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❊①❛♠♣❧❡✿ ■❢ ❛♥❞ t❤❡♥ ■❢ ✐s ❧❛r❣❡ t❤❡♥ ✇❡ ❝❛♥ ♦♠✐t ✳ ❊①❛♠♣❧❡✿ ■❢ t❤❡♥ ✳ ❊①❛♠♣❧❡✿ ■❢ t❤❡♥ ✳ Pr❛❧❧❡❧✐③❛t✐♦♥ ♦❢ s❡q✉❡♥t✐❛❧ ❝♦❞❡ ■■ � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 2 t❤❡♥ S = = 1 . 25 1 (1 − 1 2 )+ 2 2 ✺ ✴ ✼✼

■❢ ✐s ❧❛r❣❡ t❤❡♥ ✇❡ ❝❛♥ ♦♠✐t ✳ ❊①❛♠♣❧❡✿ ■❢ t❤❡♥ ✳ ❊①❛♠♣❧❡✿ ■❢ t❤❡♥ ✳ Pr❛❧❧❡❧✐③❛t✐♦♥ ♦❢ s❡q✉❡♥t✐❛❧ ❝♦❞❡ ■■ � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 2 t❤❡♥ S = = 1 . 25 1 (1 − 1 2 )+ 2 2 � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 20 t❤❡♥ S = ≈ 1 . 904 1 (1 − 1 2 )+ 2 20 ✺ ✴ ✼✼

❊①❛♠♣❧❡✿ ■❢ t❤❡♥ ✳ ❊①❛♠♣❧❡✿ ■❢ t❤❡♥ ✳ Pr❛❧❧❡❧✐③❛t✐♦♥ ♦❢ s❡q✉❡♥t✐❛❧ ❝♦❞❡ ■■ � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 2 t❤❡♥ S = = 1 . 25 1 (1 − 1 2 )+ 2 2 � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 20 t❤❡♥ S = ≈ 1 . 904 1 (1 − 1 2 )+ 2 20 � ■❢ N ✐s ❧❛r❣❡ t❤❡♥ ✇❡ ❝❛♥ ♦♠✐t P N ✳ ✺ ✴ ✼✼

❊①❛♠♣❧❡✿ ■❢ t❤❡♥ ✳ Pr❛❧❧❡❧✐③❛t✐♦♥ ♦❢ s❡q✉❡♥t✐❛❧ ❝♦❞❡ ■■ � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 2 t❤❡♥ S = = 1 . 25 1 (1 − 1 2 )+ 2 2 � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 20 t❤❡♥ S = ≈ 1 . 904 1 (1 − 1 2 )+ 2 20 � ■❢ N ✐s ❧❛r❣❡ t❤❡♥ ✇❡ ❝❛♥ ♦♠✐t P N ✳ � ❊①❛♠♣❧❡✿ ■❢ P = 3 1 4 t❤❡♥ S = 4 ) = 4 ✳ (1 − 3 ✺ ✴ ✼✼

Pr❛❧❧❡❧✐③❛t✐♦♥ ♦❢ s❡q✉❡♥t✐❛❧ ❝♦❞❡ ■■ � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 2 t❤❡♥ S = = 1 . 25 1 (1 − 1 2 )+ 2 2 � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 2 ❛♥❞ N = 20 t❤❡♥ S = ≈ 1 . 904 1 (1 − 1 2 )+ 2 20 � ■❢ N ✐s ❧❛r❣❡ t❤❡♥ ✇❡ ❝❛♥ ♦♠✐t P N ✳ � ❊①❛♠♣❧❡✿ ■❢ P = 3 1 4 t❤❡♥ S = 4 ) = 4 ✳ (1 − 3 � ❊①❛♠♣❧❡✿ ■❢ P = 1 1 6 ) = 6 6 t❤❡♥ S = 5 = 1 . 2 ✳ (1 − 1 ✺ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✻ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✼ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

❖♥❡ ♦❢ ♣♦ss✐❜❧❡ ♦♣t✐♠✐③❛t✐♦♥s ✕ ♠❛♣ ♠❛♣ ♠❛♣ ❆❧s♦ ❛♥ ✐❞❡❛ ❢♦r ❧♦♦♣s ♣❛r❛❧❧❡❧✐③❛t✐♦♥ ✭✐❢ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✳ ■♥ ❈❯❉❆ ♠✉st ❜❡ ❞❡✜♥❡❞ ❛s ❴❴❞❡✈✐❝❡❴❴ ❢✉♥❝t✐♦♥✳ ❈❯❉❆ s✉♣♣♦rts ✷❞ ❛♥❞ ✸❞ ❛rr❛②s ♦❢ t❤r❡❛❞s✳ ✳ ✳ ✳ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ♠✉st ❜❡ s✐♠✉❧❛t❡❞✳ ▼❛♣ ❉❡✜♥✐t✐♦♥ ✭▼❛♣✮ ❚❤❡ ♠❛♣ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❢✉♥❝t✐♦♥ F ✭✇❡❧❧ ❞❡✜♥❡❞ ❢♦r ❣✐✈❡♥ ✐♥♣✉t ✈❛❧✉❡s✮ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ F ( x 0 ) , F ( x 1 ) , . . . , F ( x n − 1 )] . � ❚❤✐s t❛s❦ ✐s ♦♥❡ ♦❢ ❡♠❜❛rr❛ss✐♥❣❧② ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠s✳ ✽ ✴ ✼✼

❆❧s♦ ❛♥ ✐❞❡❛ ❢♦r ❧♦♦♣s ♣❛r❛❧❧❡❧✐③❛t✐♦♥ ✭✐❢ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✳ ■♥ ❈❯❉❆ ♠✉st ❜❡ ❞❡✜♥❡❞ ❛s ❴❴❞❡✈✐❝❡❴❴ ❢✉♥❝t✐♦♥✳ ❈❯❉❆ s✉♣♣♦rts ✷❞ ❛♥❞ ✸❞ ❛rr❛②s ♦❢ t❤r❡❛❞s✳ ✳ ✳ ✳ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ♠✉st ❜❡ s✐♠✉❧❛t❡❞✳ ▼❛♣ ❉❡✜♥✐t✐♦♥ ✭▼❛♣✮ ❚❤❡ ♠❛♣ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❢✉♥❝t✐♦♥ F ✭✇❡❧❧ ❞❡✜♥❡❞ ❢♦r ❣✐✈❡♥ ✐♥♣✉t ✈❛❧✉❡s✮ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ F ( x 0 ) , F ( x 1 ) , . . . , F ( x n − 1 )] . � ❚❤✐s t❛s❦ ✐s ♦♥❡ ♦❢ ❡♠❜❛rr❛ss✐♥❣❧② ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠s✳ � ❖♥❡ ♦❢ ♣♦ss✐❜❧❡ ♦♣t✐♠✐③❛t✐♦♥s ✕ ♠❛♣ ( F ) ◦ ♠❛♣ ( G ) = ♠❛♣ ( F ◦ G ) ✽ ✴ ✼✼

■♥ ❈❯❉❆ ♠✉st ❜❡ ❞❡✜♥❡❞ ❛s ❴❴❞❡✈✐❝❡❴❴ ❢✉♥❝t✐♦♥✳ ❈❯❉❆ s✉♣♣♦rts ✷❞ ❛♥❞ ✸❞ ❛rr❛②s ♦❢ t❤r❡❛❞s✳ ✳ ✳ ✳ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ♠✉st ❜❡ s✐♠✉❧❛t❡❞✳ ▼❛♣ ❉❡✜♥✐t✐♦♥ ✭▼❛♣✮ ❚❤❡ ♠❛♣ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❢✉♥❝t✐♦♥ F ✭✇❡❧❧ ❞❡✜♥❡❞ ❢♦r ❣✐✈❡♥ ✐♥♣✉t ✈❛❧✉❡s✮ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ F ( x 0 ) , F ( x 1 ) , . . . , F ( x n − 1 )] . � ❚❤✐s t❛s❦ ✐s ♦♥❡ ♦❢ ❡♠❜❛rr❛ss✐♥❣❧② ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠s✳ � ❖♥❡ ♦❢ ♣♦ss✐❜❧❡ ♦♣t✐♠✐③❛t✐♦♥s ✕ ♠❛♣ ( F ) ◦ ♠❛♣ ( G ) = ♠❛♣ ( F ◦ G ) � ❆❧s♦ ❛♥ ✐❞❡❛ ❢♦r ❧♦♦♣s ♣❛r❛❧❧❡❧✐③❛t✐♦♥ ✭✐❢ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✳ ✽ ✴ ✼✼

❈❯❉❆ s✉♣♣♦rts ✷❞ ❛♥❞ ✸❞ ❛rr❛②s ♦❢ t❤r❡❛❞s✳ ✳ ✳ ✳ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ♠✉st ❜❡ s✐♠✉❧❛t❡❞✳ ▼❛♣ ❉❡✜♥✐t✐♦♥ ✭▼❛♣✮ ❚❤❡ ♠❛♣ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❢✉♥❝t✐♦♥ F ✭✇❡❧❧ ❞❡✜♥❡❞ ❢♦r ❣✐✈❡♥ ✐♥♣✉t ✈❛❧✉❡s✮ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ F ( x 0 ) , F ( x 1 ) , . . . , F ( x n − 1 )] . � ❚❤✐s t❛s❦ ✐s ♦♥❡ ♦❢ ❡♠❜❛rr❛ss✐♥❣❧② ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠s✳ � ❖♥❡ ♦❢ ♣♦ss✐❜❧❡ ♦♣t✐♠✐③❛t✐♦♥s ✕ ♠❛♣ ( F ) ◦ ♠❛♣ ( G ) = ♠❛♣ ( F ◦ G ) � ❆❧s♦ ❛♥ ✐❞❡❛ ❢♦r ❧♦♦♣s ♣❛r❛❧❧❡❧✐③❛t✐♦♥ ✭✐❢ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✳ � ■♥ ❈❯❉❆ F ♠✉st ❜❡ ❞❡✜♥❡❞ ❛s ❴❴❞❡✈✐❝❡❴❴ ❢✉♥❝t✐♦♥✳ ✽ ✴ ✼✼

✳ ✳ ✳ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ♠✉st ❜❡ s✐♠✉❧❛t❡❞✳ ▼❛♣ ❉❡✜♥✐t✐♦♥ ✭▼❛♣✮ ❚❤❡ ♠❛♣ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❢✉♥❝t✐♦♥ F ✭✇❡❧❧ ❞❡✜♥❡❞ ❢♦r ❣✐✈❡♥ ✐♥♣✉t ✈❛❧✉❡s✮ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ F ( x 0 ) , F ( x 1 ) , . . . , F ( x n − 1 )] . � ❚❤✐s t❛s❦ ✐s ♦♥❡ ♦❢ ❡♠❜❛rr❛ss✐♥❣❧② ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠s✳ � ❖♥❡ ♦❢ ♣♦ss✐❜❧❡ ♦♣t✐♠✐③❛t✐♦♥s ✕ ♠❛♣ ( F ) ◦ ♠❛♣ ( G ) = ♠❛♣ ( F ◦ G ) � ❆❧s♦ ❛♥ ✐❞❡❛ ❢♦r ❧♦♦♣s ♣❛r❛❧❧❡❧✐③❛t✐♦♥ ✭✐❢ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✳ � ■♥ ❈❯❉❆ F ♠✉st ❜❡ ❞❡✜♥❡❞ ❛s ❴❴❞❡✈✐❝❡❴❴ ❢✉♥❝t✐♦♥✳ � ❈❯❉❆ s✉♣♣♦rts ✷❞ ❛♥❞ ✸❞ ❛rr❛②s ♦❢ t❤r❡❛❞s✳ ✽ ✴ ✼✼

▼❛♣ ❉❡✜♥✐t✐♦♥ ✭▼❛♣✮ ❚❤❡ ♠❛♣ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❢✉♥❝t✐♦♥ F ✭✇❡❧❧ ❞❡✜♥❡❞ ❢♦r ❣✐✈❡♥ ✐♥♣✉t ✈❛❧✉❡s✮ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ F ( x 0 ) , F ( x 1 ) , . . . , F ( x n − 1 )] . � ❚❤✐s t❛s❦ ✐s ♦♥❡ ♦❢ ❡♠❜❛rr❛ss✐♥❣❧② ♣❛r❛❧❧❡❧ ♣r♦❜❧❡♠s✳ � ❖♥❡ ♦❢ ♣♦ss✐❜❧❡ ♦♣t✐♠✐③❛t✐♦♥s ✕ ♠❛♣ ( F ) ◦ ♠❛♣ ( G ) = ♠❛♣ ( F ◦ G ) � ❆❧s♦ ❛♥ ✐❞❡❛ ❢♦r ❧♦♦♣s ♣❛r❛❧❧❡❧✐③❛t✐♦♥ ✭✐❢ s✉❜s❡q✉❡♥t ✐t❡r❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✮✳ � ■♥ ❈❯❉❆ F ♠✉st ❜❡ ❞❡✜♥❡❞ ❛s ❴❴❞❡✈✐❝❡❴❴ ❢✉♥❝t✐♦♥✳ � ❈❯❉❆ s✉♣♣♦rts ✷❞ ❛♥❞ ✸❞ ❛rr❛②s ♦❢ t❤r❡❛❞s✳ � ✳ ✳ ✳ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ♠✉st ❜❡ s✐♠✉❧❛t❡❞✳ ✽ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✾ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

✶ ✈♦✐❞ r✉♥❴s✉♠❴♥❛✐✈❡✭❢❧♦❛t ✯❞❴✐❞❛t❛✱ ❢❧♦❛t ✯❞❴♦❞❛t❛✱ ✐♥t ♥✉♠❴❡❧❡♠❡♥ts✮ ✷ ④ ❞✐♠✸ t❤r❡❛❞s✭♥✉♠❴❡❧❡♠❡♥ts✴✷✮❀ ✸ ❞✐♠✸ ❣r✐❞✭✶✱✶✱✶✮❀ ✹ ✺ s✉♠❴♥❛✐✈❡❁❁❁ ❣r✐❞✱ t❤r❡❛❞s✱ ♥✉♠❴❡❧❡♠❡♥ts✯s✐③❡♦❢✭❢❧♦❛t✮ ❃❃❃✭❞❴✐❞❛t❛✱ ❞❴♦❞❛t❛✱ ✻ ♥✉♠❴❡❧❡♠❡♥ts✮❀ ✼ ⑥ ❚❛s❦✿ ❲r✐t❡ ❛♥ ❛❧❣♦r✐t❤♠ ❝❛❧❝✉❧❛t✐♥❣ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❣✐✈❡♥ ❛rr❛② ❚❤✐s t❛s❦ s❡❡♠s t♦ ❜❡ s❡q✉❡♥t✐❛❧ ✭✇✐t❤ o ( n ) ❝♦♠♣❧❡①✐t②✮ ❋✐❣✉r❡✿ ❆❧❣♦r✐t❤♠ ♦❢ ♣❛r❛❧❧❡❧ s✉♠✳ ✶✵ ✴ ✼✼

❚❛s❦✿ ❲r✐t❡ ❛♥ ❛❧❣♦r✐t❤♠ ❝❛❧❝✉❧❛t✐♥❣ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ ❛ ❣✐✈❡♥ ❛rr❛② ❚❤✐s t❛s❦ s❡❡♠s t♦ ❜❡ s❡q✉❡♥t✐❛❧ ✭✇✐t❤ o ( n ) ❝♦♠♣❧❡①✐t②✮ ❋✐❣✉r❡✿ ❆❧❣♦r✐t❤♠ ♦❢ ♣❛r❛❧❧❡❧ s✉♠✳ ✶ ✈♦✐❞ r✉♥❴s✉♠❴♥❛✐✈❡✭❢❧♦❛t ✯❞❴✐❞❛t❛✱ ❢❧♦❛t ✯❞❴♦❞❛t❛✱ ✐♥t ♥✉♠❴❡❧❡♠❡♥ts✮ ✷ ④ ❞✐♠✸ t❤r❡❛❞s✭♥✉♠❴❡❧❡♠❡♥ts✴✷✮❀ ✸ ❞✐♠✸ ❣r✐❞✭✶✱✶✱✶✮❀ ✹ ✺ s✉♠❴♥❛✐✈❡❁❁❁ ❣r✐❞✱ t❤r❡❛❞s✱ ♥✉♠❴❡❧❡♠❡♥ts✯s✐③❡♦❢✭❢❧♦❛t✮ ❃❃❃✭❞❴✐❞❛t❛✱ ❞❴♦❞❛t❛✱ ✻ ♥✉♠❴❡❧❡♠❡♥ts✮❀ ✼ ⑥ ✶✵ ✴ ✼✼

✶ ❴❴❣❧♦❜❛❧❴❴ ✈♦✐❞ s✉♠❴♥❛✐✈❡✭❢❧♦❛t ✯❣❴✐❞❛t❛✱ ❢❧♦❛t ✯❣❴♦❞❛t❛✱ ✐♥t ♥✉♠❴❡❧❡♠❡♥ts✮ ✷ ④ ❡①t❡r♥ ❴❴s❤❛r❡❞❴❴ ❢❧♦❛t t❡♠♣❬❪❀ ✸ ❝♦♥st ✐♥t ✐♥❞❡①✶ ❂ t❤r❡❛❞■❞①✳① ❁❁ ✶❀ ✹ ❝♦♥st ✐♥t ✐♥❞❡①✷ ❂ ✐♥❞❡①✶ ✰ ✶❀ ✺ ✴✴ ❝♦♣② ✐♥♣✉t t♦ s❤❛r❡❞ ♠❡♠♦r② ✻ t❡♠♣❬✐♥❞❡①✶❪ ❂ ❣❴✐❞❛t❛❬✐♥❞❡①✶❪❀ ✼ t❡♠♣❬✐♥❞❡①✷❪ ❂ ❣❴✐❞❛t❛❬✐♥❞❡①✷❪❀ ✽ ✴✴ ❞♦ t❤❡ ❥♦❜ ✾ ✐♥t ❧♦✇■♥❞❡① ❂ ✐♥❞❡①✶❀ ✶✵ ✐♥t ❤✐❣❤■♥❞❡①❀ ✶✶ ❢♦r ✭✐♥t ❞✐st❂✶❀ ❞✐st❁❂♥✉♠❴❡❧❡♠❡♥ts❀ ❞✐st❁❁❂✶✮ ✶✷ ④ ✶✸ ❤✐❣❤■♥❞❡① ❂ ❧♦✇■♥❞❡① ✰ ❞✐st❀ ✶✹ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✶✺ ✐❢ ✭ ❤✐❣❤■♥❞❡① ❁ ♥✉♠❴❡❧❡♠❡♥ts ✮ ✶✻ ④ ✶✼ t❡♠♣❬❧♦✇■♥❞❡①❪ ✰❂ t❡♠♣❬❤✐❣❤■♥❞❡①❪❀ ✶✽ ❧♦✇■♥❞❡① ❁❁❂ ✶❀ ✶✾ ⑥ ✷✵ ⑥ ✷✶ ✴✴ ✇r✐t❡ r❡s✉❧ts ❢r♦♠ ❧♦❝❛❧ s❤❛r❡❞ ♠❡♠♦r② t♦ ❣❧♦❜❛❧ ✷✷ ❣❴♦❞❛t❛❬✐♥❞❡①✶❪ ❂ t❡♠♣❬✐♥❞❡①✶❪❀ ✷✸ ❣❴♦❞❛t❛❬✐♥❞❡①✷❪ ❂ t❡♠♣❬✐♥❞❡①✷❪❀ ✷✹ ✷✺ ⑥ ✶✶ ✴ ✼✼

❉❡✜♥✐t✐♦♥ ✭Pr❡s❝❛♥✮ ❚❤❡ ♣r❡s❝❛♥ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❜✐♥❛r② ❛ss♦❝✐❛t✐✈❡ ♦♣❡r❛t♦r ✇✐t❤ ✐❞❡♥t✐t② ✱ ❛♥❞ ❛♥ ❛rr❛② ♦❢ ❡❧❡♠❡♥ts ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ Pr❡✜① s✉♠s ❉❡✜♥✐t✐♦♥ ✭❙❝❛♥ ✕ ❆rr❛② ❛❧❧✲♣r❡✜①✲s✉♠s✮ ❚❤❡ s❝❛♥ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❜✐♥❛r② ❛ss♦❝✐❛t✐✈❡ ♦♣❡r❛t♦r ⊕ ✱ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ x 0 , ( x 0 ⊕ x 1 ) , . . . , ( x 0 ⊕ x 1 · · · ⊕ x n − 1 )] . ✶✷ ✴ ✼✼

Pr❡✜① s✉♠s ❉❡✜♥✐t✐♦♥ ✭❙❝❛♥ ✕ ❆rr❛② ❛❧❧✲♣r❡✜①✲s✉♠s✮ ❚❤❡ s❝❛♥ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❜✐♥❛r② ❛ss♦❝✐❛t✐✈❡ ♦♣❡r❛t♦r ⊕ ✱ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ x 0 , ( x 0 ⊕ x 1 ) , . . . , ( x 0 ⊕ x 1 · · · ⊕ x n − 1 )] . ❉❡✜♥✐t✐♦♥ ✭Pr❡s❝❛♥✮ ❚❤❡ ♣r❡s❝❛♥ ♦♣❡r❛t✐♦♥ t❛❦❡s ❛ ❜✐♥❛r② ❛ss♦❝✐❛t✐✈❡ ♦♣❡r❛t♦r ⊕ ✇✐t❤ ✐❞❡♥t✐t② I ✱ ❛♥❞ ❛♥ ❛rr❛② ♦❢ n ❡❧❡♠❡♥ts [ x 0 , x 1 , . . . , x n − 1 ] ✱ ❛♥❞ r❡t✉r♥s t❤❡ ❛rr❛② [ I, x 0 , ( x 0 ⊕ x 1 ) , . . . , ( x 0 ⊕ x 1 · · · ⊕ x n − 2 )] . ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✶✷ ✴ ✼✼

❙❝❛♥ ✕ ♥❛✐✈❡ s♦❧✉t✐♦♥ ✶ ❢♦r d ✿❂ ✶ t♦ log 2 n ❞♦ ❢♦r❛❧❧ k ✐♥ ♣❛r❛❧❧❡❧ ❞♦ ✷ ✐❢ k ≥ 2 d t❤❡♥ ①❬ k ❪ ✿❂ ①❬ k − 2 d − 1 ❪ ✰ ①❬ k ❪ ✸ ✵ x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 � 0 � 1 � 2 � 3 � 4 � 5 � 6 � 7 ✶ 0 x i 0 x i 1 x i 2 x i 3 x i 4 x i 5 x i 6 x i � 0 � 1 � 2 � 3 � 4 � 5 � 6 � 7 ✷ 0 x i 0 x i 0 x i 0 x i 1 x i 2 x i 3 x i 4 x i � 0 � 1 � 2 � 3 � 4 � 5 � 6 � 7 ✸ 0 x i 0 x i 0 x i 0 x i 0 x i 0 x i 0 x i 0 x i ◆♦t ✇♦r❦✲❡✣❝✐❡♥t✿ O ( n log( n )) ❝♦♠♣❛r❡❞ t♦ s❡q✉❡♥t✐❛❧ O ( n ) ❍✐❧❧✐s❴❉❛t❛P❛r❛❧❧❡❧❆❧❣ ✶✸ ✴ ✼✼

❙❝❛♥ ✕ ✇♦r❦✲❡✣❝✐❡♥t s♦❧✉t✐♦♥ ✭■✮ ❯♣✲s✇❡❡♣ ✭r❡❞✉❝❡✮ ♣❤❛s❡ ✭s❝❛♥✮ ✶ ❢♦r d ✿❂ ✵ t♦ log 2 n − 1 ❞♦ ❢♦r k ❢r♦♠ ✵ t♦ n − 1 ❜② 2 d + 1 ✐♥ ♣❛r❛❧❧❡❧ ❞♦ ✷ ①❬ k + 2 d +1 − 1 ❪ ✿❂ ①❬ k + 2 d − 1 ❪ ✰ ① ❬ k + 2 d +1 − 1 ❪ ✸ � 1 � 3 � 5 � 7 x 0 0 x i x 2 0 x i x 4 4 x i x 6 0 x i � 1 � 3 � 5 � 7 x 0 0 x i x 2 0 x i x 4 4 x i x 6 4 x i � 1 � 3 � 5 � 7 x 0 0 x i x 2 2 x i x 4 4 x i x 6 6 x i x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✶✹ ✴ ✼✼

❙❝❛♥ ✕ ✇♦r❦✲❡✣❝✐❡♥t s♦❧✉t✐♦♥ ✭■■✮ ❉♦✇♥✲s✇❡❡♣ ✭r❡❞✉❝❡✮ ♣❤❛s❡ ✭♣r❡s❝❛♥✮ ✶ ①❬ n − 1 ❪ ✿❂ ✵ ✷ ❢♦r d ✿❂ log 2 n ❞♦✇♥ t♦ ✵ ❞♦ ❢♦r k ❢r♦♠ ✵ t♦ n − 1 ❜② 2 d +1 ✐♥ ♣❛r❛❧❧❡❧ ❞♦ ✸ t ✿❂ ①❬ k + 2 d − 1 ❪ ✹ ①❬ k + 2 d − 1 ❪ ✿❂ ① ❬ k + 2 d +1 − 1 ❪ ✺ ①❬ k + 2 d +1 − 1 ❪ ✿❂ t ✰ ① ❬ k + 2 d +1 − 1 ❪ ✻ � 1 � 3 � 5 x 0 0 x i x 2 0 x i x 4 4 x i x 6 0 � 1 � 5 � 3 x 0 0 x i x 2 0 x 4 4 x i x 6 0 x i � 1 � 3 � 5 x 0 0 x 2 0 x i x 4 0 x i x 6 0 x i � 1 � 2 � 3 � 4 � 5 � 6 0 x 0 0 x i 0 x i 0 x i 0 x i 0 x i 0 x i ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✶✺ ✴ ✼✼

Pr❡s❝❛♥ r❡s✉❧t ♠❛② ❜❡ ❝♦♥✈❡rt❡❞ t♦ s❝❛♥ ❜②✿ s❝❛♥ ♣r❡s❝❛♥ ♦r ❜② s❤✐❢t✐♥❣ t❤❡ r❡s✉❧t ❜② ♦♥❡ ❡❧❡♠❡♥t ❧❡❢t ❛♥❞ ❛❞❞✐♥❣ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ ♣r❡s❝❛♥ t♦ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✐♥♣✉t✳ ❙✐♠♣❧❡ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♠✉st ❜❡ ❧✐♠✐t❡❞ t♦ s✐♥❣❧❡ t❤r❡❛❞ ❜❧♦❝❦ ✕ t❤r❡❛❞s ♠✉st ❜❡ s②♥❝❤r♦♥✐③❡❞✳ ❙❝❛♥ ❛♥❞ ♣r❡s❝❛♥ ✐s ❛ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ❢♦r ♠❛♥② ❡✣❝✐❡♥t ♣❛r❛❧❧❡❧ ❛❧❣♦r✐t❤♠s✳ ❙❝❛♥ ✕ ✇♦r❦✲❡✣❝✐❡♥t s♦❧✉t✐♦♥ ✭■■■✮ � ❲♦r❦✲❡✣❝✐❡♥t O ( n ) ✶✻ ✴ ✼✼

❙✐♠♣❧❡ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♠✉st ❜❡ ❧✐♠✐t❡❞ t♦ s✐♥❣❧❡ t❤r❡❛❞ ❜❧♦❝❦ ✕ t❤r❡❛❞s ♠✉st ❜❡ s②♥❝❤r♦♥✐③❡❞✳ ❙❝❛♥ ❛♥❞ ♣r❡s❝❛♥ ✐s ❛ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ❢♦r ♠❛♥② ❡✣❝✐❡♥t ♣❛r❛❧❧❡❧ ❛❧❣♦r✐t❤♠s✳ ❙❝❛♥ ✕ ✇♦r❦✲❡✣❝✐❡♥t s♦❧✉t✐♦♥ ✭■■■✮ � ❲♦r❦✲❡✣❝✐❡♥t O ( n ) � Pr❡s❝❛♥ r❡s✉❧t ♠❛② ❜❡ ❝♦♥✈❡rt❡❞ t♦ s❝❛♥ ❜②✿ s❝❛♥ [ i ] = ♣r❡s❝❛♥ [ i ] ⊕ x i ♦r ❜② s❤✐❢t✐♥❣ t❤❡ r❡s✉❧t ❜② ♦♥❡ ❡❧❡♠❡♥t ❧❡❢t ❛♥❞ ❛❞❞✐♥❣ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ ♣r❡s❝❛♥ t♦ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✐♥♣✉t✳ ✶✻ ✴ ✼✼

❙❝❛♥ ❛♥❞ ♣r❡s❝❛♥ ✐s ❛ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ❢♦r ♠❛♥② ❡✣❝✐❡♥t ♣❛r❛❧❧❡❧ ❛❧❣♦r✐t❤♠s✳ ❙❝❛♥ ✕ ✇♦r❦✲❡✣❝✐❡♥t s♦❧✉t✐♦♥ ✭■■■✮ � ❲♦r❦✲❡✣❝✐❡♥t O ( n ) � Pr❡s❝❛♥ r❡s✉❧t ♠❛② ❜❡ ❝♦♥✈❡rt❡❞ t♦ s❝❛♥ ❜②✿ s❝❛♥ [ i ] = ♣r❡s❝❛♥ [ i ] ⊕ x i ♦r ❜② s❤✐❢t✐♥❣ t❤❡ r❡s✉❧t ❜② ♦♥❡ ❡❧❡♠❡♥t ❧❡❢t ❛♥❞ ❛❞❞✐♥❣ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ ♣r❡s❝❛♥ t♦ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✐♥♣✉t✳ � ❙✐♠♣❧❡ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♠✉st ❜❡ ❧✐♠✐t❡❞ t♦ s✐♥❣❧❡ t❤r❡❛❞ ❜❧♦❝❦ ✕ t❤r❡❛❞s ♠✉st ❜❡ s②♥❝❤r♦♥✐③❡❞✳ ✶✻ ✴ ✼✼

❙❝❛♥ ✕ ✇♦r❦✲❡✣❝✐❡♥t s♦❧✉t✐♦♥ ✭■■■✮ � ❲♦r❦✲❡✣❝✐❡♥t O ( n ) � Pr❡s❝❛♥ r❡s✉❧t ♠❛② ❜❡ ❝♦♥✈❡rt❡❞ t♦ s❝❛♥ ❜②✿ s❝❛♥ [ i ] = ♣r❡s❝❛♥ [ i ] ⊕ x i ♦r ❜② s❤✐❢t✐♥❣ t❤❡ r❡s✉❧t ❜② ♦♥❡ ❡❧❡♠❡♥t ❧❡❢t ❛♥❞ ❛❞❞✐♥❣ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ ♣r❡s❝❛♥ t♦ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✐♥♣✉t✳ � ❙✐♠♣❧❡ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♠✉st ❜❡ ❧✐♠✐t❡❞ t♦ s✐♥❣❧❡ t❤r❡❛❞ ❜❧♦❝❦ ✕ t❤r❡❛❞s ♠✉st ❜❡ s②♥❝❤r♦♥✐③❡❞✳ � ❙❝❛♥ ❛♥❞ ♣r❡s❝❛♥ ✐s ❛ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ❢♦r ♠❛♥② ❡✣❝✐❡♥t ♣❛r❛❧❧❡❧ ❛❧❣♦r✐t❤♠s✳ ✶✻ ✴ ✼✼

Pr❡s❝❛♥ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭■✮ ❍❛rr✐s❴s❝❛♥❈✉❞❛ ✶ ❴❴❣❧♦❜❛❧❴❴ ✈♦✐❞ ♣r❡s❝❛♥❴✇♦r❦❡❢❢✐❝✐❡♥t✭❢❧♦❛t ✯❣❴♦❞❛t❛✱ ❢❧♦❛t ✯❣❴✐❞❛t❛✱ ✐♥t ♥✮ ✷ ④ ✴✴ ❉②♥❛♠✐❝❛❧❧② ❛❧❧♦❝❛t❡❞ s❤❛r❡❞ ♠❡♠♦r② ❢♦r s❝❛♥ ❦❡r♥❡❧s ✸ ❡①t❡r♥ ❴❴s❤❛r❡❞❴❴ ❢❧♦❛t t❡♠♣❬❪❀ ✹ ✐♥t t❤✐❞ ❂ t❤r❡❛❞■❞①✳①❀ ✺ ✐♥t ♦❢❢s❡t ❂ ✶❀ ✻ ✴✴ ❈❛❝❤❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✇✐♥❞♦✇ ✐♥ s❤❛r❡❞ ♠❡♠♦r② ✼ t❡♠♣❬✷✯t❤✐❞❪ ❂ ❣❴✐❞❛t❛❬✷✯t❤✐❞❪❀ ✽ t❡♠♣❬✷✯t❤✐❞✰✶❪ ❂ ❣❴✐❞❛t❛❬✷✯t❤✐❞✰✶❪❀ ✾ ✶✵ ✴✴ ❜✉✐❧❞ t❤❡ s✉♠ ✐♥ ♣❧❛❝❡ ✉♣ t❤❡ tr❡❡ ✶✶ ❢♦r ✭✐♥t ❞ ❂ ♥❃❃✶❀ ❞ ❃ ✵❀ ❞ ❃❃❂ ✶✮ ✶✷ ④ ✶✸ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✶✹ ✐❢ ✭t❤✐❞ ❁ ❞✮ ✶✺ ④ ✶✻ ✐♥t ❛✐ ❂ ♦❢❢s❡t✯✭✷✯t❤✐❞✰✶✮✲✶❀ ✶✼ ✐♥t ❜✐ ❂ ♦❢❢s❡t✯✭✷✯t❤✐❞✰✷✮✲✶❀ ✶✽ t❡♠♣❬❜✐❪ ✰❂ t❡♠♣❬❛✐❪❀ ✶✾ ⑥ ✷✵ ✶✼ ✴ ✼✼ ♦❢❢s❡t ✯❂ ✷❀ ✷✶ ⑥ ✷✷

Pr❡s❝❛♥ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭■■✮ ✴✴ s❝❛♥ ❜❛❝❦ ❞♦✇♥ t❤❡ tr❡❡ ✷✸ ✷✹ ✴✴ ❝❧❡❛r t❤❡ ❧❛st ❡❧❡♠❡♥t ✷✺ ✐❢ ✭t❤✐❞ ❂❂ ✵✮ ✷✻ t❡♠♣❬♥ ✲ ✶❪ ❂ ✵❀ ✷✼ ✷✽ ✴✴ tr❛✈❡rs❡ ❞♦✇♥ t❤❡ tr❡❡ ❜✉✐❧❞✐♥❣ t❤❡ s❝❛♥ ✐♥ ♣❧❛❝❡ ✷✾ ❢♦r ✭✐♥t ❞ ❂ ✶❀ ❞ ❁ ♥❀ ❞ ✯❂ ✷✮ ✸✵ ④ ✸✶ ♦❢❢s❡t ❃❃❂ ✶❀ ✸✷ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✸✸ ✸✹ ✐❢ ✭t❤✐❞ ❁ ❞✮ ✸✺ ④ ✸✻ ✐♥t ❛✐ ❂ ♦❢❢s❡t✯✭✷✯t❤✐❞✰✶✮✲✶❀ ✸✼ ✐♥t ❜✐ ❂ ♦❢❢s❡t✯✭✷✯t❤✐❞✰✷✮✲✶❀ ✸✽ ✸✾ ❢❧♦❛t t ❂ t❡♠♣❬❛✐❪❀ ✹✵ t❡♠♣❬❛✐❪ ❂ t❡♠♣❬❜✐❪❀ ✹✶ t❡♠♣❬❜✐❪ ✰❂ t❀ ✹✷ ⑥ ✹✸ ✶✽ ✴ ✼✼ ⑥ ✹✹

✶ ✈♦✐❞ r✉♥❴s❝❛♥❴✇♦r❦❡❢❢✐❝✐❡♥t✭❢❧♦❛t ✯❞❴♦❞❛t❛✱ ❢❧♦❛t ✯❞❴✐❞❛t❛✱ ✐♥t ♥✉♠❴❡❧❡♠❡♥ts✮ ✷ ④ ✐♥t ♥✉♠❴t❤r❡❛❞s ❂ ♥✉♠❴❡❧❡♠❡♥ts✴✷❀ ✸ ❞✐♠✸ ❣r✐❞✭✶✱ ✶✱ ✶✮❀ ✹ ❞✐♠✸ t❤r❡❛❞s✭♥✉♠❴t❤r❡❛❞s✱ ✶✱ ✶✮❀ ✺ ♣r❡s❝❛♥❴✇♦r❦❡❢❢✐❝✐❡♥t❁❁❁ ❣r✐❞✱ t❤r❡❛❞s✱ s✐③❡♦❢✭❢❧♦❛t✮ ✯ ♥✉♠❴❡❧❡♠❡♥ts ❃❃❃ ✻ ✭❞❴♦❞❛t❛✱ ❞❴✐❞❛t❛✱ ♥✉♠❴❡❧❡♠❡♥ts✮❀ ✼ ✽ ⑥ Pr❡s❝❛♥ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭■■■✮ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✹✺ ✹✻ ✴✴ ✇r✐t❡ r❡s✉❧ts t♦ ❣❧♦❜❛❧ ♠❡♠♦r② ✹✼ ❣❴♦❞❛t❛❬✷✯t❤✐❞❪ ❂ t❡♠♣❬✷✯t❤✐❞❪❀ ✹✽ ❣❴♦❞❛t❛❬✷✯t❤✐❞✰✶❪ ❂ t❡♠♣❬✷✯t❤✐❞✰✶❪❀ ✹✾ ✺✵ ⑥ ✶✾ ✴ ✼✼

Pr❡s❝❛♥ ❈❯❉❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✭■■■✮ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✹✺ ✹✻ ✴✴ ✇r✐t❡ r❡s✉❧ts t♦ ❣❧♦❜❛❧ ♠❡♠♦r② ✹✼ ❣❴♦❞❛t❛❬✷✯t❤✐❞❪ ❂ t❡♠♣❬✷✯t❤✐❞❪❀ ✹✽ ❣❴♦❞❛t❛❬✷✯t❤✐❞✰✶❪ ❂ t❡♠♣❬✷✯t❤✐❞✰✶❪❀ ✹✾ ✺✵ ⑥ ✶ ✈♦✐❞ r✉♥❴s❝❛♥❴✇♦r❦❡❢❢✐❝✐❡♥t✭❢❧♦❛t ✯❞❴♦❞❛t❛✱ ❢❧♦❛t ✯❞❴✐❞❛t❛✱ ✐♥t ♥✉♠❴❡❧❡♠❡♥ts✮ ✷ ④ ✐♥t ♥✉♠❴t❤r❡❛❞s ❂ ♥✉♠❴❡❧❡♠❡♥ts✴✷❀ ✸ ❞✐♠✸ ❣r✐❞✭✶✱ ✶✱ ✶✮❀ ✹ ❞✐♠✸ t❤r❡❛❞s✭♥✉♠❴t❤r❡❛❞s✱ ✶✱ ✶✮❀ ✺ ♣r❡s❝❛♥❴✇♦r❦❡❢❢✐❝✐❡♥t❁❁❁ ❣r✐❞✱ t❤r❡❛❞s✱ s✐③❡♦❢✭❢❧♦❛t✮ ✯ ♥✉♠❴❡❧❡♠❡♥ts ❃❃❃ ✻ ✭❞❴♦❞❛t❛✱ ❞❴✐❞❛t❛✱ ♥✉♠❴❡❧❡♠❡♥ts✮❀ ✼ ✽ ⑥ ✶✾ ✴ ✼✼

❆♣♣❧✐❝❛t✐♦♥s ♦❢ ♣r❡✜① s✉♠s ❛❧❣♦r✐t❤♠ � ❈♦♠♣✉t❛t✐♦♥ ♦❢ ♠✐♥✐♠✉♠✱ ♠❛①✐♠✉♠✱ ❛✈❡r❛❣❡✱ ❡t❝✳ ♦❢ ❛♥ ❛rr❛② � ▲❡①✐❝❛❧ ❝♦♠♣❛r✐s♦♥ ♦❢ str✐♥❣s ♦❢ ❝❤❛r❛❝t❡rs � ❆❞❞✐t✐♦♥ ♦❢ ♠✉❧t✐✲♣r❡❝✐s✐♦♥ ♥✉♠❜❡rs t❤❛t ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡ ✇♦r❞ � ❊✈❛❧✉❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧s � ❙♦❧✈✐♥❣ ♦❢ r❡❝✉rr❡♥❝❡ ❡q✉❛t✐♦♥s � ❘❛❞✐① s♦rt � ◗✉✐❝❦ s♦rt � ❙♦❧✈✐♥❣ tr✐❞✐❛❣♦♥❛❧ ❧✐♥❡❛r s②st❡♠s � ❘❡♠♦✈❛❧ ♦❢ ♠❛r❦❡❞ ❡❧❡♠❡♥ts ❢r♦♠ ❛♥ ❛rr❛② � ❉②♥❛♠✐❝❛❧ ❛❧❧♦❝❛t✐♦♥ ♦❢ ♣r♦❝❡ss♦rs � ▲❡①✐❝❛❧ ❛♥❛❧②s✐s ✭♣❛rs✐♥❣ ✐♥t♦ t♦❦❡♥s✮ � ❙❡❛r❝❤✐♥❣ ❢♦r r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s � ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ s♦♠❡ tr❡❡ ♦♣❡r❛t✐♦♥s ✷✵ ✴ ✼✼

❉❡✜♥✐t✐♦♥ ✭❊♥✉♠❡r❛t❡✮ ❋♦r ❣✐✈❡♥ ✐♥♣✉t ✈❡❝t♦r ♦❢ tr✉❡✴❢❛❧s❡ ✢❛❣s ❡♥✉♠❡r❛t❡ r❡t✉r♥s ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥ ❛ ♥✉♠❜❡r ♦❢ ♣r❡❞❡❝❡❛s✐♥❣ tr✉❡ ✈❛❧✉❡s ✐♥ ✳ ❊①❛♠♣❧❡✿ ❆ ✻ ✸ ✹ ✽ ✶ ✷ ✹ ✷ ❋ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✷ ✷ ✷ ♣r❡s❝❛♥✭❋✮ ♣❛❝❦✭❆✱❋✮ ✽ ✶ ✷ P❛❝❦ ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭P❛❝❦✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s A ❛♥❞ ✢❛❣s ❛rr❛② F ✭tr✉❡✴❢❛❧s❡✮✱ ♣❛❝❦ r❡t✉r♥s ❛rr❛② ✇✐t❤ ❡❧❡♠❡♥ts ❢r♦♠ A ❛rr❛② ✇❤✐❝❤ ❛r❡ ♠❛r❦❡❞ ❛s tr✉❡ ✐♥ ✢❛❣s ❛rr❛② ♦♥❧②✳ ✷✶ ✴ ✼✼

❊①❛♠♣❧❡✿ ❆ ✻ ✸ ✹ ✽ ✶ ✷ ✹ ✷ ❋ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✷ ✷ ✷ ♣r❡s❝❛♥✭❋✮ ♣❛❝❦✭❆✱❋✮ ✽ ✶ ✷ P❛❝❦ ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭P❛❝❦✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s A ❛♥❞ ✢❛❣s ❛rr❛② F ✭tr✉❡✴❢❛❧s❡✮✱ ♣❛❝❦ r❡t✉r♥s ❛rr❛② ✇✐t❤ ❡❧❡♠❡♥ts ❢r♦♠ A ❛rr❛② ✇❤✐❝❤ ❛r❡ ♠❛r❦❡❞ ❛s tr✉❡ ✐♥ ✢❛❣s ❛rr❛② ♦♥❧②✳ ❉❡✜♥✐t✐♦♥ ✭❊♥✉♠❡r❛t❡✮ ❋♦r ❣✐✈❡♥ ✐♥♣✉t ✈❡❝t♦r ♦❢ tr✉❡✴❢❛❧s❡ ✢❛❣s F ❡♥✉♠❡r❛t❡ r❡t✉r♥s ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥ ❛ ♥✉♠❜❡r ♦❢ ♣r❡❞❡❝❡❛s✐♥❣ tr✉❡ ✈❛❧✉❡s ✐♥ F ✳ ✷✶ ✴ ✼✼

✵ ✵ ✵ ✵ ✶ ✷ ✷ ✷ ♣r❡s❝❛♥✭❋✮ ♣❛❝❦✭❆✱❋✮ ✽ ✶ ✷ P❛❝❦ ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭P❛❝❦✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s A ❛♥❞ ✢❛❣s ❛rr❛② F ✭tr✉❡✴❢❛❧s❡✮✱ ♣❛❝❦ r❡t✉r♥s ❛rr❛② ✇✐t❤ ❡❧❡♠❡♥ts ❢r♦♠ A ❛rr❛② ✇❤✐❝❤ ❛r❡ ♠❛r❦❡❞ ❛s tr✉❡ ✐♥ ✢❛❣s ❛rr❛② ♦♥❧②✳ ❉❡✜♥✐t✐♦♥ ✭❊♥✉♠❡r❛t❡✮ ❋♦r ❣✐✈❡♥ ✐♥♣✉t ✈❡❝t♦r ♦❢ tr✉❡✴❢❛❧s❡ ✢❛❣s F ❡♥✉♠❡r❛t❡ r❡t✉r♥s ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥ ❛ ♥✉♠❜❡r ♦❢ ♣r❡❞❡❝❡❛s✐♥❣ tr✉❡ ✈❛❧✉❡s ✐♥ F ✳ ❊①❛♠♣❧❡✿ ❆ ✻ ✸ ✹ ✽ ✶ ✷ ✹ ✷ ❋ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✷✶ ✴ ✼✼

♣❛❝❦✭❆✱❋✮ ✽ ✶ ✷ P❛❝❦ ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭P❛❝❦✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s A ❛♥❞ ✢❛❣s ❛rr❛② F ✭tr✉❡✴❢❛❧s❡✮✱ ♣❛❝❦ r❡t✉r♥s ❛rr❛② ✇✐t❤ ❡❧❡♠❡♥ts ❢r♦♠ A ❛rr❛② ✇❤✐❝❤ ❛r❡ ♠❛r❦❡❞ ❛s tr✉❡ ✐♥ ✢❛❣s ❛rr❛② ♦♥❧②✳ ❉❡✜♥✐t✐♦♥ ✭❊♥✉♠❡r❛t❡✮ ❋♦r ❣✐✈❡♥ ✐♥♣✉t ✈❡❝t♦r ♦❢ tr✉❡✴❢❛❧s❡ ✢❛❣s F ❡♥✉♠❡r❛t❡ r❡t✉r♥s ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥ ❛ ♥✉♠❜❡r ♦❢ ♣r❡❞❡❝❡❛s✐♥❣ tr✉❡ ✈❛❧✉❡s ✐♥ F ✳ ❊①❛♠♣❧❡✿ ❆ ✻ ✸ ✹ ✽ ✶ ✷ ✹ ✷ ❋ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✷ ✷ ✷ ♣r❡s❝❛♥✭❋✮ ✷✶ ✴ ✼✼

P❛❝❦ ♦♣❡r❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭P❛❝❦✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s A ❛♥❞ ✢❛❣s ❛rr❛② F ✭tr✉❡✴❢❛❧s❡✮✱ ♣❛❝❦ r❡t✉r♥s ❛rr❛② ✇✐t❤ ❡❧❡♠❡♥ts ❢r♦♠ A ❛rr❛② ✇❤✐❝❤ ❛r❡ ♠❛r❦❡❞ ❛s tr✉❡ ✐♥ ✢❛❣s ❛rr❛② ♦♥❧②✳ ❉❡✜♥✐t✐♦♥ ✭❊♥✉♠❡r❛t❡✮ ❋♦r ❣✐✈❡♥ ✐♥♣✉t ✈❡❝t♦r ♦❢ tr✉❡✴❢❛❧s❡ ✢❛❣s F ❡♥✉♠❡r❛t❡ r❡t✉r♥s ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥ ❛ ♥✉♠❜❡r ♦❢ ♣r❡❞❡❝❡❛s✐♥❣ tr✉❡ ✈❛❧✉❡s ✐♥ F ✳ ❊①❛♠♣❧❡✿ ❆ ✻ ✸ ✹ ✽ ✶ ✷ ✹ ✷ ❋ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✷ ✷ ✷ ♣r❡s❝❛♥✭❋✮ ♣❛❝❦✭❆✱❋✮ ✽ ✶ ✷ ✷✶ ✴ ✼✼

❘❛❞✐① s♦rt ✶ ♣r♦❝❡❞✉r❡ s♣❧✐t❴r❛❞✐①❴s♦rt✭❆✱ ♥✉♠❜❡r❴♦❢❴❜✐ts✮ ❢♦r ✐ ❢r♦♠ ✵ t♦ ✭♥✉♠❜❡r❴♦❢❴❜✐ts ✲ ✶✮ ✷ ❆ ✿❂ s♣❧✐t✭❆✱ ❆❁✐❃✮ ✸ ✺ ✼ ✸ ✶ ✹ ✷ ✼ ✷ ❆ ✶ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ❆❁✵❃ ✹ ✷ ✷ ✺ ✼ ✸ ✶ ✼ s♣❧✐t✭❆✱ ❆❁✵❃✮ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ❆❁✶❃ s♣❧✐t✭❆✱ ❆❁✶❃✮ ✹ ✺ ✶ ✷ ✷ ✼ ✸ ✼ ✶ ✶ ✵ ✵ ✵ ✶ ✵ ✶ ❆❁✷❃ s♣❧✐t✭❆✱ ❆❁✷❃✮ ✶ ✷ ✷ ✸ ✹ ✺ ✼ ✼ ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✷✷ ✴ ✼✼

❙♣❧✐t ✇✐t❤ s❝❛♥ ✶ ♣r♦❝❡❞✉r❡ s♣❧✐t✭❆✱ ❋❧❛❣s✮ ■❴❞♦✇♥ ✿❂ s✉♠❴♣r❡s❝❛♥✭♥♦t✭❋❧❛❣s✮✮ ✷ ■❴✉♣ ✿❂ ♥ ✲ s✉♠❴s❝❛♥✭r❡✈❡rs❡❴♦r❞❡r✭❋❧❛❣s✮✮ ✸ ❢♦r❛❧❧ ✐ ✐♥ ♣❛r❛❧❧❡❧ ❞♦ ✹ ✐❢ ✭❋❧❛❣s❬✐❪✮ ✺ ■♥❞❡①❬✐❪ ✿❂ ■❴✉♣❬✐❪ ✻ ❡❧s❡ ✼ ■♥❞❡①❬✐❪ ✿❂ ■❴❞♦✇♥❬✐❪ ✽ r❡s✉❧t ✿❂ ♣❡r♠✉t❡✭❆✱ ■♥❞❡①✮ ✾ ✺ ✼ ✸ ✶ ✹ ✷ ✼ ✷ ❆ ✶ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ❋❧❛❣s ✵ ✵ ✵ ✵ ✵ ✶ ✷ ✷ ■❴❞♦✇♥ ✸ ✹ ✺ ✻ ✻ ✻ ✼ ✼ ■❴✉♣ ✸ ✹ ✺ ✻ ✵ ✶ ✼ ✷ ■♥❞❡① ✹ ✷ ✷ ✺ ✼ ✸ ✶ ✼ ♣❡r♠✉t❡✭❆✱ ■♥❞❡①✮ ✷✸ ✴ ✼✼ ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵

❖r✐❣✐♥❛❧ s❝❛♥ ♠❛② ❜❡ s❡❣♠❡♥t❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ s❝❛♥ st❛rts ❛❣❛✐♥ ❛t ❡❛❝❤ s❡❣♠❡♥t ❜♦✉♥❞❛r②✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤✐s ♠❡t❤♦❞ ✐s ♠✉❝❤ s❧♦✇❡r t❤❛♥ ♦r✐❣✐♥❛❧ ✉♥s❡❣♠❡♥t❡❞ s❝❛♥✳ ❙❡❣♠❡♥t❡❞ s❝❛♥ ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ❉❡✜♥✐t✐♦♥ ✭❙❡❣♠❡♥t❡❞ s❝❛♥✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s [ a 0 , . . . , a n − 1 ] ❛♥❞ ❛rr❛② ♦❢ ✢❛❣s [ f 0 , . . . , f n − 1 ] s❡❣♠❡♥t❡❞ s❝❛♥ r❡t✉r♥s ❛rr❛② [ x 0 , . . . , x n − 1 ] s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥✿  a 0 i = 0   � x i = a i f i = 1 0 < i < n  ( x i − 1 ⊕ a i ) f i = 0  ✷✹ ✴ ✼✼

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤✐s ♠❡t❤♦❞ ✐s ♠✉❝❤ s❧♦✇❡r t❤❛♥ ♦r✐❣✐♥❛❧ ✉♥s❡❣♠❡♥t❡❞ s❝❛♥✳ ❙❡❣♠❡♥t❡❞ s❝❛♥ ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ❉❡✜♥✐t✐♦♥ ✭❙❡❣♠❡♥t❡❞ s❝❛♥✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s [ a 0 , . . . , a n − 1 ] ❛♥❞ ❛rr❛② ♦❢ ✢❛❣s [ f 0 , . . . , f n − 1 ] s❡❣♠❡♥t❡❞ s❝❛♥ r❡t✉r♥s ❛rr❛② [ x 0 , . . . , x n − 1 ] s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥✿  a 0 i = 0   � x i = a i f i = 1 0 < i < n  ( x i − 1 ⊕ a i ) f i = 0  � ❖r✐❣✐♥❛❧ s❝❛♥ ♠❛② ❜❡ s❡❣♠❡♥t❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ s❝❛♥ st❛rts ❛❣❛✐♥ ❛t ❡❛❝❤ s❡❣♠❡♥t ❜♦✉♥❞❛r②✳ ✷✹ ✴ ✼✼

❙❡❣♠❡♥t❡❞ s❝❛♥ ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ❉❡✜♥✐t✐♦♥ ✭❙❡❣♠❡♥t❡❞ s❝❛♥✮ ❋♦r ❣✐✈❡♥ ❛rr❛② ♦❢ ✐♥♣✉t ✈❛❧✉❡s [ a 0 , . . . , a n − 1 ] ❛♥❞ ❛rr❛② ♦❢ ✢❛❣s [ f 0 , . . . , f n − 1 ] s❡❣♠❡♥t❡❞ s❝❛♥ r❡t✉r♥s ❛rr❛② [ x 0 , . . . , x n − 1 ] s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥✿  a 0 i = 0   � x i = a i f i = 1 0 < i < n  ( x i − 1 ⊕ a i ) f i = 0  � ❖r✐❣✐♥❛❧ s❝❛♥ ♠❛② ❜❡ s❡❣♠❡♥t❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ s❝❛♥ st❛rts ❛❣❛✐♥ ❛t ❡❛❝❤ s❡❣♠❡♥t ❜♦✉♥❞❛r②✳ � ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤✐s ♠❡t❤♦❞ ✐s ♠✉❝❤ s❧♦✇❡r t❤❛♥ ♦r✐❣✐♥❛❧ ✉♥s❡❣♠❡♥t❡❞ s❝❛♥✳ ✷✹ ✴ ✼✼

❊①❛♠♣❧❡ ♦❢ s❡❣♠❡♥t❡❞ s❝❛♥ ✭❯♣✲s✇❡❡♣ ♣❤❛s❡✮ ❙❡♥❣✉♣t❛❴❙❝❛♥Pr✐♠✐t✐✈❡s ✶ ❢♦r d = 1 t♦ log 2 n − 1 ❞♦ ❢♦r k = 0 t♦ n − 1 ❜② 2 d +1 ✐♥ ♣❛r❛❧❧❡❧ ❞♦ ✷ ✐❢ ❢❬ k + 2 d +1 − 1 ❪ ❂ ❢❛❧s❡ t❤❡♥ ✸ ①❬ k + 2 d +1 − 1 ❪ ✿❂ ①❬ k + 2 d − 1 ❪ ✰ ①❬ k + 2 d +1 − 1 ❪ ✹ ❢❬ k + 2 d +1 − 1 ❪ ✿❂ ❢❬ k + 2 d − 1 ❪ ⑤ ❢❬ k + 2 d +1 − 1 ❪ ✺ ❢ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ � 1 � 5 � 7 ① x 0 0 x i x 2 x 3 x 4 4 x i x 6 3 x i � 1 � 5 � 7 ① x 0 0 x i x 2 x 3 x 4 4 x i x 6 4 x i � 1 � 5 � 7 ① x 0 0 x i x 2 x 3 x 4 4 x i x 6 6 x i ① x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 ✷✺ ✴ ✼✼

❊①❛♠♣❧❡ ♦❢ s❡❣♠❡♥t❡❞ s❝❛♥ ✭❉♦✇♥✲s✇❡❡♣ ♣❤❛s❡✮ ✶ ①❬ n − 1 ❪ ✿❂ ✵ ✷ ❢♦r d ✿❂ log 2 n − 1 ❞♦✇♥ t♦ ✵ ❞♦ ❢♦r k ❢r♦♠ ✵ t♦ n − 1 ❜② 2 d +1 ✐♥ ♣❛r❛❧❧❡❧ ❞♦ ✸ t ✿❂ ①❬ k + 2 d − 1 ❪ ✹ ①❬ k + 2 d − 1 ❪ ✿❂ ① ❬ k + 2 d +1 − 1 ❪ ✺ ✐❢ ❢❬ k + 2 d ❪ ❂ tr✉❡ t❤❡♥ ✻ ①❬ k + 2 d +1 − 1 ❪ ✿❂ ✵ ✼ ❡❧s❡ ✐❢ ❢❬ k + 2 d − 1 ❪ ❂ tr✉❡ t❤❡♥ ✽ ①❬ k + 2 d +1 − 1 ❪ ✿❂ t ✾ ❡❧s❡ ✶✵ ①❬ k + 2 d +1 − 1 ❪ ✿❂ t ✰ ① ❬ k + 2 d +1 − 1 ❪ ✶✶ ❢❬ k + 2 d − 1 ❪ ✿❂ ❢❛❧s❡ ✶✷ ❢ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ � 1 � 5 x 0 0 x i x 2 x 3 x 4 4 x i x 6 0 ① � 1 � 5 ① x 0 0 x i x 2 0 x 4 4 x i x 6 x 3 ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ❢ � 1 � 5 ① x 0 0 x 2 0 x i x 4 x 3 x 6 3 x i ✷✻ ✴ ✼✼ � 1 � 2 � 4 � 5 � 6 ① 0 x 0 0 x i 0 x i x 3 3 x i 3 x i 3 x i

❦❡② ✻✳✹ ✾✳✷ ✸✳✹ ✶✳✻ ✽✳✼ ✹✳✶ ✾✳✷ ✸✳✹ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ s❡❣❴✢❛❣s ♣✐✈♦ts ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ❂ ❃ ❁ ❁ ❃ ❁ ❃ ❁ ❢ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ ✸✳✹ ✶✳✻ ✹✳✶ ✸✳✹ ✻✳✹ ✾✳✷ ✽✳✼ ✾✳✷ s❡❣❴✢❛❣s ✶ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✸✳✹ ✸✳✹ ✸✳✹ ✸✳✹ ✻✳✹ ✾✳✷ ✾✳✷ ✾✳✷ ♣✐✈♦ts ❢ ❂ ❁ ❃ ❂ ❂ ❂ ❁ ❂ ✶✳✻ ✸✳✹ ✸✳✹ ✹✳✶ ✻✳✹ ✽✳✼ ✾✳✷ ✾✳✷ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ s❡❣❴✢❛❣s ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ P❛r❛❧❧❡❧ ◗✉✐❝❦s♦rt ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✶ ♣r♦❝❡❞✉r❡ q✉✐❝❦s♦rt✭❦❡②s✮ s❡❣❴❢❧❛❣s❬✵❪ ✿❂ ✶ ✷ ✇❤✐❧❡ ♥♦t❴s♦rt❡❞✭❦❡②s✮ ✸ ♣✐✈♦ts ✿❂ s❡❣❴❝♦♣②✭❦❡②s✱ s❡❣❴❢❧❛❣s✮ ✹ ❢ ✿❂ ♣✐✈♦ts ❁❂❃ ❦❡②s ✺ ❦❡②s ✿❂ s❡❣❴s♣❧✐t✭❦❡②s✱ ❢✱ s❡❣❴❢❧❛❣s✮ ✻ s❡❣❴❢❧❛❣s ✿❂ ♥❡✇❴s❡❣❴❢❧❛❣s✭❦❡②s✱ ♣✐✈♦ts✱ s❡❣❴❢❧❛❣s✮ ✼ ✷✼ ✴ ✼✼

♣✐✈♦ts ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ❂ ❃ ❁ ❁ ❃ ❁ ❃ ❁ ❢ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ ✸✳✹ ✶✳✻ ✹✳✶ ✸✳✹ ✻✳✹ ✾✳✷ ✽✳✼ ✾✳✷ s❡❣❴✢❛❣s ✶ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✸✳✹ ✸✳✹ ✸✳✹ ✸✳✹ ✻✳✹ ✾✳✷ ✾✳✷ ✾✳✷ ♣✐✈♦ts ❢ ❂ ❁ ❃ ❂ ❂ ❂ ❁ ❂ ✶✳✻ ✸✳✹ ✸✳✹ ✹✳✶ ✻✳✹ ✽✳✼ ✾✳✷ ✾✳✷ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ s❡❣❴✢❛❣s ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ P❛r❛❧❧❡❧ ◗✉✐❝❦s♦rt ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✶ ♣r♦❝❡❞✉r❡ q✉✐❝❦s♦rt✭❦❡②s✮ s❡❣❴❢❧❛❣s❬✵❪ ✿❂ ✶ ✷ ✇❤✐❧❡ ♥♦t❴s♦rt❡❞✭❦❡②s✮ ✸ ♣✐✈♦ts ✿❂ s❡❣❴❝♦♣②✭❦❡②s✱ s❡❣❴❢❧❛❣s✮ ✹ ❢ ✿❂ ♣✐✈♦ts ❁❂❃ ❦❡②s ✺ ❦❡②s ✿❂ s❡❣❴s♣❧✐t✭❦❡②s✱ ❢✱ s❡❣❴❢❧❛❣s✮ ✻ s❡❣❴❢❧❛❣s ✿❂ ♥❡✇❴s❡❣❴❢❧❛❣s✭❦❡②s✱ ♣✐✈♦ts✱ s❡❣❴❢❧❛❣s✮ ✼ ❦❡② ✻✳✹ ✾✳✷ ✸✳✹ ✶✳✻ ✽✳✼ ✹✳✶ ✾✳✷ ✸✳✹ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ s❡❣❴✢❛❣s ✷✼ ✴ ✼✼

✸✳✹ ✸✳✹ ✸✳✹ ✸✳✹ ✻✳✹ ✾✳✷ ✾✳✷ ✾✳✷ ♣✐✈♦ts ❢ ❂ ❁ ❃ ❂ ❂ ❂ ❁ ❂ ✶✳✻ ✸✳✹ ✸✳✹ ✹✳✶ ✻✳✹ ✽✳✼ ✾✳✷ ✾✳✷ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ s❡❣❴✢❛❣s ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ P❛r❛❧❧❡❧ ◗✉✐❝❦s♦rt ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✶ ♣r♦❝❡❞✉r❡ q✉✐❝❦s♦rt✭❦❡②s✮ s❡❣❴❢❧❛❣s❬✵❪ ✿❂ ✶ ✷ ✇❤✐❧❡ ♥♦t❴s♦rt❡❞✭❦❡②s✮ ✸ ♣✐✈♦ts ✿❂ s❡❣❴❝♦♣②✭❦❡②s✱ s❡❣❴❢❧❛❣s✮ ✹ ❢ ✿❂ ♣✐✈♦ts ❁❂❃ ❦❡②s ✺ ❦❡②s ✿❂ s❡❣❴s♣❧✐t✭❦❡②s✱ ❢✱ s❡❣❴❢❧❛❣s✮ ✻ s❡❣❴❢❧❛❣s ✿❂ ♥❡✇❴s❡❣❴❢❧❛❣s✭❦❡②s✱ ♣✐✈♦ts✱ s❡❣❴❢❧❛❣s✮ ✼ ❦❡② ✻✳✹ ✾✳✷ ✸✳✹ ✶✳✻ ✽✳✼ ✹✳✶ ✾✳✷ ✸✳✹ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ s❡❣❴✢❛❣s ♣✐✈♦ts ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ❂ ❃ ❁ ❁ ❃ ❁ ❃ ❁ ❢ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ ✸✳✹ ✶✳✻ ✹✳✶ ✸✳✹ ✻✳✹ ✾✳✷ ✽✳✼ ✾✳✷ s❡❣❴✢❛❣s ✶ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✷✼ ✴ ✼✼

P❛r❛❧❧❡❧ ◗✉✐❝❦s♦rt ❜❧❡❧❧♦❝❤❴♣r❡✜①❴✶✾✾✵ ✶ ♣r♦❝❡❞✉r❡ q✉✐❝❦s♦rt✭❦❡②s✮ s❡❣❴❢❧❛❣s❬✵❪ ✿❂ ✶ ✷ ✇❤✐❧❡ ♥♦t❴s♦rt❡❞✭❦❡②s✮ ✸ ♣✐✈♦ts ✿❂ s❡❣❴❝♦♣②✭❦❡②s✱ s❡❣❴❢❧❛❣s✮ ✹ ❢ ✿❂ ♣✐✈♦ts ❁❂❃ ❦❡②s ✺ ❦❡②s ✿❂ s❡❣❴s♣❧✐t✭❦❡②s✱ ❢✱ s❡❣❴❢❧❛❣s✮ ✻ s❡❣❴❢❧❛❣s ✿❂ ♥❡✇❴s❡❣❴❢❧❛❣s✭❦❡②s✱ ♣✐✈♦ts✱ s❡❣❴❢❧❛❣s✮ ✼ ❦❡② ✻✳✹ ✾✳✷ ✸✳✹ ✶✳✻ ✽✳✼ ✹✳✶ ✾✳✷ ✸✳✹ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ s❡❣❴✢❛❣s ♣✐✈♦ts ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ✻✳✹ ❂ ❃ ❁ ❁ ❃ ❁ ❃ ❁ ❢ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ ✸✳✹ ✶✳✻ ✹✳✶ ✸✳✹ ✻✳✹ ✾✳✷ ✽✳✼ ✾✳✷ s❡❣❴✢❛❣s ✶ ✵ ✵ ✵ ✶ ✶ ✵ ✵ ✸✳✹ ✸✳✹ ✸✳✹ ✸✳✹ ✻✳✹ ✾✳✷ ✾✳✷ ✾✳✷ ♣✐✈♦ts ❢ ❂ ❁ ❃ ❂ ❂ ❂ ❁ ❂ ✶✳✻ ✸✳✹ ✸✳✹ ✹✳✶ ✻✳✹ ✽✳✼ ✾✳✷ ✾✳✷ ❦❡②✿❂s♣❧✐t✭❦❡②✱ ❢✮ ✷✼ ✴ ✼✼ s❡❣❴✢❛❣s ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵

s❡❣♠❡♥t❡❞ s❝❛♥ ✐s ♠✉❝❤ ✭✹ t✐♠❡s✮ s❧♦✇❡r t❤❛♥ ♥♦r♠❛❧ s❝❛♥ ✢❛❣s ✈❡❝t♦r ♠✉st ❜❡ st♦r❡❞ ✇✐t❤ ❛❞❞✐t✐♦♥❛❧ ❝❛r❡ ❍♦✇❡✈❡r r✐s❡s ♠❛♥② ✐♠♣❧❡♠❡♥t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s✿ ❚❤❡♦r❡t✐❝❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t②✿ ◗✉✐❝❦s♦rt ♥♦t❡s � ❆ss✉r❡s ❡q✉❛❧ ❧♦❛❞ ❢♦r ❛❧❧ ♣r♦❝❡ss♦rs✳ ✷✽ ✴ ✼✼

s❡❣♠❡♥t❡❞ s❝❛♥ ✐s ♠✉❝❤ ✭✹ t✐♠❡s✮ s❧♦✇❡r t❤❛♥ ♥♦r♠❛❧ s❝❛♥ ✢❛❣s ✈❡❝t♦r ♠✉st ❜❡ st♦r❡❞ ✇✐t❤ ❛❞❞✐t✐♦♥❛❧ ❝❛r❡ ❚❤❡♦r❡t✐❝❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t②✿ ◗✉✐❝❦s♦rt ♥♦t❡s � ❆ss✉r❡s ❡q✉❛❧ ❧♦❛❞ ❢♦r ❛❧❧ ♣r♦❝❡ss♦rs✳ � ❍♦✇❡✈❡r r✐s❡s ♠❛♥② ✐♠♣❧❡♠❡♥t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s✿ ✷✽ ✴ ✼✼

✢❛❣s ✈❡❝t♦r ♠✉st ❜❡ st♦r❡❞ ✇✐t❤ ❛❞❞✐t✐♦♥❛❧ ❝❛r❡ ❚❤❡♦r❡t✐❝❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t②✿ ◗✉✐❝❦s♦rt ♥♦t❡s � ❆ss✉r❡s ❡q✉❛❧ ❧♦❛❞ ❢♦r ❛❧❧ ♣r♦❝❡ss♦rs✳ � ❍♦✇❡✈❡r r✐s❡s ♠❛♥② ✐♠♣❧❡♠❡♥t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s✿ � s❡❣♠❡♥t❡❞ s❝❛♥ ✐s ♠✉❝❤ ✭✹ t✐♠❡s✮ s❧♦✇❡r t❤❛♥ ♥♦r♠❛❧ s❝❛♥ ✷✽ ✴ ✼✼

❚❤❡♦r❡t✐❝❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t②✿ ◗✉✐❝❦s♦rt ♥♦t❡s � ❆ss✉r❡s ❡q✉❛❧ ❧♦❛❞ ❢♦r ❛❧❧ ♣r♦❝❡ss♦rs✳ � ❍♦✇❡✈❡r r✐s❡s ♠❛♥② ✐♠♣❧❡♠❡♥t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s✿ � s❡❣♠❡♥t❡❞ s❝❛♥ ✐s ♠✉❝❤ ✭✹ t✐♠❡s✮ s❧♦✇❡r t❤❛♥ ♥♦r♠❛❧ s❝❛♥ � ✢❛❣s ✈❡❝t♦r ♠✉st ❜❡ st♦r❡❞ ✇✐t❤ ❛❞❞✐t✐♦♥❛❧ ❝❛r❡ ✷✽ ✴ ✼✼

◗✉✐❝❦s♦rt ♥♦t❡s � ❆ss✉r❡s ❡q✉❛❧ ❧♦❛❞ ❢♦r ❛❧❧ ♣r♦❝❡ss♦rs✳ � ❍♦✇❡✈❡r r✐s❡s ♠❛♥② ✐♠♣❧❡♠❡♥t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s✿ � s❡❣♠❡♥t❡❞ s❝❛♥ ✐s ♠✉❝❤ ✭✹ t✐♠❡s✮ s❧♦✇❡r t❤❛♥ ♥♦r♠❛❧ s❝❛♥ � ✢❛❣s ✈❡❝t♦r ♠✉st ❜❡ st♦r❡❞ ✇✐t❤ ❛❞❞✐t✐♦♥❛❧ ❝❛r❡ � ❚❤❡♦r❡t✐❝❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t②✿ O ( n p log 2 n + log 2 2 n ) ✷✽ ✴ ✼✼

❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❍❛rr✐s❴s❝❛♥❈✉❞❛ ✷✾ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✸✵ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❙♦rt✐♥❣ ♥❡t✇♦r❦✮ ❙♦rt✐♥❣ ♥❡t✇♦r❦ ❝♦♥t❛✐♥s ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ✳ ❋♦r ❛♥② ❣✐✈❡♥ ✐♥♣✉t ✈❡❝t♦r✱ t❤❡ ♦✉t♣✉t ✈❡❝t♦r ✐s s♦rt❡❞ ✭ ✮✳ ❉❛t❛ ✢♦✇ ✐♥s✐❞❡ t❤❡ ♥❡t✇♦r❦ ❤❛s ♥♦ ❝✐r❝❧❡s✳ ❙♦rt✐♥❣ ♥❡t✇♦r❦s ❉❡✜♥✐t✐♦♥ ✭❈♦♠♣❛r❛t♦r✮ ❈♦♠♣❛r❛t♦r ✐s ❛ ❞❡✈✐❝❡ ✇✐t❤ t✇♦ ✐♥♣✉ts ✭ x ❛♥❞ y ✮ ❛♥❞ t✇♦ ♦✉t♣✉ts ✭ x ′ ❛♥❞ y ′ ✮ ❝❛❧❝✉❧❛t✐♥❣ ✐♥ t✐♠❡ O (1) t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ x ′ = min( x, y ) y ′ = max( x, y ) ❈♦♠♣❛r❛t♦r ♠❛② ❝❛❧❝✉❧❛t❡ r❡s✉❧ts ♦♥❧② ✐❢ ❜♦t❤ ✐♥♣✉t ✈❛❧✉❡s ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ✸✶ ✴ ✼✼

❙♦rt✐♥❣ ♥❡t✇♦r❦s ❉❡✜♥✐t✐♦♥ ✭❈♦♠♣❛r❛t♦r✮ ❈♦♠♣❛r❛t♦r ✐s ❛ ❞❡✈✐❝❡ ✇✐t❤ t✇♦ ✐♥♣✉ts ✭ x ❛♥❞ y ✮ ❛♥❞ t✇♦ ♦✉t♣✉ts ✭ x ′ ❛♥❞ y ′ ✮ ❝❛❧❝✉❧❛t✐♥❣ ✐♥ t✐♠❡ O (1) t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥✿ x ′ = min( x, y ) y ′ = max( x, y ) ❈♦♠♣❛r❛t♦r ♠❛② ❝❛❧❝✉❧❛t❡ r❡s✉❧ts ♦♥❧② ✐❢ ❜♦t❤ ✐♥♣✉t ✈❛❧✉❡s ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ❉❡✜♥✐t✐♦♥ ✭❙♦rt✐♥❣ ♥❡t✇♦r❦✮ ❙♦rt✐♥❣ ♥❡t✇♦r❦ ❝♦♥t❛✐♥s n ✐♥♣✉ts a 1 , . . . , a n ❛♥❞ n ♦✉t♣✉ts b 1 , . . . , b n ✳ ❋♦r ❛♥② ❣✐✈❡♥ ✐♥♣✉t ✈❡❝t♦r✱ t❤❡ ♦✉t♣✉t ✈❡❝t♦r ✐s s♦rt❡❞ ✭ b 1 ≤ b 2 ≤ · · · ≤ b n ✮✳ ❉❛t❛ ✢♦✇ ✐♥s✐❞❡ t❤❡ ♥❡t✇♦r❦ ❤❛s ♥♦ ❝✐r❝❧❡s✳ ✸✶ ✴ ✼✼

▼❡r❣❡r✱ ❜✐t♦♥✐❝ ❛♥❞ s❤❡❧❧ s♦rt✐♥❣ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ ✱ ❝♦♠♣❛r❛t♦rs✿ ❖♣t✐♠❛❧ ❆❑❙ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ ✱ ❝♦♠♣❛r❛t♦rs✿ ✭✐♠♣r❛❝t✐❝❛❧✮ ❙♦rt✐♥❣ ♥❡t✇♦r❦s ❙♦rt✐♥❣ ♥❡t✇♦r❦s ❝❛♥ ❜❡ ❝♦♠♣❛r❡❞ ❜② ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦r ❞❡♣t❤✳ � ❖❞❞✲❡✈❡♥ s♦rt✐♥❣ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ O ( n ) ✱ ❝♦♠♣❛r❛t♦rs✿ O ( n 2 ) ✸✷ ✴ ✼✼

❖♣t✐♠❛❧ ❆❑❙ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ ✱ ❝♦♠♣❛r❛t♦rs✿ ✭✐♠♣r❛❝t✐❝❛❧✮ ❙♦rt✐♥❣ ♥❡t✇♦r❦s ❙♦rt✐♥❣ ♥❡t✇♦r❦s ❝❛♥ ❜❡ ❝♦♠♣❛r❡❞ ❜② ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦r ❞❡♣t❤✳ � ❖❞❞✲❡✈❡♥ s♦rt✐♥❣ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ O ( n ) ✱ ❝♦♠♣❛r❛t♦rs✿ O ( n 2 ) � ▼❡r❣❡r✱ ❜✐t♦♥✐❝ ❛♥❞ s❤❡❧❧ s♦rt✐♥❣ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ O (log 2 n ) ✱ ❝♦♠♣❛r❛t♦rs✿ O ( n log 2 n ) ✸✷ ✴ ✼✼

❙♦rt✐♥❣ ♥❡t✇♦r❦s ❙♦rt✐♥❣ ♥❡t✇♦r❦s ❝❛♥ ❜❡ ❝♦♠♣❛r❡❞ ❜② ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦r ❞❡♣t❤✳ � ❖❞❞✲❡✈❡♥ s♦rt✐♥❣ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ O ( n ) ✱ ❝♦♠♣❛r❛t♦rs✿ O ( n 2 ) � ▼❡r❣❡r✱ ❜✐t♦♥✐❝ ❛♥❞ s❤❡❧❧ s♦rt✐♥❣ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ O (log 2 n ) ✱ ❝♦♠♣❛r❛t♦rs✿ O ( n log 2 n ) � ❖♣t✐♠❛❧ ❆❑❙ ♥❡t✇♦r❦ ✕ ❞❡♣t❤✿ O (log n ) ✱ ❝♦♠♣❛r❛t♦rs✿ O ( n log n ) ✭✐♠♣r❛❝t✐❝❛❧✮ ✸✷ ✴ ✼✼

❋✐❣✉r❡✿ ❙♦rt✐♥❣ ♥❡t✇♦r❦s ❋✐❣✉r❡✿ ✵✲✶ ♣r✐♥❝✐♣❧❡ ✸✸ ✴ ✼✼

❖❞❞✲❡✈❡♥ s♦rt ❦❡r♥❡❧ ❡①❛♠♣❧❡ ✶ ❴❴❣❧♦❜❛❧❴❴ st❛t✐❝ ✈♦✐❞ s♦rt❴s❤❛r❡❞❴♠❡♠✭❢❧♦❛t ✯❣❴✐❞❛t❛✱ ✐♥t ♥✉♠❴❡❧❡♠❡♥ts✮ ✷ ④ ❡①t❡r♥ ❴❴s❤❛r❡❞❴❴ ❢❧♦❛t t❡♠♣❬❪❀ ✸ ✉✐♥t t❤✐❞ ❂ t❤r❡❛❞■❞①✳①❀ ✹ ✉✐♥t ♠ ❂ t❤✐❞✱ ♥ ❂ ♠ ✰ ✶✱ ♦❢❢ ❂ ✵❀ ✺ ✻ t❡♠♣❬t❤✐❞❪ ❂ ❣❴✐❞❛t❛❬t❤✐❞❪❀ ✼ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✽ ✾ ✐❢ ✭✭♠ ✫ ✶✮ ❂❂ ✵✮ ✶✵ ❢♦r ✭✉♥s✐❣♥❡❞ ✐♥t ✐❂✵❀ ✐❁♥✉♠❴❡❧❡♠❡♥ts❀ ✰✰✐✮ ✶✶ ④ ✶✷ ✐❢ ✭ ♥ ❁❂ ✭♥✉♠❴❡❧❡♠❡♥ts✲✶✮ ✮ ✶✸ ✐❢ ✭t❡♠♣❬♠❪ ❃ t❡♠♣❬♥❪✮ ✶✹ s✇❛♣✭ t❡♠♣❬♠❪✱ t❡♠♣❬♥❪ ✮❀ ✶✺ ♦❢❢ ❂ ♦❢❢ ①♦r ✶❀ ✶✻ ♠ ❂ t❤✐❞ ✰ ♦❢❢❀ ✶✼ ♥ ❂ ♠ ✰ ✶❀ ✶✽ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✶✾ ⑥ ✷✵ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✷✶ ✸✹ ✴ ✼✼ ❫❫■❣❴✐❞❛t❛❬t❤✐❞❪ ❂ t❡♠♣❬t❤✐❞❪❀ ✷✷ ✷✸ ⑥

❆♥❛❧②s✐s ♦❢ ❍❛❧❢✲❈❧❡❛♥❡r❬♥❪ ♥❡t✇♦r❦ ❈♦r♠❡♥✷✵✵✶ ❑♥✉t❤✶✾✼✸ ✸✺ ✴ ✼✼

P❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❜✐t♦♥✐❝ s♦rt ❝✉❞❛❙❉❑ ✶ ❴❴❣❧♦❜❛❧❴❴ st❛t✐❝ ✈♦✐❞ ❜✐t♦♥✐❝❙♦rt✭✐♥t ✯ ✈❛❧✉❡s✮ ✷ ④ ❡①t❡r♥ ❴❴s❤❛r❡❞❴❴ ✐♥t s❤❛r❡❞❬❪❀ ✸ ❝♦♥st ✉♥s✐❣♥❡❞ ✐♥t t✐❞ ❂ t❤r❡❛❞■❞①✳①❀ ✹ s❤❛r❡❞❬t✐❞❪ ❂ ✈❛❧✉❡s❬t✐❞❪❀ ✺ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✻ ❢♦r ✭✉♥s✐❣♥❡❞ ✐♥t ❦ ❂ ✷❀ ❦ ❁❂ ◆❯▼❀ ❦ ✯❂ ✷✮ ✼ ❢♦r ✭✉♥s✐❣♥❡❞ ✐♥t ❥ ❂ ❦ ✴ ✷❀ ❥❃✵❀ ❥ ✴❂ ✷✮ ✽ ④ ✾ ✉♥s✐❣♥❡❞ ✐♥t ✐①❥ ❂ t✐❞ ❫ ❥❀ ✶✵ ✐❢ ✭✐①❥ ❃ t✐❞✮ ④ ✶✶ ✐❢ ✭✭t✐❞ ✫ ❦✮ ❂❂ ✵✮④ ✶✷ ✐❢ ✭s❤❛r❡❞❬t✐❞❪ ❃ s❤❛r❡❞❬✐①❥❪✮ ✶✸ s✇❛♣✭s❤❛r❡❞❬t✐❞❪✱ s❤❛r❡❞❬✐①❥❪✮❀ ✶✹ ⑥ ✶✺ ❡❧s❡ ④ ✶✻ ✐❢ ✭s❤❛r❡❞❬t✐❞❪ ❁ s❤❛r❡❞❬✐①❥❪✮ ✶✼ s✇❛♣✭s❤❛r❡❞❬t✐❞❪✱ s❤❛r❡❞❬✐①❥❪✮❀ ✶✽ ⑥ ✶✾ ⑥ ✷✵ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✷✶ ✸✻ ✴ ✼✼ ⑥ ✷✷ ✈❛❧✉❡s❬t✐❞❪ ❂ s❤❛r❡❞❬t✐❞❪❀ ✷✸ ✷✹ ⑥

❇✐t♦♥✐❝ s♦rt ♥❡t✇♦r❦ ✸✼ ✴ ✼✼

❙❤❡❧❧ s♦rt � ❆❧❣♦r✐t❤♠✿ � ❆rr❛♥❣❡ ❞❛t❛ ✐♥t♦ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❛rr❛② � ❙♦rt ❝♦❧✉♠♥s � ❘❡♣❡❛t ❢♦r ❢❡✇❡r ❝♦❧✉♠♥s � Pr♦♣❡rt✐❡s✿ � ❱❡r② ❣♦♦❞ ❛❧❣♦r✐t❤♠ ❢♦r ❛❧♠♦st s♦rt❡❞ s❡ts � P❛r❛❧❧❡❧✐③♠✿ � ❊❛❝❤ ❝♦❧✉♠♥ ♠❛② ❜❡ s♦rt❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② � ❍♦✇❡✈❡r✱ t❤❡ ❧❛st st❡♣ ✐s s❡q✉❡♥t✐❛❧ � Pr♦❜❧❡♠✿ � ❲❤❛t s❤♦✉❧❞ ❜❡ t❤❡ ♦♣t✐♠❛❧ s❡q✉❡♥❝❡ ♦❢ ❝♦❧✉♠♥s❄ ✸✽ ✴ ✼✼

❙❤❡❧❧ s♦rt ✐♥❝r❡♠❡♥t s❡q✉❡♥❝❡s → O ( n √ n ) � 1 , 3 , 7 , 15 , 31 , 63 , 127 , . . . , 2 k 1 − → O ( n √ n ) � 1 , 5 , 9 , 17 , 33 , 65 , 129 , . . . , 2 k + 1 − → O ( n log 2 n ) � 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 16 , . . . , 2 p 3 q − � 1 , 3 , 5 , 11 , . . . , (2 k ± 1) / 3 � 1 , 4 , 13 , 40 , . . . , (3 k − 1) / 2 � 1 , 5 , 16 , 49 , . . . , 3 h s − 1 + 1 ❲❤✐❝❤ s❡q✉❡♥❝❡ ✐s t❤❡ ❜❡st ❢♦r ●P❯ ✕ s✉❜❥❡❝t ♦❢ ❡①♣❡r✐♠❡♥ts✳ ❝♦♥❢✴❡s❛✴❙❡❞❣❡✇✐❝❦✾✻ ❑♥✉t❤✶✾✼✸ ✸✾ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✹✵ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

▼❛tr✐①✕❱❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ▼❛tr✐①✕▼❛tr✐① ♦♣❡r❛t✐♦♥ ✸ ❇❛s✐❝ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❖♣❡r❛t✐♦♥s � ❱❡❝t♦r ✕ ✈❡❝t♦r✱ ❙❝❛❧❛r ♣r♦❞✉❝t ✭❞♦t ♣r♦❞✉❝t✮ c = a · b = a T b ✹✶ ✴ ✼✼

▼❛tr✐①✕▼❛tr✐① ♦♣❡r❛t✐♦♥ ✸ ❇❛s✐❝ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❖♣❡r❛t✐♦♥s � ❱❡❝t♦r ✕ ✈❡❝t♦r✱ ❙❝❛❧❛r ♣r♦❞✉❝t ✭❞♦t ♣r♦❞✉❝t✮ c = a · b = a T b � ▼❛tr✐①✕❱❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ y = Ax ✹✶ ✴ ✼✼

✸ ❇❛s✐❝ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❖♣❡r❛t✐♦♥s � ❱❡❝t♦r ✕ ✈❡❝t♦r✱ ❙❝❛❧❛r ♣r♦❞✉❝t ✭❞♦t ♣r♦❞✉❝t✮ c = a · b = a T b � ▼❛tr✐①✕❱❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ y = Ax � ▼❛tr✐①✕▼❛tr✐① ♦♣❡r❛t✐♦♥ C = A + B D = AB E = E − 1 ✹✶ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✹✷ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

❙❝❛❧❛r ♣r♦❞✉❝t ■ � ▼✉st ✐t❡r❛t❡ ♦✈❡r ❡❧❡♠❡♥ts ♦❢ ❜♦t❤ ✈❡❝t♦rs � ❖❢t❡♥ ♠✉❧t✐♣❧✐❡❞ s❡ts ♦❢ ✈❡❝t♦rs � ◆♦t s♦ ♦❜✈✐♦✉s ✐❞❡❛✿ � ♥✉♠❜❡r ♦❢ ✈❡❝t♦rs ♣❛✐rs ❂ ♥✉♠❜❡r ♦❢ t❤r❡❛❞ ❜❧♦❝❦s � t❤r❡❛❞s ✇✐t❤✐♥ ❛ ❜❧♦❝❦ ❞✐✈✐❞❡ ✈❡❝t♦r ✐♥t♦ s❡❣♠❡♥ts � r❡s✉❧t ❢r♦♠ ❡❛❝❤ s❡❣♠❡♥t ✐s ❝❛❧❝✉❧❛t❡❞ ❜② s✐♥❣❧❡ t❤r❡❛❞ ❛♥❞ ❦❡♣t ✐♥ s❤❛r❡❞ ♠❡♠♦r② � ❛❧❧ r❡s✉❧ts ❢r♦♠ ❛ ❜❧♦❝❦ ✐s s✉♠♠❡❞ ❜② r❡❞✉❝t✐♦♥ � ♦♥❡ ♦❢ t❤❡ t❤r❡❛❞s ❢r♦♠ ❛ ❜❧♦❝❦ ✇r✐t❡s r❡s✉❧t t♦ ❣❧♦❜❛❧ ♠❡♠♦r② ✹✸ ✴ ✼✼

❙❝❛❧❛r ♣r♦❞✉❝t ■■ ✹✹ ✴ ✼✼

❙❝❛❧❛r ♣r♦❞✉❝t ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■ s❝❛❧❛rPr♦❞❈✉❞❛❙❉❑ ❙✐♥❣❧❡ t❤r❡❛❞✬s ❥♦❜ ✕ t❤❡ ♠♦st ✐♥♥❡r ♣❛rt ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✶ ❢♦r✭✐♥t ✐❚❡♠♣ ❂ t❤r❡❛❞■❞①✳①❀ ✐❚❡♠♣ ❁ ❚❡♠♣◆❀ ✐❚❡♠♣ ✰❂ ❜❧♦❝❦❉✐♠✳①✮④ ❢❧♦❛t s✉♠ ❂ ✵❀ ✷ ✸ ❢♦r✭✐♥t ♣♦s ❂ ✈❡❝t♦r❇❛s❡ ✰ ✐❚❡♠♣❀ ♣♦s ❁ ✈❡❝t♦r❊♥❞❀ ♣♦s ✰❂ ❚❡♠♣◆✮ ✹ s✉♠ ✰❂ ❞❴❆❬♣♦s❪ ✯ ❞❴❇❬♣♦s❪❀ ✺ ✻ t❡♠♣❬✐❚❡♠♣❪ ❂ s✉♠❀ ✼ ✽ ⑥ ✹✺ ✴ ✼✼

❙❝❛❧❛r ♣r♦❞✉❝t ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■■ s❝❛❧❛rPr♦❞❈✉❞❛❙❉❑ ❙✐♥❣❧❡ ❜❧♦❝❦✬s ❥♦❜ ✕ t❤❡ ♦✉t❡r ♣❛rt ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❢♦r s✐♥❣❧❡ ✈❡❝t♦r ✶ ❢♦r✭✐♥t ✈❡❝❑ ❂ ❜❧♦❝❦■❞①✳①❀ ✈❡❝❑ ❁ ✈❡❝t♦r◆❀ ✈❡❝❑ ✰❂ ❣r✐❞❉✐♠✳①✮④ ✷ ✐♥t ✈❡❝t♦r❇❛s❡ ❂ ■▼❯▲✭❡❧❡♠❡♥t▼✱ ✈❡❝❑✮❀ ✸ ✐♥t ✈❡❝t♦r❊♥❞ ❂ ✈❡❝t♦r❇❛s❡ ✰ ❡❧❡♠❡♥t▼❀ ✹ ✺ ✴✯ s✐♥❣❧❡ t❤r❡❛❞✬s ❥♦❜ ✻ ✯ ✳✳✳ ✼ ✯✴ t❡♠♣❬✐❚❡♠♣❪ ❂ s✉♠❀ ✽ ✾ ✴✴ r❡❞✉❝t✐♦♥ ♦♥ t❡♠♣ ❛rr❛② ✶✵ ❢♦r✭✐♥t str✐❞❡ ❂ ❆❈❈❯▼❴◆ ✴ ✷❀ str✐❞❡ ❃ ✵❀ str✐❞❡ ❃❃❂ ✶✮④ ✶✶ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✶✷ ❢♦r✭✐♥t ✐❆❝❝✉♠ ❂ t❤r❡❛❞■❞①✳①❀ ✐❆❝❝✉♠ ❁ str✐❞❡❀ ✐❆❝❝✉♠ ✰❂ ❜❧♦❝❦❉✐♠✳①✮ ✶✸ t❡♠♣❬✐❆❝❝✉♠❪ ✰❂ t❡♠♣❬str✐❞❡ ✰ ✐❆❝❝✉♠❪❀ ✶✹ ⑥ ✶✺ ✶✻ ✐❢✭t❤r❡❛❞■❞①✳① ❂❂ ✵✮ ❞❴❘❬✈❡❝❑❪ ❂ t❡♠♣❬✵❪❀ ✶✼ ✶✽ ⑥ ✹✻ ✴ ✼✼

✶ ❴❴❣❧♦❜❛❧❴❴ ✈♦✐❞ s❝❛❧❛rPr♦❞●P❯✭✯❞❴❈✱✯❞❴❆✱✯❞❴❇✱✈❡❝t♦r◆✱❡❧❡♠❡♥t◆✮④ ❴❴s❤❛r❡❞❴❴ ❢❧♦❛t ❛❝❝✉♠❘❡s✉❧t❬❆❈❈❯▼❴◆❪❀ ✷ ❢♦r✭✐♥t ✈❡❝ ❂ ❜❧♦❝❦■❞①✳①❀ ✈❡❝ ❁ ✈❡❝t♦r◆❀ ✈❡❝ ✰❂ ❣r✐❞❉✐♠✳①✮④ ✸ ✐♥t ✈❡❝t♦r❇❛s❡ ❂ ■▼❯▲✭❡❧❡♠❡♥t◆✱ ✈❡❝✮❀ ✹ ✐♥t ✈❡❝t♦r❊♥❞ ❂ ✈❡❝t♦r❇❛s❡ ✰ ❡❧❡♠❡♥t◆❀ ✺ ✻ ❢♦r✭✐♥t ✐❆❝❝✉♠ ❂ t❤r❡❛❞■❞①✳①❀ ✐❆❝❝✉♠ ❁ ❆❈❈❯▼❴◆❀ ✐❆❝❝✉♠ ✰❂ ❜❧♦❝❦❉✐♠✳①✮④ ✼ ❢❧♦❛t s✉♠ ❂ ✵❀ ✽ ✾ ❢♦r✭✐♥t ♣♦s ❂ ✈❡❝t♦r❇❛s❡ ✰ ✐❆❝❝✉♠❀ ♣♦s ❁ ✈❡❝t♦r❊♥❞❀ ♣♦s ✰❂ ❆❈❈❯▼❴◆✮ ✶✵ s✉♠ ✰❂ ❞❴❆❬♣♦s❪ ✯ ❞❴❇❬♣♦s❪❀ ✶✶ ✶✷ ❛❝❝✉♠❘❡s✉❧t❬✐❆❝❝✉♠❪ ❂ s✉♠❀ ✶✸ ⑥ ✶✹ ✶✺ ❢♦r✭✐♥t str✐❞❡ ❂ ❆❈❈❯▼❴◆ ✴ ✷❀ str✐❞❡ ❃ ✵❀ str✐❞❡ ❃❃❂ ✶✮④ ✶✻ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✶✼ ❢♦r✭✐♥t ✐❆❝❝✉♠ ❂ t❤r❡❛❞■❞①✳①❀ ✐❆❝❝✉♠ ❁ str✐❞❡❀ ✐❆❝❝✉♠ ✰❂ ❜❧♦❝❦❉✐♠✳①✮ ✶✽ ❛❝❝✉♠❘❡s✉❧t❬✐❆❝❝✉♠❪ ✰❂ ❛❝❝✉♠❘❡s✉❧t❬str✐❞❡ ✰ ✐❆❝❝✉♠❪❀ ✶✾ ⑥ ✷✵ ✷✶ ✐❢✭t❤r❡❛❞■❞①✳① ❂❂ ✵✮ ❞❴❈❬✈❡❝❪ ❂ ❛❝❝✉♠❘❡s✉❧t❬✵❪❀ ✷✷ ⑥ ✷✸ ✷✹ ⑥ ✹✼ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✹✽ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ✹✾ ✴ ✼✼

❇❧♦❝❦❡❞ ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ✺✵ ✴ ✼✼

❇❧♦❝❦❡❞ ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ■♠♣❧❡♠❡♥t❛t✐♦♥ ■ ✶ ❴❴❣❧♦❜❛❧❴❴ ✈♦✐❞ ♠❛tr✐①▼✉❧✭ ❢❧♦❛t✯ ❈✱ ❢❧♦❛t✯ ❆✱ ❢❧♦❛t✯ ❇✱ ✐♥t ✇❆✱ ✐♥t ✇❇✮ ✷ ④ ✐♥t ❜① ❂ ❜❧♦❝❦■❞①✳①✱ ❜② ❂ ❜❧♦❝❦■❞①✳②❀ ✸ ✹ ✐♥t t① ❂ t❤r❡❛❞■❞①✳①✱ t② ❂ t❤r❡❛❞■❞①✳②❀ ✺ ✻ ✴✴ ■♥❞❡① ♦❢ t❤❡ ❢✐rst s✉❜✲♠❛tr✐① ♦❢ ❆ ♣r♦❝❡ss❡❞ ❜② t❤❡ ❜❧♦❝❦ ✼ ✐♥t ❛❇❡❣✐♥ ❂ ✇❆ ✯ ❇▲❖❈❑❴❙■❩❊ ✯ ❜②❀ ✽ ✾ ✴✴ ■♥❞❡① ♦❢ t❤❡ ❧❛st s✉❜✲♠❛tr✐① ♦❢ ❆ ♣r♦❝❡ss❡❞ ❜② t❤❡ ❜❧♦❝❦ ✶✵ ✐♥t ❛❊♥❞ ❂ ❛❇❡❣✐♥ ✰ ✇❆ ✲ ✶❀ ✶✶ ✶✷ ✴✴ ❙t❡♣ s✐③❡ ✉s❡❞ t♦ ✐t❡r❛t❡ t❤r♦✉❣❤ t❤❡ s✉❜✲♠❛tr✐❝❡s ♦❢ ❆ ✶✸ ✐♥t ❛❙t❡♣ ❂ ❇▲❖❈❑❴❙■❩❊❀ ✶✹ ✶✺ ✴✴ ■♥❞❡① ♦❢ t❤❡ ❢✐rst s✉❜✲♠❛tr✐① ♦❢ ❇ ♣r♦❝❡ss❡❞ ❜② t❤❡ ❜❧♦❝❦ ✶✻ ✐♥t ❜❇❡❣✐♥ ❂ ❇▲❖❈❑❴❙■❩❊ ✯ ❜①❀ ✶✼ ✶✽ ✴✴ ❙t❡♣ s✐③❡ ✉s❡❞ t♦ ✐t❡r❛t❡ t❤r♦✉❣❤ t❤❡ s✉❜✲♠❛tr✐❝❡s ♦❢ ❇ ✶✾ ✐♥t ❜❙t❡♣ ❂ ❇▲❖❈❑❴❙■❩❊ ✯ ✇❇❀ ✷✵ ✺✶ ✴ ✼✼

❇❧♦❝❦❡❞ ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ■♠♣❧❡♠❡♥t❛t✐♦♥ ■■ ✴✴ ❈s✉❜ ✐s ✉s❡❞ t♦ st♦r❡ t❤❡ ❡❧❡♠❡♥t ♦❢ t❤❡ ❜❧♦❝❦ s✉❜✲♠❛tr✐① ✶ ✴✴ t❤❛t ✐s ❝♦♠♣✉t❡❞ ❜② t❤❡ t❤r❡❛❞ ✷ ❢❧♦❛t ❈s✉❜ ❂ ✵❀ ✸ ✹ ✴✴ ▲♦♦♣ ♦✈❡r ❛❧❧ t❤❡ s✉❜✲♠❛tr✐❝❡s ♦❢ ❆ ❛♥❞ ❇ t♦ ❝♦♠♣✉t❡ t❤❡ ❜❧♦❝❦ s✉❜✲♠❛tr✐① ✺ ❢♦r ✭✐♥t ❛ ❂ ❛❇❡❣✐♥✱ ❜ ❂ ❜❇❡❣✐♥❀ ✻ ❛ ❁❂ ❛❊♥❞❀ ✼ ❛ ✰❂ ❛❙t❡♣✱ ❜ ✰❂ ❜❙t❡♣✮ ④ ✽ ✾ ✴✴ ❙❤❛r❡❞ ♠❡♠♦r② ❛rr❛② ❆s ✉s❡❞ t♦ st♦r❡ t❤❡ s✉❜✲♠❛tr✐① ♦❢ ❆ ✶✵ ❴❴s❤❛r❡❞❴❴ ❢❧♦❛t ❆s❬❇▲❖❈❑❴❙■❩❊❪❬❇▲❖❈❑❴❙■❩❊❪❀ ✶✶ ✶✷ ✴✴ ❙❤❛r❡❞ ♠❡♠♦r② ❛rr❛② ❇s ✉s❡❞ t♦ st♦r❡ t❤❡ s✉❜✲♠❛tr✐① ♦❢ ❇ ✶✸ ❴❴s❤❛r❡❞❴❴ ❢❧♦❛t ❇s❬❇▲❖❈❑❴❙■❩❊❪❬❇▲❖❈❑❴❙■❩❊❪❀ ✶✹ ✶✺ ✴✴ ▲♦❛❞ t❤❡ ♠❛tr✐❝❡s t♦ s❤❛r❡❞ ♠❡♠♦r② ✶✻ ✴✴ ❡❛❝❤ t❤r❡❛❞ ❧♦❛❞s ♦♥❡ ❡❧❡♠❡♥t ♦❢ ❡❛❝❤ ♠❛tr✐① ✶✼ ❆❙✭t②✱ t①✮ ❂ ❆❬❛ ✰ ✇❆ ✯ t② ✰ t①❪❀ ✶✽ ❇❙✭t②✱ t①✮ ❂ ❇❬❜ ✰ ✇❇ ✯ t② ✰ t①❪❀ ✶✾ ✷✵ ✴✴ ❙②♥❝❤r♦♥✐③❡ t♦ ♠❛❦❡ s✉r❡ t❤❡ ♠❛tr✐❝❡s ❛r❡ ❧♦❛❞❡❞ ✷✶ ✺✷ ✴ ✼✼ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✷✷

❇❧♦❝❦❡❞ ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ■♠♣❧❡♠❡♥t❛t✐♦♥ ■■■ ✴✴ ▼✉❧t✐♣❧② t❤❡ t✇♦ ♠❛tr✐❝❡s t♦❣❡t❤❡r❀ ✶ ✴✴ ❡❛❝❤ t❤r❡❛❞ ❝♦♠♣✉t❡s ♦♥❡ ❡❧❡♠❡♥t ♦❢ t❤❡ ❜❧♦❝❦ s✉❜✲♠❛tr✐① ✷ ❢♦r ✭✐♥t ❦ ❂ ✵❀ ❦ ❁ ❇▲❖❈❑❴❙■❩❊❀ ✰✰❦✮ ✸ ❈s✉❜ ✰❂ ❆❙✭t②✱ ❦✮ ✯ ❇❙✭❦✱ t①✮❀ ✹ ✺ ✴✴ ❙②♥❝❤r♦♥✐③❡ t♦ ♠❛❦❡ s✉r❡ t❤❛t t❤❡ ♣r❡❝❡❞✐♥❣ ✻ ✴✴ ❝♦♠♣✉t❛t✐♦♥ ✐s ❞♦♥❡ ❜❡❢♦r❡ ❧♦❛❞✐♥❣ t✇♦ ♥❡✇ ✼ ✴✴ s✉❜✲♠❛tr✐❝❡s ♦❢ ❆ ❛♥❞ ❇ ✐♥ t❤❡ ♥❡①t ✐t❡r❛t✐♦♥ ✽ ❴❴s②♥❝t❤r❡❛❞s✭✮❀ ✾ ⑥ ✶✵ ✶✶ ✴✴ ❲r✐t❡ t❤❡ ❜❧♦❝❦ s✉❜✲♠❛tr✐① t♦ ❞❡✈✐❝❡ ♠❡♠♦r②❀ ✶✷ ✴✴ ❡❛❝❤ t❤r❡❛❞ ✇r✐t❡s ♦♥❡ ❡❧❡♠❡♥t ✶✸ ✐♥t ❝ ❂ ✇❇ ✯ ❇▲❖❈❑❴❙■❩❊ ✯ ❜② ✰ ❇▲❖❈❑❴❙■❩❊ ✯ ❜①❀ ✶✹ ❈❬❝ ✰ ✇❇ ✯ t② ✰ t①❪ ❂ ❈s✉❜❀ ✶✺ ✶✻ ⑥ ✺✸ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✺✹ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✺✺ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

❙❡q✉❡♥t✐❛❧ ❛❧❣♦r✐t❤♠ ✶ ❢♦r✭✐♥t ✐ ❂ ✵❀ ✐ ❁ ❇■◆❴❈❖❯◆❚❀ ✐✰✰✮ r❡s✉❧t❬✐❪ ✿❂ ✵❀ ✷ ✸ ❢♦r✭✐♥t ✐ ❂ ✵❀ ✐ ❁ ❞❛t❛◆❀ ✐✰✰✮ r❡s✉❧t❬❞❛t❛❬✐❪❪✰✰❀ ✹ ✺✻ ✴ ✼✼

❈♦♥✢✐❝t✐♥❣ ✇r✐t❡s ❜② t❤r❡❛❞s ✐♥ t❤❡ s❛♠❡ ✇❛r♣ ▲✐♠✐t❛t✐♦♥ ♦♥ ♥✉♠❜❡r ♦❢ ❜✐♥s ✭s❤❛r❡❞ ♠❡♠♦r②✮ ❉❛t❛✲❞❡♣❡♥❞❡♥t ♣❡r❢♦r♠❛♥❝❡ ✭P♦ss✐❜❧❡ ❜❛♥❦ ❝♦♥✢✐❝ts ✐♥ s❤❛r❡❞ ♠❡♠♦r②✮ ❆ ♥❛✐✈❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧ ❤✐st♦❣r❛♠ ✇✐t❤ ❣❧♦❜❛❧ ♠❡♠♦r② ♦♥❧② ✐s ❛♣r♦①✳ ✶✵✵ s❧♦✇❡r t❤❛♥ ❈P❯ ♦♥❡✳ ❙❤❛♠s❴■❈❙P❈❙❴✷✵✵✼ P♦❞❧♦③❤♥②✉❦❴❤✐st♦❣r❛♠❈✉❞❛ P❛r❛❧❧❡❧ s♦❧✉t✐♦♥ ✺✼ ✴ ✼✼

❈♦♥✢✐❝t✐♥❣ ✇r✐t❡s ❜② t❤r❡❛❞s ✐♥ t❤❡ s❛♠❡ ✇❛r♣ ▲✐♠✐t❛t✐♦♥ ♦♥ ♥✉♠❜❡r ♦❢ ❜✐♥s ✭s❤❛r❡❞ ♠❡♠♦r②✮ ❉❛t❛✲❞❡♣❡♥❞❡♥t ♣❡r❢♦r♠❛♥❝❡ ✭P♦ss✐❜❧❡ ❜❛♥❦ ❝♦♥✢✐❝ts ✐♥ s❤❛r❡❞ ♠❡♠♦r②✮ ❙❤❛♠s❴■❈❙P❈❙❴✷✵✵✼ P♦❞❧♦③❤♥②✉❦❴❤✐st♦❣r❛♠❈✉❞❛ P❛r❛❧❧❡❧ s♦❧✉t✐♦♥ ❆ ♥❛✐✈❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧ ❤✐st♦❣r❛♠ ✇✐t❤ ❣❧♦❜❛❧ ♠❡♠♦r② ♦♥❧② ✐s ❛♣r♦①✳ ✶✵✵ s❧♦✇❡r t❤❛♥ ❈P❯ ♦♥❡✳ ✺✼ ✴ ✼✼

❙❤❛♠s❴■❈❙P❈❙❴✷✵✵✼ P♦❞❧♦③❤♥②✉❦❴❤✐st♦❣r❛♠❈✉❞❛ P❛r❛❧❧❡❧ s♦❧✉t✐♦♥ � ❈♦♥✢✐❝t✐♥❣ ✇r✐t❡s ❜② t❤r❡❛❞s ✐♥ t❤❡ s❛♠❡ ✇❛r♣ � ▲✐♠✐t❛t✐♦♥ ♦♥ ♥✉♠❜❡r ♦❢ ❜✐♥s ✭s❤❛r❡❞ ♠❡♠♦r②✮ � ❉❛t❛✲❞❡♣❡♥❞❡♥t ♣❡r❢♦r♠❛♥❝❡ � ✭P♦ss✐❜❧❡ ❜❛♥❦ ❝♦♥✢✐❝ts ✐♥ s❤❛r❡❞ ♠❡♠♦r②✮ ❆ ♥❛✐✈❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧ ❤✐st♦❣r❛♠ ✇✐t❤ ❣❧♦❜❛❧ ♠❡♠♦r② ♦♥❧② ✐s ❛♣r♦①✳ ✶✵✵ s❧♦✇❡r t❤❛♥ ❈P❯ ♦♥❡✳ ✺✼ ✴ ✼✼

P❛r❛❧❧❡❧ s♦❧✉t✐♦♥ � ❈♦♥✢✐❝t✐♥❣ ✇r✐t❡s ❜② t❤r❡❛❞s ✐♥ t❤❡ s❛♠❡ ✇❛r♣ � ▲✐♠✐t❛t✐♦♥ ♦♥ ♥✉♠❜❡r ♦❢ ❜✐♥s ✭s❤❛r❡❞ ♠❡♠♦r②✮ � ❉❛t❛✲❞❡♣❡♥❞❡♥t ♣❡r❢♦r♠❛♥❝❡ � ✭P♦ss✐❜❧❡ ❜❛♥❦ ❝♦♥✢✐❝ts ✐♥ s❤❛r❡❞ ♠❡♠♦r②✮ ❆ ♥❛✐✈❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣❛r❛❧❧❡❧ ❤✐st♦❣r❛♠ ✇✐t❤ ❣❧♦❜❛❧ ♠❡♠♦r② ♦♥❧② ✐s ❛♣r♦①✳ ✶✵✵ s❧♦✇❡r t❤❛♥ ❈P❯ ♦♥❡✳ ❙❤❛♠s❴■❈❙P❈❙❴✷✵✵✼ P♦❞❧♦③❤♥②✉❦❴❤✐st♦❣r❛♠❈✉❞❛ ✺✼ ✴ ✼✼

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆❧❣♦r✐t❤♠✐❝ Pr✐♠✐t✐✈❡s ✷ ▼❛♣ ❙❝❛♥ P❛❝❦ ♦♣❡r❛t✐♦♥ ❘❛❞✐① s♦rt ◗✉✐❝❦s♦rt ❙❝❛♥ ♦❢ ❛r❜✐tr❛r② s✐③❡ ❛rr❛②s ❙♦rt✐♥❣ ❈♦♠♣❛r❛t♦rs ❖❞❞✲❡✈❡♥ s♦rt ❇✐t♦♥✐❝ s♦rt ❙❤❡❧❧ s♦rt ▲✐♥❡❛r ❆❧❣❡❜r❛ ✸ ❙❝❛❧❛r ♣r♦❞✉❝t ▼❛tr✐① ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❍✐st♦❣r❛♠ ✹ ✺✽ ✴ ✼✼ ●❡♥❡r❛❧ ✐❞❡❛ ▼✉❧t✐♣❧❡ ❜✐♥s ❤✐st♦❣r❛♠ P❤②s✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s P❛rt✐❝❧❡s ♥✲❜♦❞✐❡s ✐♥t❡r❛❝t✐♦♥

❍✐st♦❣r❛♠ ✇✐t❤ ♠✉❧t✐♣❧❡ ❜✐♥s P♦❞❧♦③❤♥②✉❦❴❤✐st♦❣r❛♠❈✉❞❛ ❊❛❝❤ t❤r❡❛❞ ❦❡❡♣s ✐ts ♦✇♥ s❡t ♦❢ ❜✐♥s ❢♦r ❡①❝❧✉s✐✈❡ ❛❝❝❡ss✳ � ❈♦♥✢✐❝t✐♥❣ ✇r✐ts ❛r❡ ♥♦✇ r❡♠♦✈❡❞ � ❉❛t❛✲❞❡♣❡♥❞❡♥❝❡ r❡♠♦✈❡❞ � ❋❡✇❡r ❜✐♥s ✭♠❛① ✻✹ ❢♦r ✸✷✲❜✐t ❝♦✉♥t❡rs✮ ❛✈❛✐❧❛❜❧❡ ✺✾ ✴ ✼✼