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❚❤❡ P❛r❛③♦❛ ❋❛♠✐❧②✿

• ❡♥❡r❛❧✐③✐♥❣ t❤❡ ❙♣♦♥❣❡ ❍❛s❤ ❋✉♥❝t✐♦♥s

❊❧❡♥❛ ❆♥❞r❡❡✈❛✱ ❇❛rt ▼❡♥♥✐♥❦ ❛♥❞ ❇❛rt Pr❡♥❡❡❧ ❑❯ ▲❡✉✈❡♥

❊❈❘❨P❚ ■■ ❍❛s❤ ❲♦r❦s❤♦♣ ✷✵✶✶ ✖ ▼❛② ✶✾✱ ✷✵✶✶

✶ ✴ ✶✺

❚❤❡ ❙♣♦♥❣❡ ❍❛s❤ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

✶ ▼❡ss❛❣❡ ♣❛❞❞❡❞ ✐♥t♦ M1, . . . , Mk ✭✇❤❡r❡ Mk = 0✮ ✷ Mi✬s ✐t❡r❛t✐✈❡❧② ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ ❛❜s♦r❜✐♥❣ ♣❤❛s❡ ✸ Pi✬s ✐t❡r❛t✐✈❡❧② ❡①tr❛❝t❡❞ ✐♥ t❤❡ ❡①tr❛❝t✐♦♥ ♣❤❛s❡ ✹ P1, . . . , Pl ❛r❡ ❝♦♥❝❛t❡♥❛t❡❞ ❛♥❞ ❝❤♦♣♣❡❞ ✐❢ ♥❡❝❡ss❛r②

• ❙♣♦♥❣❡ ❢✉♥❝t✐♦♥s ✐♥❞✐✛❡r❡♥t✐❛❜❧❡ ❢r♦♠ ❘❖ ✉♣ t♦ O(2c/2) q✉❡r✐❡s

✷ ✴ ✶✺

❙♣♦♥❣❡ ❋✉♥❝t✐♦♥s ❛♥❞ ❱❛r✐❛♥ts

• ❙♣♦♥❣❡ ❢✉♥❝t✐♦♥✿
• ❑❡❝❝❛❦

✏❙♣♦♥❣❡✲❧✐❦❡✑ ❢✉♥❝t✐♦♥s✿

• r✐♥❞❛❤❧

❙❍❆✲✸ ❝❛♥❞✐❞❛t❡s ❈✉❜❡❍❛s❤✱ ❋✉❣✉❡✱ ❍❛♠s✐✱ ❏❍✱ ▲✉✛❛

❙❡❝✉r✐t② ♦❢ s♣♦♥❣❡ ❢✉♥❝t✐♦♥s ❞♦❡s ♥♦t ❞✐r❡❝t❧② ❝❛rr② ♦✈❡r ▼✐♥♦r ♠♦❞✐✜❝❛t✐♦♥ t♦ s♣♦♥❣❡ ❞❡s✐❣♥ ❝❛♥ ♠❛❦❡ ✐t ✐♥s❡❝✉r❡

✸ ✴ ✶✺

❙♣♦♥❣❡ ❋✉♥❝t✐♦♥s ❛♥❞ ❱❛r✐❛♥ts

• ❙♣♦♥❣❡ ❢✉♥❝t✐♦♥✿
• ❑❡❝❝❛❦
• ✏❙♣♦♥❣❡✲❧✐❦❡✑ ❢✉♥❝t✐♦♥s✿
• ●r✐♥❞❛❤❧
• ❙❍❆✲✸ ❝❛♥❞✐❞❛t❡s ❈✉❜❡❍❛s❤✱ ❋✉❣✉❡✱ ❍❛♠s✐✱ ❏❍✱ ▲✉✛❛

❙❡❝✉r✐t② ♦❢ s♣♦♥❣❡ ❢✉♥❝t✐♦♥s ❞♦❡s ♥♦t ❞✐r❡❝t❧② ❝❛rr② ♦✈❡r ▼✐♥♦r ♠♦❞✐✜❝❛t✐♦♥ t♦ s♣♦♥❣❡ ❞❡s✐❣♥ ❝❛♥ ♠❛❦❡ ✐t ✐♥s❡❝✉r❡

✸ ✴ ✶✺

❙♣♦♥❣❡ ❋✉♥❝t✐♦♥s ❛♥❞ ❱❛r✐❛♥ts

• ❙♣♦♥❣❡ ❢✉♥❝t✐♦♥✿
• ❑❡❝❝❛❦
• ✏❙♣♦♥❣❡✲❧✐❦❡✑ ❢✉♥❝t✐♦♥s✿
• ●r✐♥❞❛❤❧
• ❙❍❆✲✸ ❝❛♥❞✐❞❛t❡s ❈✉❜❡❍❛s❤✱ ❋✉❣✉❡✱ ❍❛♠s✐✱ ❏❍✱ ▲✉✛❛
• ❙❡❝✉r✐t② ♦❢ s♣♦♥❣❡ ❢✉♥❝t✐♦♥s ❞♦❡s ♥♦t ❞✐r❡❝t❧② ❝❛rr② ♦✈❡r
• ▼✐♥♦r ♠♦❞✐✜❝❛t✐♦♥ t♦ s♣♦♥❣❡ ❞❡s✐❣♥ ❝❛♥ ♠❛❦❡ ✐t ✐♥s❡❝✉r❡

✸ ✴ ✶✺

■♥s❡❝✉r❡ ❙♣♦♥❣❡✲▲✐❦❡ ❋✉♥❝t✐♦♥

❆ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥ ✭❤❡r❡✱ c = r✮✿ ❉✐✛❡r❡♥t✐❛❜❧❡ ❢r♦♠ ❘❖ ❞✉❡ t♦ t❤❡ ❧❡♥❣t❤✲❡①t❡♥s✐♦♥ ❛tt❛❝❦ ■♥❥❡❝t✐♦♥ ✐♥t♦ ✉♣♣❡r ❤❛❧✈❡✱ ❡①tr❛❝t✐♦♥ ❢r♦♠ ❧♦✇❡r ❤❛❧✈❡ ❆tt❛❝❦ ❞♦❡s ♥♦t ✐♥✈❛❧✐❞❛t❡ s❡❝✉r✐t② ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s♣♦♥❣❡ ❞❡s✐❣♥

✹ ✴ ✶✺

■♥s❡❝✉r❡ ❙♣♦♥❣❡✲▲✐❦❡ ❋✉♥❝t✐♦♥

❆ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥ ✭❤❡r❡✱ c = r✮✿

• ❉✐✛❡r❡♥t✐❛❜❧❡ ❢r♦♠ ❘❖ ❞✉❡ t♦ t❤❡ ❧❡♥❣t❤✲❡①t❡♥s✐♦♥ ❛tt❛❝❦
• ■♥❥❡❝t✐♦♥ ✐♥t♦ ✉♣♣❡r ❤❛❧✈❡✱ ❡①tr❛❝t✐♦♥ ❢r♦♠ ❧♦✇❡r ❤❛❧✈❡

❆tt❛❝❦ ❞♦❡s ♥♦t ✐♥✈❛❧✐❞❛t❡ s❡❝✉r✐t② ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s♣♦♥❣❡ ❞❡s✐❣♥

✹ ✴ ✶✺

■♥s❡❝✉r❡ ❙♣♦♥❣❡✲▲✐❦❡ ❋✉♥❝t✐♦♥

❆ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥ ✭❤❡r❡✱ c = r✮✿

• ❉✐✛❡r❡♥t✐❛❜❧❡ ❢r♦♠ ❘❖ ❞✉❡ t♦ t❤❡ ❧❡♥❣t❤✲❡①t❡♥s✐♦♥ ❛tt❛❝❦
• ■♥❥❡❝t✐♦♥ ✐♥t♦ ✉♣♣❡r ❤❛❧✈❡✱ ❡①tr❛❝t✐♦♥ ❢r♦♠ ❧♦✇❡r ❤❛❧✈❡
• ❆tt❛❝❦ ❞♦❡s ♥♦t ✐♥✈❛❧✐❞❛t❡ s❡❝✉r✐t② ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s♣♦♥❣❡ ❞❡s✐❣♥

✹ ✴ ✶✺

❖r✐❣✐♥ ♦❢ t❤❡ ◆❛♠❡ ✏P❛r❛③♦❛✑

❙♣♦♥❣❡ ■♥ t❤❡ ❜✐♦❧♦❣✐❝❛❧ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♦r❣❛♥✲ ✐s♠s✱ s♣♦♥❣❡s ❛r❡ ❛ ♠❡♠❜❡r ♦❢ t❤❡ ♣❤②✲ ❧✉♠ P♦r✐❢❡r❛✱ ✇❤✐❝❤ ❜❡❧♦♥❣s t♦ t❤❡ s✉❜✲ ❦✐♥❣❞♦♠ P❛r❛③♦❛

❙♦✉r❝❡✿ ❤tt♣✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴P❛r❛③♦❛ ✺ ✴ ✶✺

❖r✐❣✐♥ ♦❢ t❤❡ ◆❛♠❡ ✏P❛r❛③♦❛✑

❙♣♦♥❣❡ ■♥ t❤❡ ❜✐♦❧♦❣✐❝❛❧ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♦r❣❛♥✲ ✐s♠s✱ s♣♦♥❣❡s ❛r❡ ❛ ♠❡♠❜❡r ♦❢ t❤❡ ♣❤②✲ ❧✉♠ P♦r✐❢❡r❛✱ ✇❤✐❝❤ ❜❡❧♦♥❣s t♦ t❤❡ s✉❜✲ ❦✐♥❣❞♦♠ P❛r❛③♦❛

❙♦✉r❝❡✿ ❤tt♣✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴✇✐❦✐✴P❛r❛③♦❛ ✺ ✴ ✶✺

❚❤❡ P❛r❛③♦❛ ❍❛s❤ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

✶ M ♣❛❞❞❡❞ ✐♥t♦ M1, . . . , Mk ✷

✬s ✐t❡r❛t✐✈❡❧② ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ ❛❜s♦r❜✐♥❣ ♣❤❛s❡

✸

✬s ✐t❡r❛t✐✈❡❧② ❡①tr❛❝t❡❞ ✐♥ t❤❡ ❡①tr❛❝t✐♦♥ ♣❤❛s❡

✹

❣❡♥❡r❛t❡❞ ❢r♦♠ ✐♥ t❤❡ ✜♥❛❧✐③❛t✐♦♥

✻ ✴ ✶✺

❚❤❡ P❛r❛③♦❛ ❍❛s❤ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

✶ M ♣❛❞❞❡❞ ✐♥t♦ M1, . . . , Mk ✷ Mi✬s ✐t❡r❛t✐✈❡❧② ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ ❛❜s♦r❜✐♥❣ ♣❤❛s❡ ✸

✬s ✐t❡r❛t✐✈❡❧② ❡①tr❛❝t❡❞ ✐♥ t❤❡ ❡①tr❛❝t✐♦♥ ♣❤❛s❡

✹

❣❡♥❡r❛t❡❞ ❢r♦♠ ✐♥ t❤❡ ✜♥❛❧✐③❛t✐♦♥

✻ ✴ ✶✺

❚❤❡ P❛r❛③♦❛ ❍❛s❤ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

✶ M ♣❛❞❞❡❞ ✐♥t♦ M1, . . . , Mk ✷ Mi✬s ✐t❡r❛t✐✈❡❧② ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ ❛❜s♦r❜✐♥❣ ♣❤❛s❡ ✸ Pi✬s ✐t❡r❛t✐✈❡❧② ❡①tr❛❝t❡❞ ✐♥ t❤❡ ❡①tr❛❝t✐♦♥ ♣❤❛s❡ ✹

❣❡♥❡r❛t❡❞ ❢r♦♠ ✐♥ t❤❡ ✜♥❛❧✐③❛t✐♦♥

✻ ✴ ✶✺

❚❤❡ P❛r❛③♦❛ ❍❛s❤ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

✶ M ♣❛❞❞❡❞ ✐♥t♦ M1, . . . , Mk ✷ Mi✬s ✐t❡r❛t✐✈❡❧② ❝♦♠♣r❡ss❡❞ ✐♥ t❤❡ ❛❜s♦r❜✐♥❣ ♣❤❛s❡ ✸ Pi✬s ✐t❡r❛t✐✈❡❧② ❡①tr❛❝t❡❞ ✐♥ t❤❡ ❡①tr❛❝t✐♦♥ ♣❤❛s❡ ✹ h ❣❡♥❡r❛t❡❞ ❢r♦♠ P1, . . . , Pl ✐♥ t❤❡ ✜♥❛❧✐③❛t✐♦♥

✻ ✴ ✶✺

❚❤❡ P❛r❛③♦❛ ❍❛s❤ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

• ❚❤❡ ❢✉♥❝t✐♦♥s f✱ g✱ fin ❛♥❞ pad ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ ♠♦r❡ ❞❡t❛✐❧
• π ✐s ❛♥ s✲❜✐ts ♣❡r♠✉t❛t✐♦♥
• ❆ss✉♠❡❞ t♦ ❜❡❤❛✈❡ ❧✐❦❡ r❛♥❞♦♠ ♣r✐♠✐t✐✈❡

✼ ✴ ✶✺

❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ f

❲❡ r❡q✉✐r❡✿ ❋♦r ✜①❡❞ ✱ ❛ ❞✐st✐♥❝t r❡s✉❧ts ✐♥ ❛ ❞✐st✐♥❝t ■❢ s❤❛r❡ s♦♠❡ ♣r❡✐♠❛❣❡ ✉♥❞❡r ✱ t❤❡② s❤❛r❡ ❛❧❧ ♣r❡✐♠❛❣❡s ❋♦r ✜①❡❞ ✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❜✐❥❡❝t✐♦♥ ♦♥ t❤❡ st❛t❡ ❙t❛♥❞❛r❞ ❢✉♥❝t✐♦♥s ❛♥❞ s❛t✐s❢② t❤❡s❡ r❡q✉✐r❡♠❡♥ts

✽ ✴ ✶✺

❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ f

❲❡ r❡q✉✐r❡✿

• ❋♦r ✜①❡❞ vi−1✱ ❛ ❞✐st✐♥❝t Mi r❡s✉❧ts ✐♥ ❛ ❞✐st✐♥❝t x = Lin(vi−1, Mi)

■❢ s❤❛r❡ s♦♠❡ ♣r❡✐♠❛❣❡ ✉♥❞❡r ✱ t❤❡② s❤❛r❡ ❛❧❧ ♣r❡✐♠❛❣❡s ❋♦r ✜①❡❞ ✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❜✐❥❡❝t✐♦♥ ♦♥ t❤❡ st❛t❡ ❙t❛♥❞❛r❞ ❢✉♥❝t✐♦♥s ❛♥❞ s❛t✐s❢② t❤❡s❡ r❡q✉✐r❡♠❡♥ts

✽ ✴ ✶✺

❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ f

❲❡ r❡q✉✐r❡✿

• ❋♦r ✜①❡❞ vi−1✱ ❛ ❞✐st✐♥❝t Mi r❡s✉❧ts ✐♥ ❛ ❞✐st✐♥❝t x = Lin(vi−1, Mi)
• ■❢ x, x′ s❤❛r❡ s♦♠❡ ♣r❡✐♠❛❣❡ vi−1 ✉♥❞❡r Lin✱ t❤❡② s❤❛r❡ ❛❧❧ ♣r❡✐♠❛❣❡s

❋♦r ✜①❡❞ ✱ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❜✐❥❡❝t✐♦♥ ♦♥ t❤❡ st❛t❡ ❙t❛♥❞❛r❞ ❢✉♥❝t✐♦♥s ❛♥❞ s❛t✐s❢② t❤❡s❡ r❡q✉✐r❡♠❡♥ts

✽ ✴ ✶✺

❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ f

❲❡ r❡q✉✐r❡✿

• ❋♦r ✜①❡❞ vi−1✱ ❛ ❞✐st✐♥❝t Mi r❡s✉❧ts ✐♥ ❛ ❞✐st✐♥❝t x = Lin(vi−1, Mi)
• ■❢ x, x′ s❤❛r❡ s♦♠❡ ♣r❡✐♠❛❣❡ vi−1 ✉♥❞❡r Lin✱ t❤❡② s❤❛r❡ ❛❧❧ ♣r❡✐♠❛❣❡s
• ❋♦r ✜①❡❞ vi−1, Mi✱ t❤❡ ❢✉♥❝t✐♦♥ Lout ✐s ❛ ❜✐❥❡❝t✐♦♥ ♦♥ t❤❡ st❛t❡

❙t❛♥❞❛r❞ ❢✉♥❝t✐♦♥s ❛♥❞ s❛t✐s❢② t❤❡s❡ r❡q✉✐r❡♠❡♥ts

✽ ✴ ✶✺

❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ f

❲❡ r❡q✉✐r❡✿

• ❋♦r ✜①❡❞ vi−1✱ ❛ ❞✐st✐♥❝t Mi r❡s✉❧ts ✐♥ ❛ ❞✐st✐♥❝t x = Lin(vi−1, Mi)
• ■❢ x, x′ s❤❛r❡ s♦♠❡ ♣r❡✐♠❛❣❡ vi−1 ✉♥❞❡r Lin✱ t❤❡② s❤❛r❡ ❛❧❧ ♣r❡✐♠❛❣❡s
• ❋♦r ✜①❡❞ vi−1, Mi✱ t❤❡ ❢✉♥❝t✐♦♥ Lout ✐s ❛ ❜✐❥❡❝t✐♦♥ ♦♥ t❤❡ st❛t❡

❙t❛♥❞❛r❞ ❢✉♥❝t✐♦♥s Lin ❛♥❞ Lout s❛t✐s❢② t❤❡s❡ r❡q✉✐r❡♠❡♥ts

✽ ✴ ✶✺

❊①tr❛❝t✐♦♥ ❋✉♥❝t✐♦♥ g

❲❡ r❡q✉✐r❡✿ Lex ✐s ❜❛❧❛♥❝❡❞ ❘❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❣❡♥❡r❛❧ ✿

✾ ✴ ✶✺

❊①tr❛❝t✐♦♥ ❋✉♥❝t✐♦♥ g

❲❡ r❡q✉✐r❡✿ Lex ✐s ❜❛❧❛♥❝❡❞ ❘❡s✉❧t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❣❡♥❡r❛❧ g✿

✾ ✴ ✶✺

❋✐♥❛❧✐③❛t✐♦♥ ❋✉♥❝t✐♦♥ fin

❲❡ r❡q✉✐r❡✿ fin ✐s ❜❛❧❛♥❝❡❞

• P❛r❛③♦❛ ❢✉♥❝t✐♦♥s ❛❧s♦ ❛❧❧♦✇ ❢♦r ❛r❜✐tr❛r✐❧② ❧♦♥❣ ♦✉t♣✉ts
• ❙♣♦♥❣❡ ❞❡s✐❣♥✿

fin(P1, . . . , Pl) = choplp−n(P1 · · · Pl)

✶✵ ✴ ✶✺

P❛❞❞✐♥❣ ❋✉♥❝t✐♦♥ pad

❲❡ r❡q✉✐r❡✿ pad ✐s ❛♥② ✐♥❥❡❝t✐✈❡ ♣❛❞❞✐♥❣ ❢✉♥❝t✐♦♥ s✳t✳✿

• ❊✐t❤❡r l = 1 ✭♦♥❧② ♦♥❡ ❡①tr❛❝t✐♦♥ r♦✉♥❞✮✱ ♦r
• ▲❛st ❜❧♦❝❦ Mk s❛t✐s✜❡s ❢♦r ❛♥② x, v′, M′✿

Lin(x, Mk) = x ❛♥❞ Lin(Lout(x, v′, M′), Mk) = x ✭❢♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ ✏❧❛st ❜❧♦❝❦ ✐s ♥♦♥✲③❡r♦✑✮

✶✶ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M

✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿

❋♦r ✜①❡❞ ❛♥❞ ✿ ❛t ♠♦st ♣♦ss✐❜❧❡ t✉♣❧❡s ❋♦r ✜①❡❞ ❛♥❞ ✿ ❛t ♠♦st ♣♦ss✐❜❧❡ t✉♣❧❡s

■♥t✉✐t✐✈❡❧②✱ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑ ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ ❛♥❞ ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ ❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)

■♥t✉✐t✐✈❡❧②✱ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑ ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ ❛♥❞ ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ ❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑

❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ ❛♥❞ ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ ❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿

❛♥❞ ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ ❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿

❛♥❞ ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ ❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿

❛♥❞ ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ ❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ d = 0 ❛♥❞ s − d − p = c

❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ ❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ d = 0 ❛♥❞ s − d − p = c
• ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿

❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ d = 0 ❛♥❞ s − d − p = c
• ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿

❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ d = 0 ❛♥❞ s − d − p = c
• ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿

❛♥❞

✶✷ ✴ ✶✺

P❛r❛♠❡t❡r d

• ❈♦♥s✐❞❡r t✉♣❧❡s (v, x) s✳t✳ Lin(v, M) = x ❢♦r s♦♠❡ M
• d ≥ 0 ✐s t❤❡ ♠✐♥✐♠❛❧ ✈❛❧✉❡ s✉❝❤ t❤❛t✿
• ❋♦r ✜①❡❞ x ❛♥❞ P := Lex(v)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ❋♦r ✜①❡❞ v ❛♥❞ P := Lex(x)✿ ❛t ♠♦st 2d ♣♦ss✐❜❧❡ t✉♣❧❡s (v, x)
• ■♥t✉✐t✐✈❡❧②✱ s − d − p ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ✏❝❛♣❛❝✐t②✑
• ❋♦r s♣♦♥❣❡ ❢✉♥❝t✐♦♥s✿ d = 0 ❛♥❞ s − d − p = c
• ❋♦r t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❢✉♥❝t✐♦♥✿ d = r ❛♥❞ s − d − p = 0

✶✷ ✴ ✶✺

❙❡❝✉r✐t② ❆♥❛❧②s✐s

P❛r❛③♦❛ ❢✉♥❝t✐♦♥s ❛r❡ O (Kq)2 2s−d−p

• ✐♥❞✐✛❡r❡♥t✐❛❜❧❡ ❢r♦♠ ❘❖

✭✇❤❡r❡ t❤❡ ❞✐st✐♥❣✉✐s❤❡r ♠❛❦❡s ❛t ♠♦st q q✉❡r✐❡s ♦❢ K ❜❧♦❝❦s✮ s✿ ✐t❡r❛t❡❞ st❛t❡ s✐③❡ d✿ q✉❛♥t✐t② ✐♥❤❡r❡♥t t♦ t❤❡ s♣❡❝✐✜❝ ♣❛r❛③♦❛ ❞❡s✐❣♥ p✿ ♥✉♠❜❡r ♦❢ ❜✐ts ❡①tr❛❝t❡❞ ✐♥ ♦♥❡ ❡①❡❝✉t✐♦♥ ♦❢ g ❜❡❤❛✈❡s ❧✐❦❡ ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❘❡s✉❧t ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ t♦ ✉s❡ ♦❢ ♠✉❧t✐♣❧❡ r❛♥❞♦♠ ♣r✐♠✐t✐✈❡s

✶✸ ✴ ✶✺

❙❡❝✉r✐t② ❆♥❛❧②s✐s

P❛r❛③♦❛ ❢✉♥❝t✐♦♥s ❛r❡ O (Kq)2 2s−d−p

• ✐♥❞✐✛❡r❡♥t✐❛❜❧❡ ❢r♦♠ ❘❖

✭✇❤❡r❡ t❤❡ ❞✐st✐♥❣✉✐s❤❡r ♠❛❦❡s ❛t ♠♦st q q✉❡r✐❡s ♦❢ K ❜❧♦❝❦s✮ s✿ ✐t❡r❛t❡❞ st❛t❡ s✐③❡ d✿ q✉❛♥t✐t② ✐♥❤❡r❡♥t t♦ t❤❡ s♣❡❝✐✜❝ ♣❛r❛③♦❛ ❞❡s✐❣♥ p✿ ♥✉♠❜❡r ♦❢ ❜✐ts ❡①tr❛❝t❡❞ ✐♥ ♦♥❡ ❡①❡❝✉t✐♦♥ ♦❢ g

• π ❜❡❤❛✈❡s ❧✐❦❡ ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥
• ❘❡s✉❧t ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ t♦ ✉s❡ ♦❢ ♠✉❧t✐♣❧❡ r❛♥❞♦♠ ♣r✐♠✐t✐✈❡s

✶✸ ✴ ✶✺

■♠♣❧✐❝❛t✐♦♥s ❢♦r ❊①✐st✐♥❣ ❉❡s✐❣♥s

❆❧❣♦r✐t❤♠ (s, m, p) d ■♥❞✐✛✳ q ≈ ❆ss✉♠♣t✐♦♥ ❙♣♦♥❣❡ (r + c, r, r) 2c/2 π ✐❞❡❛❧

• r✐♥❞❛❤❧

(s, m, n) m 2(s−m−n)/2 π ✐❞❡❛❧ ◗✉❛r❦ (r + c, r, r) 2c/2 π ✐❞❡❛❧ P❍❖❚❖◆✲(r′ ≤ r) (r + c, r, r′) r − r′ 2c/2 π ✐❞❡❛❧ P❍❖❚❖◆✲(r′ ≥ r) (r + c, r, r′) 2(c+r−r′)/2 π ✐❞❡❛❧ ❙P❖◆●❊◆❚ (r + c, r, r) 2c/2 π ✐❞❡❛❧ ❈✉❜❡❍❛s❤✲n (1024, 257, n) 1 2(1023−n)/2 P 16 ✐❞❡❛❧ ❋✉❣✉❡✲✭n ≤ 256✮ (960, 32, n) m 2(928−n)/2 π, π′ ✐❞❡❛❧ ❋✉❣✉❡✲✭n > 256✮ (1152, 32, n) m 2(1120−n)/2 π, π′ ✐❞❡❛❧ ❏❍✲n (1024, 512, n) m 2(512−n)/2 π ✐❞❡❛❧ ❑❡❝❝❛❦✲n (1600, s − 2n, n) s − 3n 2n π ✐❞❡❛❧ ▲✉✛❛✲✭n ≤ 256✮ (768, 256, 256) 2256 Q1 · · · Q3 ✐❞❡❛❧ ▲✉✛❛✲384 (1024, 256, 256) 2384 Q1 · · · Q4 ✐❞❡❛❧ ▲✉✛❛✲512 (1280, 256, 256) 2512 Q1 · · · Q5 ✐❞❡❛❧ s = ✐♥t❡r♥❛❧ st❛t❡✱ m = ♠❡ss❛❣❡ ✐♥❥❡❝t✐♦♥✱ p = ✐s ❞✐❣❡st ❡①tr❛❝t✐♦♥✱ n = ♦✉t♣✉t s✐③❡ ❋♦r ❙❍❆✲✸ ❝❛♥❞✐❞❛t❡s✿ n ∈ {224, 256, 384, 512}

▼♦♦❞② ❡t ❛❧✳ ✭✷✵✶✷✮✿ ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ ❏❍ ✉♣ t♦ q✉❡r✐❡s

❉❡s✐❣♥✲s♣❡❝✐✜❝ ♣r♦♦❢s ♠❛② r❡s✉❧t ✐♥ ❜❡tt❡r ❜♦✉♥❞s

✶✹ ✴ ✶✺

■♠♣❧✐❝❛t✐♦♥s ❢♦r ❊①✐st✐♥❣ ❉❡s✐❣♥s

❆❧❣♦r✐t❤♠ (s, m, p) d ■♥❞✐✛✳ q ≈ ❆ss✉♠♣t✐♦♥ ❙♣♦♥❣❡ (r + c, r, r) 2c/2 π ✐❞❡❛❧

• r✐♥❞❛❤❧

(s, m, n) m 2(s−m−n)/2 π ✐❞❡❛❧ ◗✉❛r❦ (r + c, r, r) 2c/2 π ✐❞❡❛❧ P❍❖❚❖◆✲(r′ ≤ r) (r + c, r, r′) r − r′ 2c/2 π ✐❞❡❛❧ P❍❖❚❖◆✲(r′ ≥ r) (r + c, r, r′) 2(c+r−r′)/2 π ✐❞❡❛❧ ❙P❖◆●❊◆❚ (r + c, r, r) 2c/2 π ✐❞❡❛❧ ❈✉❜❡❍❛s❤✲n (1024, 257, n) 1 2(1023−n)/2 P 16 ✐❞❡❛❧ ❋✉❣✉❡✲✭n ≤ 256✮ (960, 32, n) m 2(928−n)/2 π, π′ ✐❞❡❛❧ ❋✉❣✉❡✲✭n > 256✮ (1152, 32, n) m 2(1120−n)/2 π, π′ ✐❞❡❛❧ ❏❍✲n (1024, 512, n) m 2(512−n)/2 π ✐❞❡❛❧ ❑❡❝❝❛❦✲n (1600, s − 2n, n) s − 3n 2n π ✐❞❡❛❧ ▲✉✛❛✲✭n ≤ 256✮ (768, 256, 256) 2256 Q1 · · · Q3 ✐❞❡❛❧ ▲✉✛❛✲384 (1024, 256, 256) 2384 Q1 · · · Q4 ✐❞❡❛❧ ▲✉✛❛✲512 (1280, 256, 256) 2512 Q1 · · · Q5 ✐❞❡❛❧ s = ✐♥t❡r♥❛❧ st❛t❡✱ m = ♠❡ss❛❣❡ ✐♥❥❡❝t✐♦♥✱ p = ✐s ❞✐❣❡st ❡①tr❛❝t✐♦♥✱ n = ♦✉t♣✉t s✐③❡ ❋♦r ❙❍❆✲✸ ❝❛♥❞✐❞❛t❡s✿ n ∈ {224, 256, 384, 512}

• ▼♦♦❞② ❡t ❛❧✳ ✭✷✵✶✷✮✿ ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ ❏❍ ✉♣ t♦ 2256 q✉❡r✐❡s

❉❡s✐❣♥✲s♣❡❝✐✜❝ ♣r♦♦❢s ♠❛② r❡s✉❧t ✐♥ ❜❡tt❡r ❜♦✉♥❞s

✶✹ ✴ ✶✺

■♠♣❧✐❝❛t✐♦♥s ❢♦r ❊①✐st✐♥❣ ❉❡s✐❣♥s

❆❧❣♦r✐t❤♠ (s, m, p) d ■♥❞✐✛✳ q ≈ ❆ss✉♠♣t✐♦♥ ❙♣♦♥❣❡ (r + c, r, r) 2c/2 π ✐❞❡❛❧

• r✐♥❞❛❤❧

(s, m, n) m 2(s−m−n)/2 π ✐❞❡❛❧ ◗✉❛r❦ (r + c, r, r) 2c/2 π ✐❞❡❛❧ P❍❖❚❖◆✲(r′ ≤ r) (r + c, r, r′) r − r′ 2c/2 π ✐❞❡❛❧ P❍❖❚❖◆✲(r′ ≥ r) (r + c, r, r′) 2(c+r−r′)/2 π ✐❞❡❛❧ ❙P❖◆●❊◆❚ (r + c, r, r) 2c/2 π ✐❞❡❛❧ ❈✉❜❡❍❛s❤✲n (1024, 257, n) 1 2(1023−n)/2 P 16 ✐❞❡❛❧ ❋✉❣✉❡✲✭n ≤ 256✮ (960, 32, n) m 2(928−n)/2 π, π′ ✐❞❡❛❧ ❋✉❣✉❡✲✭n > 256✮ (1152, 32, n) m 2(1120−n)/2 π, π′ ✐❞❡❛❧ ❏❍✲n (1024, 512, n) m 2(512−n)/2 π ✐❞❡❛❧ ❑❡❝❝❛❦✲n (1600, s − 2n, n) s − 3n 2n π ✐❞❡❛❧ ▲✉✛❛✲✭n ≤ 256✮ (768, 256, 256) 2256 Q1 · · · Q3 ✐❞❡❛❧ ▲✉✛❛✲384 (1024, 256, 256) 2384 Q1 · · · Q4 ✐❞❡❛❧ ▲✉✛❛✲512 (1280, 256, 256) 2512 Q1 · · · Q5 ✐❞❡❛❧ s = ✐♥t❡r♥❛❧ st❛t❡✱ m = ♠❡ss❛❣❡ ✐♥❥❡❝t✐♦♥✱ p = ✐s ❞✐❣❡st ❡①tr❛❝t✐♦♥✱ n = ♦✉t♣✉t s✐③❡ ❋♦r ❙❍❆✲✸ ❝❛♥❞✐❞❛t❡s✿ n ∈ {224, 256, 384, 512}

• ▼♦♦❞② ❡t ❛❧✳ ✭✷✵✶✷✮✿ ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦❢ ❏❍ ✉♣ t♦ 2256 q✉❡r✐❡s
• ❉❡s✐❣♥✲s♣❡❝✐✜❝ ♣r♦♦❢s ♠❛② r❡s✉❧t ✐♥ ❜❡tt❡r ❜♦✉♥❞s

✶✹ ✴ ✶✺

❈♦♥❝❧✉s✐♦♥s

• P❛r❛③♦❛ ❤❛s❤ ❢✉♥❝t✐♦♥s✿ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ s♣♦♥❣❡ ❤❛s❤ ❢✉♥❝t✐♦♥s
• P❛r❛③♦❛ ❢✉♥❝t✐♦♥s ❝♦✈❡r ❛✳♦✳ s♣♦♥❣❡s✱ ●r✐♥❞❛❤❧✱ P❍❖❚❖◆✱ ❛♥❞ s❡✈❡r❛❧

❙❍❆✲✸ ❝❛♥❞✐❞❛t❡s

• P❛r❛③♦❛ ❢✉♥❝t✐♦♥s ❛r❡ ♣r♦✈❡♥ ✐♥❞✐✛❡r❡♥t✐❛❜❧❡ ❢r♦♠ ❘❖
• ❋✉rt❤❡r r❡s❡❛r❝❤
• ❚✐❣❤t♥❡ss ♦❢ t❤❡ ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❜♦✉♥❞❄
• ■♠♣r♦✈❡❞ ❝♦❧❧✐s✐♦♥✴♣r❡✐♠❛❣❡ r❡s✐st❛♥❝❡ ♦❢ t❤❡ ♣❛r❛③♦❛ ❞❡s✐❣♥❄
• ●❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❛♥✐♠❛❧✐❛ ❢✉♥❝t✐♦♥s ♦r ❡✉❦❛r②♦t❛ ❢✉♥❝t✐♦♥s❄

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✶✺ ✴ ✶✺

■♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥

❲❤❛t ❛❜♦✉t t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥❄

• ❚❤✐s ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥ ❢❛❧❧s ✇✐t❤✐♥ t❤❡ ♣❛r❛③♦❛ ❢r❛♠❡✇♦r❦

❇✉t ♣❛r❛♠❡t❡r ✱ ❛♥❞ t❤✉s

❖✉r ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② r❡s✉❧t ✐♠♣❧✐❡s ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❜♦✉♥❞

✶✻ ✴ ✶✺

■♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥

❲❤❛t ❛❜♦✉t t❤❡ ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥❄

• ❚❤✐s ✐♥s❡❝✉r❡ s♣♦♥❣❡✲❧✐❦❡ ❞❡s✐❣♥ ❢❛❧❧s ✇✐t❤✐♥ t❤❡ ♣❛r❛③♦❛ ❢r❛♠❡✇♦r❦
• ❇✉t ♣❛r❛♠❡t❡r d = s − p✱ ❛♥❞ t❤✉s s − d − p = 0

→ ❖✉r ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② r❡s✉❧t ✐♠♣❧✐❡s O(1) ✐♥❞✐✛❡r❡♥t✐❛❜✐❧✐t② ❜♦✉♥❞

✶✻ ✴ ✶✺