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❆rt✉r ❏❡➺
❯♥✐✈❡rs✐t② ♦❢ ❲r♦❝➟❛✇
❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶ ✴ ✸✼
❘❡❝♦♠♣r❡ss✐♦♥
■ ✇✐❧❧ t❛❧❦ ❛❜♦✉t r❡❝♦♠♣r❡ss✐♦♥ t❡❝❤♥✐q✉❡ ✐❣♥♦r❡s ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt✐❡s ❢♦❝✉s❡s ♦♥ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ✭✐♠♣❧✐❝✐t✴❝♦♠♣r❡ss❡❞✮ ✐♥♣✉t s✉r♣r✐s✐♥❣❧② ✇✐❞❡✲❛♣♣❧✐❝❛❜❧❡ q✉✐t❡ s✉❝❝❡ss❢✉❧
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷ ✴ ✸✼
❘❡❝♦♠♣r❡ss✐♦♥
■ ✇✐❧❧ t❛❧❦ ❛❜♦✉t r❡❝♦♠♣r❡ss✐♦♥ t❡❝❤♥✐q✉❡ ✐❣♥♦r❡s ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt✐❡s ❢♦❝✉s❡s ♦♥ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ✭✐♠♣❧✐❝✐t✴❝♦♠♣r❡ss❡❞✮ ✐♥♣✉t s✉r♣r✐s✐♥❣❧② ✇✐❞❡✲❛♣♣❧✐❝❛❜❧❡ q✉✐t❡ s✉❝❝❡ss❢✉❧
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷ ✴ ✸✼
s❦❡t❝❤ t❤❡ ♠❛✐♥ ✐❞❡❛ ♦♥ ❛♥ ❡①❛♠♣❧❡ ❞✐s❝✉ss ❡①t❡♥s✐♦♥s✱ ✈❛r✐❛♥ts✱ ❣❡♥❡r❛❧✐s❛t✐♦♥s ❛♥❞ ♦❜t❛✐♥❡❞ r❡s✉❧ts s❦❡t❝❤ ♣♦ss✐❜❧❡ ❢✉t✉r❡ ✇♦r❦
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸ ✴ ✸✼
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✹ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❙▲P✿ ❙tr❛✐❣❤t ▲✐♥❡ Pr♦❣r❛♠♠❡✮
❈❋● ❣❡♥❡r❛t✐♥❣ ❡①❛❝t❧② ♦♥❡ ✇♦r❞ r✉❧❡s✿ ❳✐ → ❳❥❳❦ ♦r ❳✐ → ❛ ❝♦♥✈❡♥t✐♦♥✿ ♥ ✖ s✐③❡ ♦❢ ❙▲P✱ ◆ ✖ s✐③❡ ♦❢ str✐♥❣ ❳✵ ❛✱ ❳✶ ❜✱ ❳♥
✶
❳♥
✶❳♥ ✷
❛✱ ❜✱ ❜❛✱ ❜❛❜✱ ❜❛❜❜❛✱ ❜❛❜❜❛❜❛❜❜✱ ✳ ✳ ✳
Pr♦❜❧❡♠✿ ❊q✉❛❧✐t② ♦❢ ❙▲Ps
❋✐rst ❛❧❣♦r✐t❤♠✿ ♥✹ ✳ Pr♦❜❧❡♠❛t✐❝✿ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r t❤❡ s❛♠❡ str✐♥❣✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✺ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❙▲P✿ ❙tr❛✐❣❤t ▲✐♥❡ Pr♦❣r❛♠♠❡✮
❈❋● ❣❡♥❡r❛t✐♥❣ ❡①❛❝t❧② ♦♥❡ ✇♦r❞ r✉❧❡s✿ ❳✐ → ❳❥❳❦ ♦r ❳✐ → ❛ ❝♦♥✈❡♥t✐♦♥✿ ♥ ✖ s✐③❡ ♦❢ ❙▲P✱ ◆ ✖ s✐③❡ ♦❢ str✐♥❣ ❳✵ = ❛✱ ❳✶ = ❜✱ ❳♥+✶ = ❳♥−✶❳♥−✷ ❛✱ ❜✱ ❜❛✱ ❜❛❜✱ ❜❛❜❜❛✱ ❜❛❜❜❛❜❛❜❜✱ ✳ ✳ ✳
Pr♦❜❧❡♠✿ ❊q✉❛❧✐t② ♦❢ ❙▲Ps
❋✐rst ❛❧❣♦r✐t❤♠✿ ♥✹ ✳ Pr♦❜❧❡♠❛t✐❝✿ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r t❤❡ s❛♠❡ str✐♥❣✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✺ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❙▲P✿ ❙tr❛✐❣❤t ▲✐♥❡ Pr♦❣r❛♠♠❡✮
❈❋● ❣❡♥❡r❛t✐♥❣ ❡①❛❝t❧② ♦♥❡ ✇♦r❞ r✉❧❡s✿ ❳✐ → ❳❥❳❦ ♦r ❳✐ → ❛ ❝♦♥✈❡♥t✐♦♥✿ ♥ ✖ s✐③❡ ♦❢ ❙▲P✱ ◆ ✖ s✐③❡ ♦❢ str✐♥❣ ❳✵ = ❛✱ ❳✶ = ❜✱ ❳♥+✶ = ❳♥−✶❳♥−✷ ❛✱ ❜✱ ❜❛✱ ❜❛❜✱ ❜❛❜❜❛✱ ❜❛❜❜❛❜❛❜❜✱ ✳ ✳ ✳
Pr♦❜❧❡♠✿ ❊q✉❛❧✐t② ♦❢ ❙▲Ps
❋✐rst ❛❧❣♦r✐t❤♠✿ ♥✹ ✳ Pr♦❜❧❡♠❛t✐❝✿ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r t❤❡ s❛♠❡ str✐♥❣✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✺ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❙▲P✿ ❙tr❛✐❣❤t ▲✐♥❡ Pr♦❣r❛♠♠❡✮
❈❋● ❣❡♥❡r❛t✐♥❣ ❡①❛❝t❧② ♦♥❡ ✇♦r❞ r✉❧❡s✿ ❳✐ → ❳❥❳❦ ♦r ❳✐ → ❛ ❝♦♥✈❡♥t✐♦♥✿ ♥ ✖ s✐③❡ ♦❢ ❙▲P✱ ◆ ✖ s✐③❡ ♦❢ str✐♥❣ ❳✵ = ❛✱ ❳✶ = ❜✱ ❳♥+✶ = ❳♥−✶❳♥−✷ ❛✱ ❜✱ ❜❛✱ ❜❛❜✱ ❜❛❜❜❛✱ ❜❛❜❜❛❜❛❜❜✱ ✳ ✳ ✳
Pr♦❜❧❡♠✿ ❊q✉❛❧✐t② ♦❢ ❙▲Ps
❋✐rst ❛❧❣♦r✐t❤♠✿ O(♥✹)✳ Pr♦❜❧❡♠❛t✐❝✿ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r t❤❡ s❛♠❡ str✐♥❣✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✺ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❙▲P✿ ❙tr❛✐❣❤t ▲✐♥❡ Pr♦❣r❛♠♠❡✮
❈❋● ❣❡♥❡r❛t✐♥❣ ❡①❛❝t❧② ♦♥❡ ✇♦r❞ r✉❧❡s✿ ❳✐ → ❳❥❳❦ ♦r ❳✐ → ❛ ❝♦♥✈❡♥t✐♦♥✿ ♥ ✖ s✐③❡ ♦❢ ❙▲P✱ ◆ ✖ s✐③❡ ♦❢ str✐♥❣ ❳✵ = ❛✱ ❳✶ = ❜✱ ❳♥+✶ = ❳♥−✶❳♥−✷ ❛✱ ❜✱ ❜❛✱ ❜❛❜✱ ❜❛❜❜❛✱ ❜❛❜❜❛❜❛❜❜✱ ✳ ✳ ✳
Pr♦❜❧❡♠✿ ❊q✉❛❧✐t② ♦❢ ❙▲Ps
❋✐rst ❛❧❣♦r✐t❤♠✿ O(♥✹)✳ Pr♦❜❧❡♠❛t✐❝✿ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r t❤❡ s❛♠❡ str✐♥❣✳
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣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✦
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✻ ✴ ✸✼
■❞❡❛
❋♦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 ❜♦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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✼ ✴ ✸✼
■♥ ♦♥❡ ♣❤❛s❡
▲ ←❧✐st ♦❢ ❧❡tt❡rs✱ P ←❧✐st ♦❢ ♣❛✐rs ♦❢ ❧❡tt❡rs ❢♦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❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼
■♥ ♦♥❡ ♣❤❛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 ❞♦ ❣♦ t♦ ♥❡①t ♣❤❛s❡✱ ✐❢ ♥♦♥tr✐✈✐❛❧✳ ❈♦♥s✐❞❡r ❛❜ ✐♥ t❤❡ t❡①t✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ ✐s ❝♦♠♣r❡ss❡❞✳ ❆❧s♦✿ ❧♦❣ ◆ ♣❤❛s❡s✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼
■♥ ♦♥❡ ♣❤❛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♦✿ ❧♦❣ ◆ ♣❤❛s❡s✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼
■♥ ♦♥❡ ♣❤❛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♦✿ ❧♦❣ ◆ ♣❤❛s❡s✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼
■♥ ♦♥❡ ♣❤❛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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼
■♥ ♦♥❡ ♣❤❛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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✽ ✴ ✸✼
■♥ ♦♥❡ ♣❤❛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✉❧❡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❡ ❣❡♥❡r❛❧ r✉❧❡s✿ ❳✐ → ✉❳❥✈❳❦✇, ❥, ❦ < ✐✳ ❡①♣❧✐❝✐t ❧❡tt❡rs
▲❡♠♠❛
❚❤❡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❡ ❣❡♥❡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❡ ❣❡♥❡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✳
❆✳ ❏❡➺ ❘❡❝♦♠♣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✳
❆✳ ❏❡➺ ❘❡❝♦♠♣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✳
❆✳ ❏❡➺ ❘❡❝♦♠♣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✳
❆✳ ❏❡➺ ❘❡❝♦♠♣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✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✵ ✴ ✸✼
❲❤❡♥ ❛❜ ❤❛s ❛ ❵❝r♦ss✐♥❣✬ ❛♣♣❡❛r❛♥❝❡✿ ❛❳✐ ♦r ❳✐❜ ❳✐ ❞❡✜♥❡s ❜✇✿ ❝❤❛♥❣❡ ✐t t♦ ✇✱ r❡♣❧❛❝❡ ❳✐ ❜② ❜❳✐ s②♠♠❡tr✐❝❛❧❧② ❢♦r ❡♥❞✐♥❣ ❛
❯♥❝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✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✶ ✴ ✸✼
❲❤❡♥ ❛❜ ❤❛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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✶ ✴ ✸✼
❊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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✷ ✴ ✸✼
❊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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✷ ✴ ✸✼
❊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✐♥❣
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 ❛ ✐ ❛♥❞ ❛✲s✉✣① ❛r✐ ♦❢ ❳✐
✸✿
r❡♣❧❛❝❡ ❡❛❝❤ ❳✐ ✐♥ r✉❧❡s ❜♦❞✐❡s ❜② ❛ ✐❳✐❛r✐
▲❡♠♠❛
❆❢t❡r ✉♥❝r♦ss✐♥❣ ❛ ✐t ❤❛s ♥♦ ❝r♦ss✐♥❣ ❜❧♦❝❦✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✸ ✴ ✸✼
❯♥❝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♦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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✸ ✴ ✸✼
✇❤✐❧❡ ♥♦♥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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼
✇❤✐❧❡ ♥♦♥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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼
✇❤✐❧❡ ♥♦♥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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼
✇❤✐❧❡ ♥♦♥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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✹ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O(♥✷)
❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s ✶ ❧❡tt❡r ♣❡r r✉❧❡ ✭❧♦♥❣ ❜❧♦❝❦ ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ♥ ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦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
❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳
✹ ●
❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛s❡ s✐③❡ ♦❢ ❣r❛♠♠❛r ✐♥❝r❡❛s❡s ❜② O(♥✷)
◮ ❡❛❝❤ ✉♥❝r♦ss✐♥❣ ✐♥tr♦❞✉❝❡s O(✶) ❧❡tt❡r ♣❡r r✉❧❡
✭❧♦♥❣ ❜❧♦❝❦ → ❛ s✐♥❣❧❡ ❧❡tt❡r✮ ♥ ❝r♦ss✐♥❣ ♣❛✐rs ❛♥❞ ❜❧♦❝❦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
❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳
✹ ●
❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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
❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳
✹ ●
❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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
❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳
✹ ●
❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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
❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳
✹ ●
❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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
❙♦ ❧♦❣ ◆ ♣❤❛s❡s✳
✹ ●
❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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✳
✹ ●
❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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✳ |● ′| ≤ ✸
✹|●| + ❝♥✷
♥✷ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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✳ |● ′| ≤ ✸
✹|●| + ❝♥✷
❘✉♥♥✐♥❣ t✐♠❡ ✐s ♥✷ ❧♦❣ ◆
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
■♥ ❡❛❝❤ ♣❤❛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✐❞❡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✳ |● ′| ≤ ✸
✹|●| + ❝♥✷
❘✉♥♥✐♥❣ t✐♠❡ ✐s O(♥✷ ❧♦❣ ◆)
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✺ ✴ ✸✼
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✻ ✴ ✸✼
❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✼ ✴ ✸✼
❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✼ ✴ ✸✼
❈❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣❛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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✼ ✴ ✸✼
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 ♥ ❧♦❣ ◆ t✐♠❡ ❧♦❣ ♥ ♣❛rt✐t✐♦♥s ♦❢ ❛❧❧ ❧❡tt❡rs ♥ ❧♦❣ ◆ ❧♦❣ ♥ t✐♠❡
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✽ ✴ ✸✼
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 ♥ ❧♦❣ ◆ t✐♠❡ ❧♦❣ ♥ ♣❛rt✐t✐♦♥s ♦❢ ❛❧❧ ❧❡tt❡rs ♥ ❧♦❣ ◆ ❧♦❣ ♥ t✐♠❡
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✽ ✴ ✸✼
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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✽ ✴ ✸✼
❑✳ ▼❡❤❧❤♦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❤ ✶ ♣❛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❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✾ ✴ ✸✼
❑✳ ▼❡❤❧❤♦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❤ ✶ ♣❛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❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✾ ✴ ✸✼
❑✳ ▼❡❤❧❤♦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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✶✾ ✴ ✸✼
❛❧❧♦✇ ❡①♣♦♥❡♥t✐❛t✐♦♥✿ r✉❧❡s ❳ → ❨ ℓ ✭s✐③❡ O(✶)✮ ❛♥❛❧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❛♠❡
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✵ ✴ ✸✼
❛❧❧♦✇ ❡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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✵ ✴ ✸✼
❛❧❧♦✇ ❡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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✵ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮
■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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮
■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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮
■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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼
❉❡✜♥✐t✐♦♥ ✭❲♦r❞ ❡q✉❛t✐♦♥s✮
■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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✶ ✴ ✸✼
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 ❧♦❣ ♥ ❣ ❛♣♣r♦①✐♠❛t✐♦♥
❑✳ ▼❡❤❧❤♦r♥✱ ❘✳ ❙✉♥❞❛r ❛♥❞ ❈❤✳ ❯❤r✐❣ ▼❛✐♥t❛✐♥✐♥❣ ❉②♥❛♠✐❝ ❙❡q✉❡♥❝❡s ✉♥❞❡r ❊q✉❛❧✐t② ❚❡sts ✐♥ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❚✐♠❡✱ ❵✾✼
❧♦❣ ♥ ❣ ❧♦❣ ♥ ❛♣♣r♦①✐♠❛t✐♦♥✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✷ ✴ ✸✼
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❤♠✐❝ ❚✐♠❡✱ ❵✾✼
❧♦❣ ♥ ❣ ❧♦❣ ♥ ❛♣♣r♦①✐♠❛t✐♦♥✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✷ ✴ ✸✼
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✐♦♥✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼
Pr♦❜❧❡♠✿ ❲♦r❞ ❡q✉❛t✐♦♥s ✇✐t❤ ♦♥❡ ✈❛r✐❛❜❧❡
❆✵❳❆✶ . . . ❆❦−✶❳❆❦ = ❳❇✶ . . . ❇ℓ−✶❳❇ℓ✱ ♥❛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✐♦♥✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼
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♠ ✉✈❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼
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♠ ✉✈❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼
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♠ ✉✈❦ ✇ ✇❡ ❞❡❛❧ ✇❡❧❧ ✇✐t❤ t❤♦s❡ ✭❣r❛♠♠❛r ✇✐t❤ ❡①♣♦♥❡♥t✐❛t✐♦♥✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✸ ✴ ✸✼
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❡s❡♥t ✇ ❛s ✇ = ❢✶❢✷❢✸ . . . ❢ℓ✳ ❊❛❝❤ ❢ = ✇[✐ . . ✐ + |❢ |) ✐s
◮ ❛ ❧❡tt❡r ♦r ◮ ❡q✉❛❧ t♦ ✇[❥ . . ❥ + |❢ |) ❢♦r s♦♠❡ ❥ < ✐✳
❝♦♠♣r❡ss❡❞ r❡♣r❡s❡♥t❛t✐♦♥✿ (✵, ❛) ♦r (❞✐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❡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 a a c c d a b 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❡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 a a c c d a b 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❞❡r❡❞ ❧❛❜❡❧❧❡❞ ❛❧♣❤❛❜❡t Σ r❛♥❦❡❞ r❛♥❦ : Σ → N ♥♦❞❡ ❛ ❤❛s r❛♥❦(❛) ❝❤✐❧❞r❡♥
f b c c a c a c
❉✐✛❡r❡♥t ❢r♦♠ str✐♥❣s✿ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❡r♠ r❡✇r✐t✐♥❣
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✺ ✴ ✸✼
▼♦❞❡❧
r♦♦t❡❞ ♦r❞❡r❡❞ ❧❛❜❡❧❧❡❞ ❛❧♣❤❛❜❡t Σ r❛♥❦❡❞ r❛♥❦ : Σ → N ♥♦❞❡ ❛ ❤❛s r❛♥❦(❛) ❝❤✐❧❞r❡♥
f b c c a c a c
❉✐✛❡r❡♥t ❢r♦♠ str✐♥❣s✿ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❡r♠ r❡✇r✐t✐♥❣
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✺ ✴ ✸✼
❚❙▲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✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✻ ✴ ✸✼
❚❙▲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✳
A → f(f(y1, c), y2) a c a c c c c c c c c a c a f f f f f f A f f c
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✻ ✴ ✸✼
❚❙▲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✳
A → f(f(y1, c), y2) a c a c c c c c c c c a c a f f f f f f A f f c
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✻ ✴ ✸✼
❚❤❡♦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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✼ ✴ ✸✼
❚❤❡♦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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✼ ✴ ✸✼
❑❡❡♣ ♣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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✽ ✴ ✸✼
❑❡❡♣ ♣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✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✽ ✴ ✸✼
❑❡❡♣ ♣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 b c c a c f b′ a′ b′′ a c a′ b c
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✽ ✴ ✸✼
❚❤❡ s❛♠❡ ❢♦r ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ s✐♠✐❧❛r ❢♦r ❧❡❛✈❡s r❡♣❧❛❝❡ ❨ ✇✐t❤ ❝ ✭♣♦♣ ✉♣✮ r❡♣❧❛❝❡ ❳ ✇✐t❤ ❳ ❢ ❳✶ ❳✐
✶
❳✐
✶
❳ ✭♣♦♣ ❞♦✇♥✮ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❳✶ ❳✐
✶ ❳✐ ✶
❳ ❛♣♣r♦♣r✐❛t❡ r✉❧❡s
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✾ ✴ ✸✼
❚❤❡ s❛♠❡ ❢♦r ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ s✐♠✐❧❛r ❢♦r ❧❡❛✈❡s
f c f Y X c f X Y c
r❡♣❧❛❝❡ ❨ ✇✐t❤ ❝ ✭♣♦♣ ✉♣✮ r❡♣❧❛❝❡ ❳ ✇✐t❤ ❳ ❢ ❳✶ ❳✐
✶
❳✐
✶
❳ ✭♣♦♣ ❞♦✇♥✮ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❳✶ ❳✐
✶ ❳✐ ✶
❳ ❛♣♣r♦♣r✐❛t❡ r✉❧❡s
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✾ ✴ ✸✼
❚❤❡ s❛♠❡ ❢♦r ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ s✐♠✐❧❛r ❢♦r ❧❡❛✈❡s
f c X X1 X2 c f c f X X1 X2
r❡♣❧❛❝❡ ❨ ✇✐t❤ ❝ ✭♣♦♣ ✉♣✮ r❡♣❧❛❝❡ ❳ ✇✐t❤ ❳(❢ (❳✶, . . . ❳✐−✶, ·, ❳✐+✶, . . . , ❳ℓ)) ✭♣♦♣ ❞♦✇♥✮ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❳✶ ❳✐
✶ ❳✐ ✶
❳ ❛♣♣r♦♣r✐❛t❡ r✉❧❡s
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✾ ✴ ✸✼
❚❤❡ s❛♠❡ ❢♦r ♣❛✐r ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥ s✐♠✐❧❛r ❢♦r ❧❡❛✈❡s
f c X X1 X2 c f c f X X1 X2
r❡♣❧❛❝❡ ❨ ✇✐t❤ ❝ ✭♣♦♣ ✉♣✮ r❡♣❧❛❝❡ ❳ ✇✐t❤ ❳(❢ (❳✶, . . . ❳✐−✶, ·, ❳✐+✶, . . . , ❳ℓ)) ✭♣♦♣ ❞♦✇♥✮ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❳✶, . . . , ❳✐−✶, ❳✐+✶, . . . , ❳ℓ ❛♣♣r♦♣r✐❛t❡ r✉❧❡s
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✷✾ ✴ ✸✼
✐t❡r❛t❡ ♣❛✐r ❝♦♠♣r❡ss✐♦♥✱ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥✱ ❧❡❛❢ ❝♦♠♣r❡ss✐♦♥ ✭✐♥ ♣❤❛s❡s✮ t❤❡ ❛♥❛❧②s✐s ✐s ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ❚❙▲P ❛❧♦♥❡ ✭♥♦t ❦♥♦✇♥ t♦ t❤❡ ❛❧❣♦r✐t❤♠✮ ❛❞❞✐t✐♦♥❛❧ ❞✐✣❝✉❧t②✿ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❝♦♥tr♦❧ t❤❡♠ s❡♣❛r❛t❡❧② ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ r ❧♦❣ ♥ r❣
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✵ ✴ ✸✼
✐t❡r❛t❡ ♣❛✐r ❝♦♠♣r❡ss✐♦♥✱ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥✱ ❧❡❛❢ ❝♦♠♣r❡ss✐♦♥ ✭✐♥ ♣❤❛s❡s✮ t❤❡ ❛♥❛❧②s✐s ✐s ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ❚❙▲P ❛❧♦♥❡ ✭♥♦t ❦♥♦✇♥ t♦ t❤❡ ❛❧❣♦r✐t❤♠✮ ❛❞❞✐t✐♦♥❛❧ ❞✐✣❝✉❧t②✿ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❝♦♥tr♦❧ t❤❡♠ s❡♣❛r❛t❡❧② ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ r ❧♦❣ ♥ r❣
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✵ ✴ ✸✼
✐t❡r❛t❡ ♣❛✐r ❝♦♠♣r❡ss✐♦♥✱ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥✱ ❧❡❛❢ ❝♦♠♣r❡ss✐♦♥ ✭✐♥ ♣❤❛s❡s✮ t❤❡ ❛♥❛❧②s✐s ✐s ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ❚❙▲P ❛❧♦♥❡ ✭♥♦t ❦♥♦✇♥ t♦ t❤❡ ❛❧❣♦r✐t❤♠✮ ❛❞❞✐t✐♦♥❛❧ ❞✐✣❝✉❧t②✿ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❝♦♥tr♦❧ t❤❡♠ s❡♣❛r❛t❡❧② ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ r ❧♦❣ ♥ r❣
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✵ ✴ ✸✼
✐t❡r❛t❡ ♣❛✐r ❝♦♠♣r❡ss✐♦♥✱ ❜❧♦❝❦ ❝♦♠♣r❡ss✐♦♥✱ ❧❡❛❢ ❝♦♠♣r❡ss✐♦♥ ✭✐♥ ♣❤❛s❡s✮ t❤❡ ❛♥❛❧②s✐s ✐s ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ ❚❙▲P ❛❧♦♥❡ ✭♥♦t ❦♥♦✇♥ t♦ t❤❡ ❛❧❣♦r✐t❤♠✮ ❛❞❞✐t✐♦♥❛❧ ❞✐✣❝✉❧t②✿ ♥❡✇ ♥♦♥t❡r♠✐♥❛❧s ❝♦♥tr♦❧ t❤❡♠ s❡♣❛r❛t❡❧② ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ O(r ❧♦❣(♥/r❣))
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✵ ✴ ✸✼
❉❡❝✐❞❛❜✐❧✐t② st❛t✉s ✉♥❦♥♦✇♥ ❢♦r s♦♠❡ t✐♠❡✳ ❱❛r✐❛❜❧❡s ❤❛✈❡ ❛r❣✉♠❡♥ts t❤❛t ❝❛♥ ❜❡ ✉s❡❞✳ ❚❡r♠s ✇✐t❤ ❵❤♦❧❡s✬✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✶ ✴ ✸✼
❉❡❝✐❞❛❜✐❧✐t② st❛t✉s ✉♥❦♥♦✇♥ ❢♦r s♦♠❡ t✐♠❡✳ ❱❛r✐❛❜❧❡s ❤❛✈❡ ❛r❣✉♠❡♥ts t❤❛t ❝❛♥ ❜❡ ✉s❡❞✳ ❚❡r♠s ✇✐t❤ ❵❤♦❧❡s✬✳
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✶ ✴ ✸✼
❉❡❝✐❞❛❜✐❧✐t② st❛t✉s ✉♥❦♥♦✇♥ ❢♦r s♦♠❡ t✐♠❡✳ ❱❛r✐❛❜❧❡s ❤❛✈❡ ❛r❣✉♠❡♥ts t❤❛t ❝❛♥ ❜❡ ✉s❡❞✳ ❚❡r♠s ✇✐t❤ ❵❤♦❧❡s✬✳
X = h c c c f f f X(t) = h c c c f f f t c c c c
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✶ ✴ ✸✼
❊①❛♠♣❧❡
❢ ✿ ❛r✐t② ✷✱ ❝, ❝′✿ ❛r✐t② ✵ ❳(❝) = ❨ (❝′) ✐♥ P❙P❆❈❊ s✐♠✐❧❛r ❛❧❣♦r✐t❤♠ ❛s ❢♦r ✇♦r❞ ❡q✉❛t✐♦♥s ✭s♦♠❡ ❝❛r❡ ✇✐t❤ ♥❡✇ ✈❛r✐❛❜❧❡s✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✷ ✴ ✸✼
❊①❛♠♣❧❡
❢ ✿ ❛r✐t② ✷✱ ❝, ❝′✿ ❛r✐t② ✵ ❳(❝) = ❨ (❝′)
f X = f Y = f X(c) = f Y (c′) = c′ c′ c c′ c c
✐♥ P❙P❆❈❊ s✐♠✐❧❛r ❛❧❣♦r✐t❤♠ ❛s ❢♦r ✇♦r❞ ❡q✉❛t✐♦♥s ✭s♦♠❡ ❝❛r❡ ✇✐t❤ ♥❡✇ ✈❛r✐❛❜❧❡s✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✷ ✴ ✸✼
❊①❛♠♣❧❡
❢ ✿ ❛r✐t② ✷✱ ❝, ❝′✿ ❛r✐t② ✵ ❳(❝) = ❨ (❝′)
f X = f Y = f X(c) = f Y (c′) = c′ c′ c c′ c c
✐♥ P❙P❆❈❊ s✐♠✐❧❛r ❛❧❣♦r✐t❤♠ ❛s ❢♦r ✇♦r❞ ❡q✉❛t✐♦♥s ✭s♦♠❡ ❝❛r❡ ✇✐t❤ ♥❡✇ ✈❛r✐❛❜❧❡s✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✷ ✴ ✸✼
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✸ ✴ ✸✼
❬❙t❡❢❛♥ ❇ött❝❤❡r✱ ❘✐t❛ ❍❛rt❡❧✱ ❚❤♦♠❛s ❏❛❝♦❜s✱ ❙❡❜❛st✐❛♥ ▼❛♥❡t❤❪ ❣r❛♠♠❛r ❝♦♠♣r❡ss✐♦♥ ✇✐t❤ ✉♣❞❛t❡s ❝❤❛♥❣❡s t♦ ❘❡✲P❛✐r ♣❡♥❛❧t✐❡s ❢♦r ❝r♦ss✐♥❣ ♣❛✐rs ✭▲❩✼✼✮ ❆♥②t❤✐♥❣ ❡❧s❡❄
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✹ ✴ ✸✼
❬❙t❡❢❛♥ ❇ött❝❤❡r✱ ❘✐t❛ ❍❛rt❡❧✱ ❚❤♦♠❛s ❏❛❝♦❜s✱ ❙❡❜❛st✐❛♥ ▼❛♥❡t❤❪ ❣r❛♠♠❛r ❝♦♠♣r❡ss✐♦♥ ✇✐t❤ ✉♣❞❛t❡s ❝❤❛♥❣❡s t♦ ❘❡✲P❛✐r ♣❡♥❛❧t✐❡s ❢♦r ❝r♦ss✐♥❣ ♣❛✐rs ✭▲❩✼✼✮ ❆♥②t❤✐♥❣ ❡❧s❡❄
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✹ ✴ ✸✼
❬❙t❡❢❛♥ ❇ött❝❤❡r✱ ❘✐t❛ ❍❛rt❡❧✱ ❚❤♦♠❛s ❏❛❝♦❜s✱ ❙❡❜❛st✐❛♥ ▼❛♥❡t❤❪ ❣r❛♠♠❛r ❝♦♠♣r❡ss✐♦♥ ✇✐t❤ ✉♣❞❛t❡s ❝❤❛♥❣❡s t♦ ❘❡✲P❛✐r ♣❡♥❛❧t✐❡s ❢♦r ❝r♦ss✐♥❣ ♣❛✐rs ✭▲❩✼✼✮ ❆♥②t❤✐♥❣ ❡❧s❡❄
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✹ ✴ ✸✼
t❤❡ ♠❡t❤♦❞ ✐❣♥♦r❡s str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❜✉t ❞♦❡s ♥♦t ❤❛✈❡ t♦ ✐t ♣r❡s❡r✈❡s s♦♠❡ str✉❝t✉r❡ ✉s❡❞ ♦♥❧② ♦♥❝❡ ✭♦♥❡✲✈❛r✐❛❜❧❡ ✇♦r❞ ❡q✉❛t✐♦♥s✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✺ ✴ ✸✼
t❤❡ ♠❡t❤♦❞ ✐❣♥♦r❡s str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❜✉t ❞♦❡s ♥♦t ❤❛✈❡ t♦ ✐t ♣r❡s❡r✈❡s s♦♠❡ str✉❝t✉r❡ ✉s❡❞ ♦♥❧② ♦♥❝❡ ✭♦♥❡✲✈❛r✐❛❜❧❡ ✇♦r❞ ❡q✉❛t✐♦♥s✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✺ ✴ ✸✼
t❤❡ ♠❡t❤♦❞ ✐❣♥♦r❡s str✐♥❣ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❜✉t ❞♦❡s ♥♦t ❤❛✈❡ t♦ ✐t ♣r❡s❡r✈❡s s♦♠❡ str✉❝t✉r❡ ✉s❡❞ ♦♥❧② ♦♥❝❡ ✭♦♥❡✲✈❛r✐❛❜❧❡ ✇♦r❞ ❡q✉❛t✐♦♥s✮
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✺ ✴ ✸✼
s✉❝❝❡ss✿ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥ ♠❡t❤♦❞ ✐s ❞✐✛❡r❡♥t t❤❛♥ ✉s✉❛❧✿ ❜♦tt♦♠✲✉♣ tr❛❞✐t✐♦♥❛❧❧②✿ t♦♣✲❞♦✇♥ ❝❛♥ ✐t ❜❡ ✉s❡❞ ❡❧s❡✇❤❡r❡❄ ✭s❛②✱ ♦♥❡✲✈❛r✐❛❜❧❡ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥✮
❊①t❡♥s✐♦♥s✴❣❡♥❡r❛❧✐s❛t✐♦♥s ♦❢ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥
❜❛r❡❧② t♦✉❝❤❡❞ ❚❤❡♦r② ♦❢ ♦♥❡✲st❡♣ t❡r♠ r❡✇r✐t✐♥❣
♣♦s✐t✐✈❡ t❤❡♦r② ❡①✐st❡♥t✐❛❧ t❤❡♦r②
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✻ ✴ ✸✼
s✉❝❝❡ss✿ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥ ♠❡t❤♦❞ ✐s ❞✐✛❡r❡♥t t❤❛♥ ✉s✉❛❧✿ ❜♦tt♦♠✲✉♣ tr❛❞✐t✐♦♥❛❧❧②✿ t♦♣✲❞♦✇♥ ❝❛♥ ✐t ❜❡ ✉s❡❞ ❡❧s❡✇❤❡r❡❄ ✭s❛②✱ ♦♥❡✲✈❛r✐❛❜❧❡ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥✮
❊①t❡♥s✐♦♥s✴❣❡♥❡r❛❧✐s❛t✐♦♥s ♦❢ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥
❜❛r❡❧② t♦✉❝❤❡❞ ❚❤❡♦r② ♦❢ ♦♥❡✲st❡♣ t❡r♠ r❡✇r✐t✐♥❣
♣♦s✐t✐✈❡ t❤❡♦r② ❡①✐st❡♥t✐❛❧ t❤❡♦r②
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✻ ✴ ✸✼
s✉❝❝❡ss✿ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥ ♠❡t❤♦❞ ✐s ❞✐✛❡r❡♥t t❤❛♥ ✉s✉❛❧✿ ❜♦tt♦♠✲✉♣ tr❛❞✐t✐♦♥❛❧❧②✿ t♦♣✲❞♦✇♥ ❝❛♥ ✐t ❜❡ ✉s❡❞ ❡❧s❡✇❤❡r❡❄ ✭s❛②✱ ♦♥❡✲✈❛r✐❛❜❧❡ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥✮
❊①t❡♥s✐♦♥s✴❣❡♥❡r❛❧✐s❛t✐♦♥s ♦❢ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥
❜❛r❡❧② t♦✉❝❤❡❞ ❚❤❡♦r② ♦❢ ♦♥❡✲st❡♣ t❡r♠ r❡✇r✐t✐♥❣
♣♦s✐t✐✈❡ t❤❡♦r② ❡①✐st❡♥t✐❛❧ t❤❡♦r②
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✻ ✴ ✸✼
s✉❝❝❡ss✿ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥ ♠❡t❤♦❞ ✐s ❞✐✛❡r❡♥t t❤❛♥ ✉s✉❛❧✿ ❜♦tt♦♠✲✉♣ tr❛❞✐t✐♦♥❛❧❧②✿ t♦♣✲❞♦✇♥ ❝❛♥ ✐t ❜❡ ✉s❡❞ ❡❧s❡✇❤❡r❡❄ ✭s❛②✱ ♦♥❡✲✈❛r✐❛❜❧❡ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥✮
❊①t❡♥s✐♦♥s✴❣❡♥❡r❛❧✐s❛t✐♦♥s ♦❢ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥
❜❛r❡❧② t♦✉❝❤❡❞ ❚❤❡♦r② ♦❢ ♦♥❡✲st❡♣ t❡r♠ r❡✇r✐t✐♥❣
♣♦s✐t✐✈❡ t❤❡♦r② ❡①✐st❡♥t✐❛❧ t❤❡♦r②
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✻ ✴ ✸✼
s✉❝❝❡ss✿ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥ ♠❡t❤♦❞ ✐s ❞✐✛❡r❡♥t t❤❛♥ ✉s✉❛❧✿ ❜♦tt♦♠✲✉♣ tr❛❞✐t✐♦♥❛❧❧②✿ t♦♣✲❞♦✇♥ ❝❛♥ ✐t ❜❡ ✉s❡❞ ❡❧s❡✇❤❡r❡❄ ✭s❛②✱ ♦♥❡✲✈❛r✐❛❜❧❡ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥✮
❊①t❡♥s✐♦♥s✴❣❡♥❡r❛❧✐s❛t✐♦♥s ♦❢ ❝♦♥t❡①t ✉♥✐✜❝❛t✐♦♥
❜❛r❡❧② t♦✉❝❤❡❞ ❚❤❡♦r② ♦❢ ♦♥❡✲st❡♣ t❡r♠ r❡✇r✐t✐♥❣
◮ ♣♦s✐t✐✈❡ t❤❡♦r② ◮ ❡①✐st❡♥t✐❛❧ t❤❡♦r② ❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✻ ✴ ✸✼
◗✉❡st✐♦♥s❄ ❈♦♠♠❡♥ts❄
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✼ ✴ ✸✼
◗✉❡st✐♦♥s❄ ❈♦♠♠❡♥ts❄
❆✳ ❏❡➺ ❘❡❝♦♠♣r❡ss✐♦♥ ❉❛❣st✉❤❧✱ ✷✺✳✶✵✳✷✵✶✻ ✸✼ ✴ ✸✼