❆♥❛❧②s✐s ♦❢ r❛♥❞♦♠ ✇❛❧❦s ✉s✐♥❣ t❛❜✉ ❧✐sts ❑✳ ❆❧t✐s❡♥✱ ❙✳ ❉❡✈✐s♠❡s✱ ❆✳ ●❡r❜❛✉❞ ❛♥❞ P✳ ▲❛❢♦✉r❝❛❞❡ ▲❛❜♦r❛t♦✐r❡ ❱❡r✐♠❛❣ ❯♥✐✈❡rs✐té ❏♦s❡♣❤ ❋♦✉r✐❡r✱ ●r❡♥♦❜❧❡ ✶ ✶ ✴ ✶✽
❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st ✶ ❚❡r♠✐♥❛t✐♦♥ ✷ ❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠ ✲❢r❡❡ ❣r❛♣❤s ✸ ✷ ✴ ✶✽
❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st ✶ ❚❡r♠✐♥❛t✐♦♥ ✷ ❖♣t✐♠❛❧ ✉♣❞❛t❡ r✉❧❡s ❢♦r ♠ ✲❢r❡❡ ❣r❛♣❤s ✸ ✸ ✴ ✶✽
❙✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ r♦✉t✐♥❣ destination 0 6 1/4 8 1/4 ❆❧❧ ❣r❛♣❤s ❛r❡ ✜♥✐t❡✱ s✐♠♣❧❡✱ 1/4 2 5 ❝♦♥♥❡❝t❡❞✱ ✇✐t❤ ❛t ❧❡❛st t✇♦ ✈❡rt✐❝❡s✳ 9 1/4 source 1 7 3 4 ✹ ✴ ✶✽
❙✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ r♦✉t✐♥❣ destination 0 6 1/4 8 ❆t ❡❛❝❤ st❡♣✱ t❤❡ ✇❛❧❦❡r ♠♦✈❡s t♦ ❛ 1/4 ♥❡✐❣❤❜♦r ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ 1/4 2 5 ❚❤❡ ✇❛❧❦❡r r❡♣r❡s❡♥ts ❛ ♣❛❝❦❡t 9 1/4 ❢♦r✇❛r❞❡❞ ❢r♦♠ ♥♦❞❡ t♦ ♥♦❞❡✳ source 1 7 3 4 ✹ ✴ ✶✽
❙✐♠♣❧❡ r❛♥❞♦♠ ✇❛❧❦ r♦✉t✐♥❣ destination 0 6 ❆❞✈❛♥t❛❣❡s✿ � ♥♦ r♦✉t✐♥❣ t❛❜❧❡❀ 1/4 8 1/4 � ❧♦❝❛❧ ❝♦♠♣✉t❛t✐♦♥s❀ 1/4 2 5 � s❝❛❧❛❜❧❡❀ � r♦❜✉st❀ ❛♥❞ 9 1/4 � ❧♦❛❞✲❜❛❧❛♥❝❡❞✳ source 1 ❉r❛✇❜❛❝❦ ✿ s❧♦✇✳ 7 3 4 ✹ ✴ ✶✽
▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ❍✐tt✐♥❣ t✐♠❡✿ ♥✉♠❜❡r ♦❢ st❡♣s t♦ ❞❡❧✐✈❡r t❤❡ ♣❛❝❦❡t t♦ ✐ts ✜♥❛❧ ❞❡st✐♥❛t✐♦♥✳ ▼♦t✐✈❛t✐♦♥✿ r❡❞✉❝✐♥❣ t❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡s✳ ✺ ✴ ✶✽
▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s ▲❡t ( ❳ ♥ ) ♥ � ✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ● ✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① ❛♥❞ ② ♦❢ ● ✿ ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❍ ② ( ● ) = ✐♥❢ { ♥ � ✵ : ❳ ♥ = ② } . ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ ✈❛❧✉❡ ❊ ① ❍ ② ( ● ) ✳ ✺ ✴ ✶✽
▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s 3 ▲❡t ( ❳ ♥ ) ♥ � ✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ● ✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① ❛♥❞ ② ♦❢ ● ✿ 2 ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ 4 1 r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❍ ② ( ● ) = ✐♥❢ { ♥ � ✵ : ❳ ♥ = ② } . 0 ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ 5 ✈❛❧✉❡ ❊ ① ❍ ② ( ● ) ✳ ❊ ✶ ❍ ✵ ( ❑ ✻ ) = ✺ ✺ ✴ ✶✽
▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s 3 ▲❡t ( ❳ ♥ ) ♥ � ✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ● ✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① ❛♥❞ ② ♦❢ ● ✿ 2 ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ 4 1 r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❍ ② ( ● ) = ✐♥❢ { ♥ � ✵ : ❳ ♥ = ② } . 0 ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ 5 ✈❛❧✉❡ ❊ ① ❍ ② ( ● ) ✳ ❊ ✶ ❍ ✵ ( ❙ ✻ ) = ✷ × ✺ ✺ ✴ ✶✽
▼❡❛♥ ❤✐tt✐♥❣ t✐♠❡s 0 6 ▲❡t ( ❳ ♥ ) ♥ � ✵ ❜❡ ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ ❛ ❣r❛♣❤ ● ✳ ❋♦r ❡✈❡r② t✇♦ ✈❡rt✐❝❡s ① 8 ❛♥❞ ② ♦❢ ● ✿ ❚❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② ✐s t❤❡ 2 5 r❛♥❞♦♠ ✈❛r✐❛❜❧❡ 9 ❍ ② ( ● ) = ✐♥❢ { ♥ � ✵ : ❳ ♥ = ② } . 1 ❚❤❡ ♠❡❛♥ ❤✐tt✐♥❣ t✐♠❡ ♦❢ ② 7 st❛rt✐♥❣ ❢r♦♠ ① ✐s t❤❡ ♠❡❛♥ 3 ✈❛❧✉❡ ❊ ① ❍ ② ( ● ) ✳ 4 ❊ ✶ ❍ ✵ ( ● ) = ✸✾ ✺ ✴ ✶✽
❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st ❚❤❡ ✇❛❧❦❡r ❤❛s ❛ ✜♥✐t❡ ♠❡♠♦r② ✐♥ ✇❤✐❝❤ ✐t st♦r❡s ♦❝❝✉rr❡♥❝❡s ♦❢ ✈❡rt✐❝❡s ❛❧r❡❛❞② ✈✐s✐t❡❞✱ ❝❛❧❧❡❞ t❛❜✉ ❧✐st✳ ❚❤❡ ♦❝❝✉rr❡♥❝❡s ❛r❡ s♦rt❡❞ ✐♥ ❝❤r♦♥♦❧♦❣✐❝❛❧ ♦r❞❡r✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❜❧♦❝❦s ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ t❛❜✉ ❧✐st✳ ❚❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧ ✐❢ ❛❧❧ ❜❧♦❝❦s ❛r❡ ❛❧❧♦❝❛t❡❞✳ ❚❤❡ ♣♦❧✐❝② t♦ ✐♥s❡rt ♦r r❡♠♦✈❡ ♦❝❝✉rr❡♥❝❡s ♦❢ ✈❡rt✐❝❡s ✐♥ t❤❡ t❛❜✉ ❧✐st ✐s ❝❛❧❧❡❞ t❤❡ ✉♣❞❛t❡ r✉❧❡✳ 2 last occurence inserted 5 3 1 last occurence 1 inserted 2 4 ✭❛✮ ❚❤❡ t❛❜✉ ❧✐st ✭❜✮ ❚❤❡ t❛❜✉ ❧✐st ❤❛s ❧❡♥❣t❤ ✸ ✇✐t❤ ❤❛s ❧❡♥❣t❤ ✺ ❛♥❞ ✐s ✷ ❛❧❧♦❝❛t❡❞ ❜❧♦❝❦s✳ ❢✉❧❧✳ ✻ ✴ ✶✽
❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ♥♦t ❢✉❧❧✱ t❤❡♥ t❤r❡❡ ❝❛s❡s ♠❛② ♦❝❝✉r✿ x t 1 x x t 1 x x t 1 t 1 x t 1 x t 1 x x t 3 x t 3 t 3 t 3 t 3 t 3 ✭❝✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✭❞✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✭❡✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞✳ ✐♥s❡rt❡❞ ❛♥❞ ❛♥ ♦❧❞❡r ♦❝✲ ♥♦t ✐♥s❡rt❡❞✳ ❝✉rr❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ❞✐s❝❛r❞❡❞✳ ✻ ✴ ✶✽
❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧✱ t❤❡♥ t❤r❡❡ ❝❛s❡s ♠❛② ♦❝❝✉r✿ x t 1 x x t 1 x x t 1 t 1 x t 1 x t 1 x x t 3 x t 3 t 3 t 3 t 3 t 4 t 3 t 4 t 4 t 4 t 4 t 5 t 5 t 5 t 5 t 5 t 5 ✭❢✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✭❣✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✭❤✮ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞ ❛♥❞ ❛♥ ♦❝❝✉r✲ ✐♥s❡rt❡❞ ❛♥❞ ❛♥ ♦❧❞❡r ♦❝✲ ♥♦t ✐♥s❡rt❡❞✳ r❡♥❝❡ ♦❢ ❛♥♦t❤❡r ✈❡rt❡① ✐s ❝✉rr❡♥❝❡ ♦❢ t❤❡ ❝✉rr❡♥t ❞✐s❝❛r❞❡❞✳ ✈❡rt❡① ✐s ❞✐s❝❛r❞❡❞✳ ✻ ✴ ✶✽
❘❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ t❛❜✉ ❧✐st ❆t ❡❛❝❤ st❡♣✱ t❤❡ ✇❛❧❦❡r ♠♦✈❡s t♦ ❛ ♥❡✐❣❤❜♦r ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠ ❛♠♦♥❣ t❤♦s❡ ✇✐t❤♦✉t ♦❝❝✉rr❡♥❝❡ ✐♥ t❤❡ t❛❜✉ ❧✐st✳ ■❢ ❛❧❧ ♥❡✐❣❤❜♦rs ❤❛✈❡ ❛♥ ♦❝❝✉rr❡♥❝❡ ✐♥ t❤❡ t❛❜✉ ❧✐st✱ t❤❡ ✇❛❧❦❡r ♠♦✈❡s t♦ ❛ ♥❡✐❣❤❜♦r ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ 0 0 6 6 1/2 8 1/4 8 1/2 1/4 2 2 1/4 2 5 2 5 5 8 9 9 1/4 2 6 1 1 7 7 7 7 3 3 6 4 4 ✻ ✴ ✶✽
❋✐rst ■♥ ❋✐rst ❖✉t ❚❤❡ ❋■❋❖ ♠ ✉♣❞❛t❡ r✉❧❡s✱ ♠ � ✵✳ x t 1 x x t 1 x t 2 t 1 t 2 t 1 almost surely almost surely t 2 t 3 t 2 t 4 t 3 t 5 t 4 ❍❡r❡✱ ♠ = ✺✳ ❚❤❡ ❝✉rr❡♥t ✈❡rt❡① ✐s ✐♥s❡rt❡❞ ❛❧♠♦st s✉r❡❧②✳ ■❢ t❤❡ t❛❜✉ ❧✐st ✐s ❢✉❧❧✱ t❤❡♥ t❤❡ ♦❝❝✉r❡♥❝❡ ♦❢ t❤❡ ✈❡rt❡① st♦r❡❞ ✐♥ t❤❡ ❧❛st ❜❧♦❝❦ ✐s ❞✐s❝❛r❞❡❞✱ ❛❧♠♦st s✉r❡❧②✳ ✼ ✴ ✶✽
❋✐rst ■♥ ❋✐rst ❖✉t ❚❤❡ ❋■❋❖ ✹✵ ✉♣❞❛t❡ r✉❧❡✳ ✼ ✴ ✶✽
❋✐rst ■♥ ❋✐rst ❖✉t ❈♦♥tr❛r✐❧② t♦ t❤❡ s♥❛❦❡✱ t❤❡ ✇❛❧❦❡r ♥❡✈❡r ❞✐❡s✳ ✼ ✴ ✶✽
Recommend
More recommend