❤❛✐♥s ♦❢ ✉♥❛r② ♥♦❞❡s ▲❡❛✈❡s ❝♦♠♣r❡ss✐♦♥ ❵❆❜s♦r❜✬ ❛ ❧❡❛❢ ❜② ✐ts ❢❛t❤❡r ✭❛♥❞ ❝❤❛♥❣❡ ❢❛t❤❡r✬s ❧❛❜❡❧✮✳ ❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛r❛❧❧❡❧ f f b b b ′ b ′′ a a ′ c c c c a a ′ c ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✽ ✴ ✸✼

♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❳ ✶ ❳ ✐ ✶ ❳ ✐ ❳ ✶ ❛♣♣r♦♣r✐❛t❡ r✉❧❡s r❡♣❧❛❝❡ ❳ ✇✐t❤ ❳ ❢ ❳ ✶ ❳ ✐ ❳ ✐ ❳ ✭♣♦♣ ❞♦✇♥✮ ✶ ✶ r❡♣❧❛❝❡ ❨ ✇✐t❤ ❝ ✭♣♦♣ ✉♣✮ ❯♥❝r♦ss✐♥❣ ❚❤❡ s❛♠❡ ❢♦r ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ s✐♠✐❧❛r ❢♦r ❧❡❛✈❡s ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✾ ✴ ✸✼

♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❳ ✶ ❳ ✐ ✶ ❳ ✐ ❳ ✶ ❛♣♣r♦♣r✐❛t❡ r✉❧❡s r❡♣❧❛❝❡ ❨ ✇✐t❤ ❝ ✭♣♦♣ ✉♣✮ r❡♣❧❛❝❡ ❳ ✇✐t❤ ❳ ❢ ❳ ✶ ❳ ✐ ❳ ✐ ❳ ✭♣♦♣ ❞♦✇♥✮ ✶ ✶ ❯♥❝r♦ss✐♥❣ ❚❤❡ s❛♠❡ ❢♦r ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ s✐♠✐❧❛r ❢♦r ❧❡❛✈❡s f c X X f f Y c c Y ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✾ ✴ ✸✼