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  1. ❆❧❣♦r✐t❤♠ ✭ g = ✶ ❤❛♥❞❧❡✮ ❈♦♠♣✉t❡ ❛ s❤♦rt❡st ♦♥❡✲✈❡rt❡① ❝✉t ❣r❛♣❤❀ → O ( n ✷ ❧♦❣ n ) → ❆❝t✉❛❧❧②✱ O ( n ❧♦❣ n ) ✳ ❝♦✉♥t t❤❡ ❛❧❣❡❜r❛✐❝ ♥✉♠❜❡r ♦❢ ❝r♦ss✐♥❣s p ✇✐t❤ t❤❡ ✏❤♦r✐③♦♥t❛❧✑ ❝②❝❧❡ ❛♥❞ q ✇✐t❤ t❤❡ ✏✈❡rt✐❝❛❧✑ ❝②❝❧❡❀ → p , q = O ( k ) ✇❤❡r❡ k ❂❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✐♥♣✉t ♣❛t❤✳ ❣❧✉❡ p × q ❝♦♣✐❡s ♦❢ t❤❡ sq✉❛r❡❀ → O ( pqn ) = O ( k ✷ n ) ✳ → ❆❝t✉❛❧❧②✱ O ( p + q ) ❝♦♣✐❡s s✉✣❝❡✿ O ( kn ) ✳ ❝♦♠♣✉t❡ ❛ s❤♦rt❡st ♣❛t❤ ❜❡t✇❡❡♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts ♦❢ t❤❡ ❣r✐❞✳ → O ( k ✷ n ❧♦❣ ( k ✷ n )) ✭❉✐❥❦str❛✮✳ → ❆❝t✉❛❧❧②✱ O ( kn ) ✳ ❚♦t❛❧✿ O ( n ✷ ❧♦❣ n + k ✷ n ❧♦❣ kn ) ✳ ❆❝t✉❛❧❧②✿ O ( n ❧♦❣ n + kn ) ❬❈❞❱✱ ❊r✐❝❦s♦♥✱ ✷✵✵✻❪ ✰ ❬❈❞❱✱ ❏♦✉❤❡t✱ ∞ ❪ ✳

  2. ❖❦❛②✱ ❜✉t ✇❤❛t ❛❜♦✉t g ≥ ✷ ❤❛♥❞❧❡s❄ ❚❤❡ ❧✐❢ts ♦❢ ❛ s❤♦rt❡st ❝✉t ❣r❛♣❤ ❛r❡ ♥♦t s❤♦rt❡st ♣❛t❤s✳ ❲❤❛t ✐s ❛ ❣♦♦❞ r❡♣❧❛❝❡♠❡♥t ❢♦r t❤❡ s❤♦rt❡st ❝✉t ❣r❛♣❤❄ ❲❤❛t ❞♦❡s t❤❡ ✉♥✐✈❡rs❛❧ ❝♦✈❡r ❧♦♦❦ ❧✐❦❡❄

  3. ❖❦❛②✱ ❜✉t ✇❤❛t ❛❜♦✉t g ≥ ✷ ❤❛♥❞❧❡s❄ ❚❤❡ ❧✐❢ts ♦❢ ❛ s❤♦rt❡st ❝✉t ❣r❛♣❤ ❛r❡ ♥♦t s❤♦rt❡st ♣❛t❤s✳ ❲❤❛t ✐s ❛ ❣♦♦❞ r❡♣❧❛❝❡♠❡♥t ❢♦r t❤❡ s❤♦rt❡st ❝✉t ❣r❛♣❤❄ ❲❤❛t ❞♦❡s t❤❡ ✉♥✐✈❡rs❛❧ ❝♦✈❡r ❧♦♦❦ ❧✐❦❡❄

  4. ❖❦❛②✱ ❜✉t ✇❤❛t ❛❜♦✉t g ≥ ✷ ❤❛♥❞❧❡s❄ ❚❤❡ ❧✐❢ts ♦❢ ❛ s❤♦rt❡st ❝✉t ❣r❛♣❤ ❛r❡ ♥♦t s❤♦rt❡st ♣❛t❤s✳ ❲❤❛t ✐s ❛ ❣♦♦❞ r❡♣❧❛❝❡♠❡♥t ❢♦r t❤❡ s❤♦rt❡st ❝✉t ❣r❛♣❤❄ ❲❤❛t ❞♦❡s t❤❡ ✉♥✐✈❡rs❛❧ ❝♦✈❡r ❧♦♦❦ ❧✐❦❡❄ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡❛❝❤ ♣❛t❤ ❧✐❢ts t♦ ❛ s❤♦rt❡st ♣❛t❤

  5. ❖❦❛②✱ ❜✉t ✇❤❛t ❛❜♦✉t g ≥ ✷ ❤❛♥❞❧❡s❄ ❚❤❡ ❧✐❢ts ♦❢ ❛ s❤♦rt❡st ❝✉t ❣r❛♣❤ ❛r❡ ♥♦t s❤♦rt❡st ♣❛t❤s✳ ❲❤❛t ✐s ❛ ❣♦♦❞ r❡♣❧❛❝❡♠❡♥t ❢♦r t❤❡ s❤♦rt❡st ❝✉t ❣r❛♣❤❄ ❲❤❛t ❞♦❡s t❤❡ ✉♥✐✈❡rs❛❧ ❝♦✈❡r ❧♦♦❦ ❧✐❦❡❄ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡❛❝❤ ♣❛t❤ ❧✐❢ts t♦ ❛ s❤♦rt❡st ♣❛t❤

  6. ❚❡❝❤♥✐❝❛❧ ▲❡♠♠❛ ❛♥❞ ❈♦♥✈❡①✐t② ▲❡♠♠❛ ▲❡t γ ❜❡ ❛ t✐❣❤t ❝②❝❧❡✱ ❧❡t p ❜❡ ❛ ♣❛t❤ ✏✇r❛♣♣✐♥❣ ❛r♦✉♥❞✑ γ ✳ ❚❤❡♥ p ✐s t✐❣❤t✳

  7. ❚❡❝❤♥✐❝❛❧ ▲❡♠♠❛ ❛♥❞ ❈♦♥✈❡①✐t② ▲❡♠♠❛ ▲❡t γ ❜❡ ❛ t✐❣❤t ❝②❝❧❡✱ ❧❡t p ❜❡ ❛ ♣❛t❤ ✏✇r❛♣♣✐♥❣ ❛r♦✉♥❞✑ γ ✳ ❚❤❡♥ p ✐s t✐❣❤t✳ ❈♦♥✈❡①✐t② ❧✐♥❡✿ ❧✐❢t ♦❢ ❛ ❝②❝❧❡ ❤❛❧❢✲♣❧❛♥❡✿ ❞❡❧✐♠✐t❡❞ ❜② ❛ ❧✐♥❡ ❝♦♥✈❡①✿ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❤❛❧❢✲♣❧❛♥❡s t❤❡ s❤♦rt❡st ♣♦✐♥t ✇✐t❤ ❡♥❞♣♦✐♥ts ✐♥ ❛ ❝♦♥✈❡① s❡t C st❛②s ✐♥ C ✳

  8. ❚❡❝❤♥✐❝❛❧ ▲❡♠♠❛ ❛♥❞ ❈♦♥✈❡①✐t② ▲❡♠♠❛ ▲❡t γ ❜❡ ❛ t✐❣❤t ❝②❝❧❡✱ ❧❡t p ❜❡ ❛ ♣❛t❤ ✏✇r❛♣♣✐♥❣ ❛r♦✉♥❞✑ γ ✳ ❚❤❡♥ p ✐s t✐❣❤t✳ ❈♦♥✈❡①✐t② ❧✐♥❡✿ ❧✐❢t ♦❢ ❛ ❝②❝❧❡ ❤❛❧❢✲♣❧❛♥❡✿ ❞❡❧✐♠✐t❡❞ ❜② ❛ ❧✐♥❡ ❝♦♥✈❡①✿ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❤❛❧❢✲♣❧❛♥❡s t❤❡ s❤♦rt❡st ♣♦✐♥t ✇✐t❤ ❡♥❞♣♦✐♥ts ✐♥ ❛ ❝♦♥✈❡① s❡t C st❛②s ✐♥ C ✳

  9. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  10. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  11. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  12. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  13. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  14. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  15. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  16. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S

  17. ˜ ❇✉✐❧❞✐♥❣ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ♦❢ S ✱ ❞❡t❛✐❧s ■♥❝r❡♠❡♥t❛❧ ❝♦♥str✉❝t✐♦♥ ❙t❛rt ✇✐t❤ ❛ ❝♦♣② ♦❢ t❤❡ ♦❝t❛❣♦♥ ❝♦♥t❛✐♥✐♥❣ t❤❡ s♦✉r❝❡ ♦❢ ˜ p ✳ ❲❤❡♥ ˜ p ❝r♦ss❡s ❛ ♥❡✇ ❧✐♥❡✱ ❛✉❣♠❡♥t t❤❡ ❝♦♥✈❡① r❡❣✐♦♥✳ ♦❧❞ ❝♦♥✈❡① r❡❣✐♦♥

  18. ❈♦♠♣❧❡①✐t② ❙✐③❡ ♦❢ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥✿ ❤②♣❡r❜♦❧✐❝✐t② ■❢ ˜ c ❝r♦ss❡s m ✏❧✐♥❡s✑✱ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ❝♦♥t❛✐♥s O ( m ) ♦❝t❛❣♦♥s✳ Pr♦♦❢ ❆r❡❛❂❖✭♣❡r✐♠❡t❡r✮✿ ■♥❞❡❡❞✱ ✐t❡r❛t✐✈❡❧② r❡♠♦✈❡ ❛♥ ♦❝t❛❣♦♥ ❢❛rt❤❡st ❢r♦♠ t❤❡ ✏❝❡♥t❡r✑✳ ❚❤❛t ♦❝t❛❣♦♥ ❤❛s ≥ ✺ s✐❞❡s ♦♥ t❤❡ ❜♦✉♥❞❛r②✱ s♦ ❡❛❝❤ st❡♣ ❞❡❝r❡❛s❡s t❤❡ ♣❡r✐♠❡t❡r❀ ♣❡r✐♠❡t❡r ✐s O ( m ) ✱ ❜❡❝❛✉s❡✿ ❛t ♠♦st ✷ m ✢❛t ✈❡rt✐❝❡s ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ❝♦♥✈❡① r❡❣✐♦♥ ❛t ♠♦st ✻ ❝♦r♥❡r ✈❡rt✐❝❡s ❜❡t✇❡❡♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ✢❛t ✈❡rt✐❝❡s✳

  19. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥✿ t❡❝❤♥✐❝❛❧ ❧❡♠♠❛ ❆r❝✿ ♣❛t❤ ✇✐t❤ ❡♥❞♣♦✐♥ts ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ s✉r❢❛❝❡ ▲❡♠♠❛ ❖♥ ❛ s✉r❢❛❝❡✱ ❧❡t α ❜❡ ❛ t✐❣❤t s✐♠♣❧❡ ❝✉r✈❡ t❤❛t ✐s ❡✐t❤❡r ❛♥ ❛r❝ ♦r ❛ ♥♦♥✲❝♦♥tr❛❝t✐❜❧❡ ❝②❝❧❡✱ β ❜❡ ❛ s✐♠♣❧❡ ♣❛t❤ ♦r ❝②❝❧❡ ❞✐s❥♦✐♥t ❢r♦♠ α ✳ ❚❤❡♥ β ✐s t✐❣❤t ♦♥ S ✐✛ ✐t ✐s t✐❣❤t ♦♥ S \ \ α ✳

  20. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  21. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  22. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  23. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  24. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  25. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  26. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  27. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  28. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  29. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  30. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  31. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  32. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  33. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  34. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  35. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  36. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  37. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  38. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  39. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  40. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  41. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  42. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  43. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  44. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  45. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  46. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  47. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  48. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  49. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  50. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  51. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  52. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  53. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  54. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  55. ❇✉✐❧❞✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ s❦✐♣ ❖♥ ❛ s✉r❢❛❝❡ ✇✐t❤ ❝♦♠♣❧❡①✐t② n ✱ ❣❡♥✉s g ✱ ❛♥❞ b ❜♦✉♥❞❛r✐❡s✿ ✶ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❝②❝❧❡✿ O ( n ✷ ❧♦❣ n ) ✷ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❛r❝ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦❢ t❤❡ ❜♦✉♥❞❛r② ✭♦r ❜❡t✇❡❡♥ t✇♦ ❞✐st✐♥❝t ❜♦✉♥❞❛r✐❡s✮✿ O ( n ❧♦❣ n ) ❬❉✐❥❦str❛❪ ✸ ✐❢ b ≥ ✶✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ♥♦♥✲s❡♣❛r❛t✐♥❣ ❛r❝✿ O ( n ❧♦❣ n ) ❬❊r✐❝❦s♦♥✱ ❍❛r✲P❡❧❡❞✱ ✷✵✵✷❪ ✹ ✐❢ g = ✵ ❛♥❞ b ∈ { ✷ , ✸ } ✱ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ s❤♦rt❡st ❝②❝❧❡ ❤♦♠♦t♦♣✐❝ t♦ ❛ ❣✐✈❡♥ ❜♦✉♥❞❛r②✿ O ( n ❧♦❣ n ) ❬❈❞❱✱ ▲❛③❛r✉s✱ ✷✵✵✸❪ ✱ ❬❋r❡❞❡r✐❝❦s♦♥✱ ✶✾✽✼❪

  56. ❈♦♠♣❧❡①✐t② ❬❈❞❱✱ ❊r✐❝❦s♦♥✱ ✷✵✵✻❪ Pr❡♣r♦❝❡ss✐♥❣ st❡♣✿ ❈♦♠♣✉t✐♥❣ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ O ( gn ❧♦❣ n ) ✭✐♠♣r♦✈❡♠❡♥t ✉s✐♥❣ ❬❈❛❜❡❧❧♦ ❡t ❛❧✳✱ ✷✵✵✽❪ ✮✳ ❉❡t❛✐❧s✿ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ●r❛♣❤ G ❝❡❧❧✉❧❛r❧② ❡♠❜❡❞❞❡❞ ♦♥ S P❛t❤ p ✐♥ G ✇✐t❤ k ❡❞❣❡s ❖❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ✭✐♥ ❣❡♥❡r❛❧ ♣♦s✐t✐♦♥ ✇✳r✳t✳ G ∗ ✮ ❆❞♠✐tt❡❞✿ ❊❛❝❤ ❝②❝❧❡ ♦❢ t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ❡♥t❡rs O ( ✶ ) t✐♠❡s ❡✈❡r② ❢❛❝❡ ♦❢ G ∗ ✳ ❚✐❣❤t❡♥✐♥❣ ❛❧❣♦r✐t❤♠ O ( gnk ) ✱ ✇❤❡r❡ k ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✐♥♣✉t ♣❛t❤✱ ❜❡❝❛✉s❡ p ❝r♦ss❡s t❤❡ ♦❝t❛❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ O ( gk ) t✐♠❡s✱ ❡❛❝❤ ♦❝t❛❣♦♥ ❤❛s ❝♦♠♣❧❡①✐t② O ( n ) ✳ ❊①t❡♥s✐♦♥✿ ❝❧♦s❡❞ ❝✉r✈❡s ✭❛❧❧♦✇✐♥❣ t❤❡ ❜❛s❡♣♦✐♥t t♦ ♠♦✈❡✮✳

  57. ▼✐♥✐♠✉♠ ❝✉t ❛❧❣♦r✐t❤♠

  58. Pr♦❜❧❡♠ ❚❤❡ ♠✐♥✐♠✉♠ ❝✉t ♣r♦❜❧❡♠ ●✐✈❡♥ G = ( V , E ) ✿ ❛ ✇❡✐❣❤t❡❞✱ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤❀ s , t ✿ t✇♦ ✈❡rt✐❝❡s ♦❢ G ✱ ❝♦♠♣✉t❡ W ⊂ V ❝♦♥t❛✐♥✐♥❣ s ❜✉t ♥♦t t t❤❛t ♠✐♥✐♠✐③❡s t❤❡ s✉♠ ♦❢ t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ❡❞❣❡s ❜❡t✇❡❡♥ W ❛♥❞ V \ W ✳ ❚❤❡♦r❡♠ ❬❈❤❛♠❜❡rs✱ ❊r✐❝❦s♦♥✱ ◆❛②②❡r✐✱ ✷✵✵✾❪ ■❢ G ✐s ❡♠❜❡❞❞❡❞ ♦♥ ❛ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s g ✱ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ O ( g O ( g ) n ❧♦❣ n ) t✐♠❡✳ ❇❡st r❡s✉❧t ❦♥♦✇♥ ❜❡❢♦r❡ ❆❧❣♦r✐t❤♠s ❢♦r s♣❛rs❡ ❣r❛♣❤s ✐♥ O ( n ✷ ❧♦❣ n ) ❬❙❧❡❛t♦r✱ ❚❛r❥❛♥✱ ✶✾✽✸❪ ❛♥❞ O ( n ✸ / ✷ ❧♦❣ n ❧♦❣ C ) ❬●♦❧❞❜❡r❣✱ ❘❛♦✱ ✶✾✾✽❪ ✳

  59. Pr♦❜❧❡♠ ❚❤❡ ♠✐♥✐♠✉♠ ❝✉t ♣r♦❜❧❡♠ ●✐✈❡♥ G = ( V , E ) ✿ ❛ ✇❡✐❣❤t❡❞✱ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤❀ s , t ✿ t✇♦ ✈❡rt✐❝❡s ♦❢ G ✱ ❝♦♠♣✉t❡ W ⊂ V ❝♦♥t❛✐♥✐♥❣ s ❜✉t ♥♦t t t❤❛t ♠✐♥✐♠✐③❡s t❤❡ s✉♠ ♦❢ t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ❡❞❣❡s ❜❡t✇❡❡♥ W ❛♥❞ V \ W ✳ ❚❤❡♦r❡♠ ❬❈❤❛♠❜❡rs✱ ❊r✐❝❦s♦♥✱ ◆❛②②❡r✐✱ ✷✵✵✾❪ ■❢ G ✐s ❡♠❜❡❞❞❡❞ ♦♥ ❛ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s g ✱ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ O ( g O ( g ) n ❧♦❣ n ) t✐♠❡✳ ❇❡st r❡s✉❧t ❦♥♦✇♥ ❜❡❢♦r❡ ❆❧❣♦r✐t❤♠s ❢♦r s♣❛rs❡ ❣r❛♣❤s ✐♥ O ( n ✷ ❧♦❣ n ) ❬❙❧❡❛t♦r✱ ❚❛r❥❛♥✱ ✶✾✽✸❪ ❛♥❞ O ( n ✸ / ✷ ❧♦❣ n ❧♦❣ C ) ❬●♦❧❞❜❡r❣✱ ❘❛♦✱ ✶✾✾✽❪ ✳

  60. Pr♦❜❧❡♠ ❚❤❡ ♠✐♥✐♠✉♠ ❝✉t ♣r♦❜❧❡♠ ●✐✈❡♥ G = ( V , E ) ✿ ❛ ✇❡✐❣❤t❡❞✱ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤❀ s , t ✿ t✇♦ ✈❡rt✐❝❡s ♦❢ G ✱ ❝♦♠♣✉t❡ W ⊂ V ❝♦♥t❛✐♥✐♥❣ s ❜✉t ♥♦t t t❤❛t ♠✐♥✐♠✐③❡s t❤❡ s✉♠ ♦❢ t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ❡❞❣❡s ❜❡t✇❡❡♥ W ❛♥❞ V \ W ✳ ❚❤❡♦r❡♠ ❬❈❤❛♠❜❡rs✱ ❊r✐❝❦s♦♥✱ ◆❛②②❡r✐✱ ✷✵✵✾❪ ■❢ G ✐s ❡♠❜❡❞❞❡❞ ♦♥ ❛ s✉r❢❛❝❡ ♦❢ ❣❡♥✉s g ✱ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ O ( g O ( g ) n ❧♦❣ n ) t✐♠❡✳ ❇❡st r❡s✉❧t ❦♥♦✇♥ ❜❡❢♦r❡ ❆❧❣♦r✐t❤♠s ❢♦r s♣❛rs❡ ❣r❛♣❤s ✐♥ O ( n ✷ ❧♦❣ n ) ❬❙❧❡❛t♦r✱ ❚❛r❥❛♥✱ ✶✾✽✸❪ ❛♥❞ O ( n ✸ / ✷ ❧♦❣ n ❧♦❣ C ) ❬●♦❧❞❜❡r❣✱ ❘❛♦✱ ✶✾✾✽❪ ✳

  61. Pr❡❧✐♠✐♥❛r✐❡s✿ ❝❛s❡ ♦❢ t❤❡ ♣❧❛♥❡ ✭❂♦❢ t❤❡ s♣❤❡r❡✮ ▼✐♥✐♠✉♠ ❝✉t ✐♥ G ↔ ❝②❝❧❡ γ s❡♣❛r❛t✐♥❣ s ❢r♦♠ t ✱ ♦❢ ♠✐♥✐♠✉♠ ❧❡♥❣t❤ ✐♥ G ∗ ✱ ♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ γ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O ( n ❧♦❣ n ) t✐♠❡ ❬❘❡✐❢ ✶✾✽✸❀ ❋r❡❞❡r✐❝❦s♦♥ ✶✾✽✼❀ ❍❡♥③✐♥❣❡r ❡t ❛❧✳ ✶✾✾✼❪ ✳

  62. Pr❡❧✐♠✐♥❛r✐❡s✿ ❝❛s❡ ♦❢ t❤❡ ♣❧❛♥❡ ✭❂♦❢ t❤❡ s♣❤❡r❡✮ ▼✐♥✐♠✉♠ ❝✉t ✐♥ G ↔ ❝②❝❧❡ γ s❡♣❛r❛t✐♥❣ s ❢r♦♠ t ✱ ♦❢ ♠✐♥✐♠✉♠ ❧❡♥❣t❤ ✐♥ G ∗ ✱ ♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ γ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O ( n ❧♦❣ n ) t✐♠❡ ❬❘❡✐❢ ✶✾✽✸❀ ❋r❡❞❡r✐❝❦s♦♥ ✶✾✽✼❀ ❍❡♥③✐♥❣❡r ❡t ❛❧✳ ✶✾✾✼❪ ✳

  63. Pr❡❧✐♠✐♥❛r✐❡s✿ ❝❛s❡ ♦❢ t❤❡ ♣❧❛♥❡ ✭❂♦❢ t❤❡ s♣❤❡r❡✮ ▼✐♥✐♠✉♠ ❝✉t ✐♥ G ↔ ❝②❝❧❡ γ s❡♣❛r❛t✐♥❣ s ❢r♦♠ t ✱ ♦❢ ♠✐♥✐♠✉♠ ❧❡♥❣t❤ ✐♥ G ∗ ✱ ♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ γ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O ( n ❧♦❣ n ) t✐♠❡ ❬❘❡✐❢ ✶✾✽✸❀ ❋r❡❞❡r✐❝❦s♦♥ ✶✾✽✼❀ ❍❡♥③✐♥❣❡r ❡t ❛❧✳ ✶✾✾✼❪ ✳

  64. Pr❡❧✐♠✐♥❛r✐❡s✿ ❝❛s❡ ♦❢ t❤❡ ♣❧❛♥❡ ✭❂♦❢ t❤❡ s♣❤❡r❡✮ ▼✐♥✐♠✉♠ ❝✉t ✐♥ G ↔ ❝②❝❧❡ γ s❡♣❛r❛t✐♥❣ s ❢r♦♠ t ✱ ♦❢ ♠✐♥✐♠✉♠ ❧❡♥❣t❤ ✐♥ G ∗ ✱ ♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ γ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O ( n ❧♦❣ n ) t✐♠❡ ❬❘❡✐❢ ✶✾✽✸❀ ❋r❡❞❡r✐❝❦s♦♥ ✶✾✽✼❀ ❍❡♥③✐♥❣❡r ❡t ❛❧✳ ✶✾✾✼❪ ✳

  65. Pr❡❧✐♠✐♥❛r✐❡s✿ ❝❛s❡ ♦❢ t❤❡ ♣❧❛♥❡ ✭❂♦❢ t❤❡ s♣❤❡r❡✮ ▼✐♥✐♠✉♠ ❝✉t ✐♥ G ↔ ❝②❝❧❡ γ s❡♣❛r❛t✐♥❣ s ❢r♦♠ t ✱ ♦❢ ♠✐♥✐♠✉♠ ❧❡♥❣t❤ ✐♥ G ∗ ✱ ♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ γ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O ( n ❧♦❣ n ) t✐♠❡ ❬❘❡✐❢ ✶✾✽✸❀ ❋r❡❞❡r✐❝❦s♦♥ ✶✾✽✼❀ ❍❡♥③✐♥❣❡r ❡t ❛❧✳ ✶✾✾✼❪ ✳

  66. Pr❡❧✐♠✐♥❛r✐❡s✿ ❝❛s❡ ♦❢ t❤❡ ♣❧❛♥❡ ✭❂♦❢ t❤❡ s♣❤❡r❡✮ ▼✐♥✐♠✉♠ ❝✉t ✐♥ G ↔ ❝②❝❧❡ γ s❡♣❛r❛t✐♥❣ s ❢r♦♠ t ✱ ♦❢ ♠✐♥✐♠✉♠ ❧❡♥❣t❤ ✐♥ G ∗ ✱ ♦r ❡q✉✐✈❛❧❡♥t❧② ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ γ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ O ( n ❧♦❣ n ) t✐♠❡ ❬❘❡✐❢ ✶✾✽✸❀ ❋r❡❞❡r✐❝❦s♦♥ ✶✾✽✼❀ ❍❡♥③✐♥❣❡r ❡t ❛❧✳ ✶✾✾✼❪ ✳

  67. ❈❛s❡ ♦❢ s✉r❢❛❝❡s t s ❈✉t ✐♥ G → ❢❛♠✐❧② Γ ♦❢ ❞✐s❥♦✐♥t ❝②❝❧❡s ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ ❈♦♠♣✉t❡ ❛ ❝✉t ❣r❛♣❤ ❜❛s❡❞ ❛t s ✱ ♦❜t❛✐♥✐♥❣ ❧♦♦♣s ℓ ✶ , . . . , ℓ ✷ g ✳ ▲❡t p ❜❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✳ Γ ✐s ❛ ❝✉t ✐✛ ✐t ❝r♦ss❡s ❡✈❡r② ℓ i ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ t✐♠❡s ❛♥❞ p ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

  68. ❈❛s❡ ♦❢ s✉r❢❛❝❡s t s ❈✉t ✐♥ G → ❢❛♠✐❧② Γ ♦❢ ❞✐s❥♦✐♥t ❝②❝❧❡s ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ ❈♦♠♣✉t❡ ❛ ❝✉t ❣r❛♣❤ ❜❛s❡❞ ❛t s ✱ ♦❜t❛✐♥✐♥❣ ❧♦♦♣s ℓ ✶ , . . . , ℓ ✷ g ✳ ▲❡t p ❜❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✳ Γ ✐s ❛ ❝✉t ✐✛ ✐t ❝r♦ss❡s ❡✈❡r② ℓ i ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ t✐♠❡s ❛♥❞ p ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

  69. ❈❛s❡ ♦❢ s✉r❢❛❝❡s ℓ ✶ t s ℓ ✷ ❈✉t ✐♥ G → ❢❛♠✐❧② Γ ♦❢ ❞✐s❥♦✐♥t ❝②❝❧❡s ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ ❈♦♠♣✉t❡ ❛ ❝✉t ❣r❛♣❤ ❜❛s❡❞ ❛t s ✱ ♦❜t❛✐♥✐♥❣ ❧♦♦♣s ℓ ✶ , . . . , ℓ ✷ g ✳ ▲❡t p ❜❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✳ Γ ✐s ❛ ❝✉t ✐✛ ✐t ❝r♦ss❡s ❡✈❡r② ℓ i ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ t✐♠❡s ❛♥❞ p ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

  70. ❈❛s❡ ♦❢ s✉r❢❛❝❡s ℓ ✶ t s p ℓ ✷ ❈✉t ✐♥ G → ❢❛♠✐❧② Γ ♦❢ ❞✐s❥♦✐♥t ❝②❝❧❡s ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ ❈♦♠♣✉t❡ ❛ ❝✉t ❣r❛♣❤ ❜❛s❡❞ ❛t s ✱ ♦❜t❛✐♥✐♥❣ ❧♦♦♣s ℓ ✶ , . . . , ℓ ✷ g ✳ ▲❡t p ❜❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✳ Γ ✐s ❛ ❝✉t ✐✛ ✐t ❝r♦ss❡s ❡✈❡r② ℓ i ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ t✐♠❡s ❛♥❞ p ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

  71. ❈❛s❡ ♦❢ s✉r❢❛❝❡s ℓ ✶ t s p ℓ ✷ ❈✉t ✐♥ G → ❢❛♠✐❧② Γ ♦❢ ❞✐s❥♦✐♥t ❝②❝❧❡s ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ ❈♦♠♣✉t❡ ❛ ❝✉t ❣r❛♣❤ ❜❛s❡❞ ❛t s ✱ ♦❜t❛✐♥✐♥❣ ❧♦♦♣s ℓ ✶ , . . . , ℓ ✷ g ✳ ▲❡t p ❜❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✳ Γ ✐s ❛ ❝✉t ✐✛ ✐t ❝r♦ss❡s ❡✈❡r② ℓ i ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ t✐♠❡s ❛♥❞ p ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

  72. ❈❛s❡ ♦❢ s✉r❢❛❝❡s ℓ ✶ ℓ ✶ ℓ ✷ t p s p ℓ ✶ p ℓ ✷ ℓ ✷ ❈✉t ✐♥ G → ❢❛♠✐❧② Γ ♦❢ ❞✐s❥♦✐♥t ❝②❝❧❡s ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ ❈♦♠♣✉t❡ ❛ ❝✉t ❣r❛♣❤ ❜❛s❡❞ ❛t s ✱ ♦❜t❛✐♥✐♥❣ ❧♦♦♣s ℓ ✶ , . . . , ℓ ✷ g ✳ ▲❡t p ❜❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✳ Γ ✐s ❛ ❝✉t ✐✛ ✐t ❝r♦ss❡s ❡✈❡r② ℓ i ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ t✐♠❡s ❛♥❞ p ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

  73. ❈❛s❡ ♦❢ s✉r❢❛❝❡s ℓ ✶ ℓ ✶ ℓ ✷ t p s p ℓ ✶ p ℓ ✷ ℓ ✷ ❈✉t ✐♥ G → ❢❛♠✐❧② Γ ♦❢ ❞✐s❥♦✐♥t ❝②❝❧❡s ✐♥ t❤❡ ❝r♦ss✲♠❡tr✐❝ s✉r❢❛❝❡ ❞❡✜♥❡❞ ❜② G ✳ ❈♦♠♣✉t❡ ❛ ❝✉t ❣r❛♣❤ ❜❛s❡❞ ❛t s ✱ ♦❜t❛✐♥✐♥❣ ❧♦♦♣s ℓ ✶ , . . . , ℓ ✷ g ✳ ▲❡t p ❜❡ ❛ s❤♦rt❡st ♣❛t❤ ❢r♦♠ s t♦ t ✳ Γ ✐s ❛ ❝✉t ✐✛ ✐t ❝r♦ss❡s ❡✈❡r② ℓ i ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ t✐♠❡s ❛♥❞ p ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳

  74. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p p ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  75. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p p ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  76. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p p ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  77. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p p ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  78. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p p ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  79. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p p ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  80. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p ℓ ✶ p ℓ ✶ ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  81. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p ℓ ✶ ℓ ✶ p ℓ ✶ ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  82. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p ℓ ✶ ℓ ✷ ℓ ✶ p ℓ ✶ ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  83. ❆❧❣♦r✐t❤♠ ℓ ✶ ℓ ✷ p ℓ ✶ ℓ ✷ ℓ ✶ p ℓ ✶ ℓ ✶ ℓ ✷ ▲❡♠♠❛ ✳ ■❢ Γ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠✐♥✐♠✉♠ ❝✉t✱ t❤❡♥ ✐t ❝r♦ss❡s O ( g ) t✐♠❡s p ❛♥❞ ❡❛❝❤ ℓ i ✳ ❊♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ ♣❛tt❡r♥s t❤❛t ❢♦r♠ ❞✐s❥♦✐♥t ❝②❝❧❡s✳ ❚❤❡r❡ ❛r❡ g O ( g ) ♣♦ss✐❜✐❧✐t✐❡s✳ ❋♦r ❡❛❝❤ ♣❛tt❡r♥✱ ❛♥❞ ❡❛❝❤ ❝②❝❧❡ ❛♣♣❡❛r✐♥❣ ✐♥ t❤✐s ♣❛tt❡r♥✱ ❝♦♠♣✉t❡ t❤❡ s❤♦rt❡st ❝②❝❧❡ t❤❛t ❝r♦ss❡s p ❛♥❞ t❤❡ ℓ i ✐♥ t❤❡ s❛♠❡ ♦r❞❡r✳ ❚❤✐s ❜♦✐❧s ❞♦✇♥ t♦ t❤❡ ♣❧❛♥❛r ❝❛s❡✦ ❈♦♠♣❧❡①✐t② O ( g ✷ n ❧♦❣ n ) ✳ ❚♦t❛❧ r✉♥♥✐♥❣ t✐♠❡✿ O ( g O ( g ) n ❧♦❣ n ) ✳

  84. Pr♦♦❢ ♦❢ t❤❡ ❧❡♠♠❛ ▲❡♠♠❛ Γ ❤❛s O ( g ) ❝r♦ss✐♥❣s ✇✐t❤ p ❛♥❞ ✇✐t❤ ❡❛❝❤ ♦❢ t❤❡ ℓ i ✳ t ∗ s ∗ ▲❡t ✉s s❤r✐♥❦ t❤❡ s❤♦rt❡st ♣❛t❤ t♦ ❛ s✐♥❣❧❡ ✈❡rt❡① v ✿ Γ ❜❡❝♦♠❡s ❛ ❢❛♠✐❧② ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ❧♦♦♣s ❜❛s❡❞ ❛t v ✱ s✉❝❤ t❤❛t ♥♦ ❢❛❝❡ ♦❢ t❤✐s ❣r❛♣❤ ✐s ❛ ❞✐s❦ ❜♦✉♥❞❡❞ ❜② ♦♥❡ ♦r t✇♦ ❧♦♦♣s✳

  85. Pr♦♦❢ ♦❢ t❤❡ ❧❡♠♠❛ ▲❡♠♠❛ Γ ❤❛s O ( g ) ❝r♦ss✐♥❣s ✇✐t❤ p ❛♥❞ ✇✐t❤ ❡❛❝❤ ♦❢ t❤❡ ℓ i ✳ t ∗ s ∗ ▲❡t ✉s s❤r✐♥❦ t❤❡ s❤♦rt❡st ♣❛t❤ t♦ ❛ s✐♥❣❧❡ ✈❡rt❡① v ✿ Γ ❜❡❝♦♠❡s ❛ ❢❛♠✐❧② ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ❧♦♦♣s ❜❛s❡❞ ❛t v ✱ s✉❝❤ t❤❛t ♥♦ ❢❛❝❡ ♦❢ t❤✐s ❣r❛♣❤ ✐s ❛ ❞✐s❦ ❜♦✉♥❞❡❞ ❜② ♦♥❡ ♦r t✇♦ ❧♦♦♣s✳

  86. Pr♦♦❢ ♦❢ t❤❡ ❧❡♠♠❛ ▲❡♠♠❛ Γ ❤❛s O ( g ) ❝r♦ss✐♥❣s ✇✐t❤ p ❛♥❞ ✇✐t❤ ❡❛❝❤ ♦❢ t❤❡ ℓ i ✳ t ∗ s ∗ ▲❡t ✉s s❤r✐♥❦ t❤❡ s❤♦rt❡st ♣❛t❤ t♦ ❛ s✐♥❣❧❡ ✈❡rt❡① v ✿ Γ ❜❡❝♦♠❡s ❛ ❢❛♠✐❧② ♦❢ ♥♦♥✲❝r♦ss✐♥❣ ❧♦♦♣s ❜❛s❡❞ ❛t v ✱ s✉❝❤ t❤❛t ♥♦ ❢❛❝❡ ♦❢ t❤✐s ❣r❛♣❤ ✐s ❛ ❞✐s❦ ❜♦✉♥❞❡❞ ❜② ♦♥❡ ♦r t✇♦ ❧♦♦♣s✳

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