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❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

A S P W P P P P S S P W P S S S P

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❈◆❘❙ ❛♥❞ ▲■P◆✱ ❯♥✐✈❡rs✐té P❛r✐s ✶✸✱ ❱✐❧❧❡t❛♥❡✉s❡ ❆♦❢❆ ✷✵✶✾✱ ❈■❘▼ ▲✉♠✐♥②✱ ❏✉♥❡ ✷✹t❤ ✷✵✶✾

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❧❣♦r✐t❤♠s t❤❛t ✇♦r❦ ❜❡tt❡r ✇❤❡♥ t❤❡ s✐③❡ ✐s ♥♦t ✜①❡❞✳✳✳

▲❡t C =

N CN ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡s✱

✇❤❡r❡ C ∈ CN ✐❢ ✐ts s✐③❡ |C| ✐s N✳ ❊✈❡r② CN ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♠❡❛s✉r❡ µN ✭♦❢t❡♥ ❥✉st t❤❡ ✉♥✐❢♦r♠ ♠❡❛s✉r❡✮ ❨♦✉r ❣♦❛❧ ✐s t♦ ❞❡✈✐s❡ ❛❧❣♦r✐t❤♠s ❢♦r ❡①❛❝t❧② s❛♠♣❧✐♥❣ ❢r♦♠ µN✱ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❜❡st ♣♦ss✐❜❧❡ ❝♦♠♣❧❡①✐t② ✐♥ N ■t ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✏♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠✑ ❢♦r s❛♠♣❧✐♥❣ ❢r♦♠ µ[α] :=

N αNµN ✭✇✐t❤ αN ≥ ✵ ❛♥❞

N αN = ✶✮

❈❛❧❧ t❤✐s t❤❡ ✏❇♦❧t③♠❛♥♥ ❝❛s❡✑ ❛♥❞ ②♦✉ ❛r❡ t❡♠♣❡❞ t♦ ✉s❡ t❤❡ ♦❜✈✐♦✉s ❛❧❣♦r✐t❤♠ r❡♣❡❛t C ← µ[α] ✉♥t✐❧ |C| = N❀ r❡t✉r♥ C

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❧❣♦r✐t❤♠s t❤❛t ✇♦r❦ ❜❡tt❡r ✇❤❡♥ t❤❡ s✐③❡ ✐s ♥♦t ✜①❡❞✳✳✳

▲❡t C =

N CN ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡s✱

✇❤❡r❡ C ∈ CN ✐❢ ✐ts s✐③❡ |C| ✐s N✳ ❊✈❡r② CN ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♠❡❛s✉r❡ µN ✭♦❢t❡♥ ❥✉st t❤❡ ✉♥✐❢♦r♠ ♠❡❛s✉r❡✮ ❨♦✉r ❣♦❛❧ ✐s t♦ ❞❡✈✐s❡ ❛❧❣♦r✐t❤♠s ❢♦r ❡①❛❝t❧② s❛♠♣❧✐♥❣ ❢r♦♠ µN✱ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❜❡st ♣♦ss✐❜❧❡ ❝♦♠♣❧❡①✐t② ✐♥ N ■t ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✏♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠✑ ❢♦r s❛♠♣❧✐♥❣ ❢r♦♠ µ[α] :=

N αNµN ✭✇✐t❤ αN ≥ ✵ ❛♥❞

N αN = ✶✮

❈❛❧❧ t❤✐s t❤❡ ✏❇♦❧t③♠❛♥♥ ❝❛s❡✑ ❛♥❞ ②♦✉ ❛r❡ t❡♠♣❡❞ t♦ ✉s❡ t❤❡ ♦❜✈✐♦✉s ❛❧❣♦r✐t❤♠ r❡♣❡❛t C ← µ[α] ✉♥t✐❧ |C| = N❀ r❡t✉r♥ C

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❊①❛♠♣❧❡✿ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣ ♦❢ ✷✲t❡r♠✐♥❛❧ s❡r✐❡s✲♣❛r❛❧❧❡❧ ❣r❛♣❤s✱ ♦r✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ♦❢ r❡❝✉rs✐✈❡ ♠✐♥♦r✲❝❧♦s❡❞ ❢❛♠✐❧✐❡s ♦❢ ❣r❛♣❤s✳ ❍❡r❡ ✇❡ ♣r❡s❡♥t t❤❡ ❝❛s❡ ♦❢ W✹✲❢r❡❡ ❣r❛♣❤s✱ t❤❛t ✐s✱ ❣r❛♣❤s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛s ❛ ♠✐♥♦r✳ ❆ ❣r❛♣❤ G ✐s ✐♥ t❤✐s ❝❧❛ss ✐✛ ∃ ✭♦r ∀✮ ❡❞❣❡ (uv) ∈ G t❤❡ ✷✲t❡r♠✐♥❛❧ ❣r❛♣❤ G[uv] ✭r♦♦t❡❞ ❛t u ❛♥❞ v✮ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ♦♥❡ ❡❞❣❡ ❜② s❡r✐❡s✱ ♣❛r❛❧❧❡❧ ❛♥❞ ❵✇❤❡❛tst♦♥❡ ❜r✐❞❣❡✬ r❡❞✉❝t✐♦♥s✿ G[uv] G u v

• w

− →

• p

− →

• s

− → ■♥ t❤✐s ❝❛s❡ αn = A(z)−✶[ζn]A(ζ)✱ ❢♦r A(z) s♦❧✈✐♥❣ ❛ ❝❡rt❛✐♥ ❡q✉❛t✐♦♥✱ ❛♥❞ z < z∗ r♦♦t ♦❢ ❛ ❝❡rt❛✐♥ ♣♦❧②♥♦♠✐❛❧✳✳✳

✭♠♦r❡ ❞❡t❛✐❧s ❧❛t❡r ♦♥✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❧❣♦r✐t❤♠s t❤❛t ✇♦r❦ ❜❡tt❡r ✇❤❡♥ t❤❡ s✐③❡ ✐s ♥♦t ✜①❡❞✳✳✳

◆♦✇✱ ❧❡t CN =

M CN,M ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡s✱

✇❤❡r❡ C ∈ CN,M ✐❢ |C| = N ❛♥❞ s♦♠❡ st❛t✐st✐❝s m(C) ∈ Zd ✐s ❡q✉❛❧ t♦ M✳ ❊✈❡r② CN,M ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♠❡❛s✉r❡ µN,M ❆❣❛✐♥✱ ②♦✉ ♠✉st ❞❡✈✐s❡ ❛❧❣♦r✐t❤♠s ❢♦r ❡①❛❝t❧② s❛♠♣❧✐♥❣ ❢r♦♠ µN,M✱ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❜❡st ♣♦ss✐❜❧❡ ❝♦♠♣❧❡①✐t② ✐♥ N ❨♦✉ ♦❢t❡♥ ❤❛✈❡ ❛ ✏♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠✑ ❢♦r s❛♠♣❧✐♥❣ ❢r♦♠ µN,[α] :=

M αN,MµN,M ✭✇✐t❤ αN,M ≥ ✵ ❛♥❞

M αN,M = ✶ ❢♦r ❛❧❧ N✮

❈❛❧❧ t❤✐s t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ❛♥❞ ②♦✉ ❛r❡ t❡♠♣❡❞ t♦ ✉s❡ t❤❡ ♦❜✈✐♦✉s ❛❧❣♦r✐t❤♠ r❡♣❡❛t C ← µN,[α] ✉♥t✐❧ m(C) = M❀ r❡t✉r♥ C

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆❧❣♦r✐t❤♠s t❤❛t ✇♦r❦ ❜❡tt❡r ✇❤❡♥ t❤❡ s✐③❡ ✐s ♥♦t ✜①❡❞✳✳✳

◆♦✇✱ ❧❡t CN =

M CN,M ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡s✱

✇❤❡r❡ C ∈ CN,M ✐❢ |C| = N ❛♥❞ s♦♠❡ st❛t✐st✐❝s m(C) ∈ Zd ✐s ❡q✉❛❧ t♦ M✳ ❊✈❡r② CN,M ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♠❡❛s✉r❡ µN,M ❆❣❛✐♥✱ ②♦✉ ♠✉st ❞❡✈✐s❡ ❛❧❣♦r✐t❤♠s ❢♦r ❡①❛❝t❧② s❛♠♣❧✐♥❣ ❢r♦♠ µN,M✱ ✇❤✐❝❤ ❤❛✈❡ t❤❡ ❜❡st ♣♦ss✐❜❧❡ ❝♦♠♣❧❡①✐t② ✐♥ N ❨♦✉ ♦❢t❡♥ ❤❛✈❡ ❛ ✏♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠✑ ❢♦r s❛♠♣❧✐♥❣ ❢r♦♠ µN,[α] :=

M αN,MµN,M ✭✇✐t❤ αN,M ≥ ✵ ❛♥❞

M αN,M = ✶ ❢♦r ❛❧❧ N✮

❈❛❧❧ t❤✐s t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ❛♥❞ ②♦✉ ❛r❡ t❡♠♣❡❞ t♦ ✉s❡ t❤❡ ♦❜✈✐♦✉s ❛❧❣♦r✐t❤♠ r❡♣❡❛t C ← µN,[α] ✉♥t✐❧ m(C) = M❀ r❡t✉r♥ C

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❊①❛♠♣❧❡✿ ❆r❡❛✲✇❡✐❣❤t❡❞ ✐♥t❡❣❡r ♣❛rt✐t✐♦♥s ✐♥ ❛ N × M ❜♦①✱ ✇❤✐❝❤ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s (N + ✶)✲t✉♣❧❡s (x✵, x✶, . . . , xN)✱ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✈❡rt✐❝❛❧ ✐♥❝r❡♠❡♥ts ❛❧♦♥❣ t❤❡ ✈❛r✐♦✉s ❝♦❧✉♠♥s✿ µN,M(x) ∝ q

• k kxk δM,

k xk

❚❤❡ ♥❛t✉r❛❧ ♣r♦❜❧❡♠✱ ✐♥ ✇❤✐❝❤ t❤❡ δ✲❝♦♥str❛✐♥t ✐s tr❛❞❡❞ ❢♦r ❛ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ω✱ ❤❛s ♠❡❛s✉r❡ µN,[α](x) ∝ q

• k kxkω
• k xk

❛♥❞ t❤❡ ❜❡st ②♦✉ ❝❛♥ ❞♦ ✐s t✉♥❡ ω ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐s❡ αN,M✳ ■♥ t❤✐s ❝❛s❡

✶

✇✐t❤

✵

✶ ✶

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❊①❛♠♣❧❡✿ ❆r❡❛✲✇❡✐❣❤t❡❞ ✐♥t❡❣❡r ♣❛rt✐t✐♦♥s ✐♥ ❛ N × M ❜♦①✱ ✇❤✐❝❤ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s (N + ✶)✲t✉♣❧❡s (x✵, x✶, . . . , xN)✱ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ✈❡rt✐❝❛❧ ✐♥❝r❡♠❡♥ts ❛❧♦♥❣ t❤❡ ✈❛r✐♦✉s ❝♦❧✉♠♥s✿ µN,M(x) ∝ q

• k kxk δM,

k xk

❚❤❡ ♥❛t✉r❛❧ ♣r♦❜❧❡♠✱ ✐♥ ✇❤✐❝❤ t❤❡ δ✲❝♦♥str❛✐♥t ✐s tr❛❞❡❞ ❢♦r ❛ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ω✱ ❤❛s ♠❡❛s✉r❡ µN,[α](x) ∝ q

• k kxkω
• k xk

❛♥❞ t❤❡ ❜❡st ②♦✉ ❝❛♥ ❞♦ ✐s t✉♥❡ ω ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐s❡ αN,M✳

x= ✸ ✹ ✷ ✵ ✶ ✵ ✵ ✷ ✶ ✶ ✵ ✵ ✵ ✶ ✵ ✵

■♥ t❤✐s ❝❛s❡ αN,M = A(ω)−✶[zM]A(ωz) ✇✐t❤ A(ω) =

N

• k=✵

✶ ✶ − ωqk

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

◆❛ï✈❡ ❝♦♠♣❧❡①✐t②✿ ❇r✐❞❣❡ ❝❛s❡

❙❛② t❤❛t ②♦✉ ❛r❡ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡✱ ❛♥❞ t❤❛t s❛♠♣❧✐♥❣ ❢r♦♠ µN,[α] t❛❦❡s t✐♠❡ TN ∼ τN✳ ❚❤❡♥ ♦❢ ❝♦✉rs❡ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ♦❢ ♦r❞❡r T ∼ τN/αN,M✳ ❊✈❡♥ ✐♥ t❤❡ ✏❜❡st r❡❛❧✐st✐❝ ❝❛s❡✑ ♦❢ ●❛✉ss✐❛♥ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ m(C) ∈ Zd ❛r♦✉♥❞ M ✇✐t❤ ✈❛r✐❛♥❝❡ ✭t❡♥s♦r s♣❡❝tr✉♠✮ ❧✐♥❡❛r ✐♥ N✱ t❤✐s ❣✐✈❡s T ∼ N✶+ d

✷

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✐♥✈❡♥t ❛ ❜❡tt❡r ❛❧❣♦r✐t❤♠ ✐♥ ♦r❞❡r t♦ ❦✐❧❧ t❤❡ ❡①tr❛ ❢❛❝t♦r N

d ✷ ❛s ♠✉❝❤ ❛s ✇❡ ❝❛♥✳

❚❤✐s ✇✐❧❧ ❜❡ ❞♦♥❡✱ ✐♥ ❛ r❡str✐❝t❡❞ ❢❛♠✐❧② ♦❢ ♠♦❞❡❧s✱ t❤r♦✉❣❤ t❤❡ tr✐❝❦ ♦❢ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✇❤✐❝❤ ✐s ✐♥ ♠② ✏●❆❙❈♦♠✷✵✶✽✑ ♣❛♣❡r✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

◆❛ï✈❡ ❝♦♠♣❧❡①✐t②✿ ❇♦❧t③♠❛♥♥ ❝❛s❡

❙❛② t❤❛t ②♦✉ ❛r❡ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡✱ t❤❛t s❛♠♣❧✐♥❣ ❢r♦♠ µ[α] t❛❦❡s t✐♠❡ Tn ∼ τn ✇❤❡♥ t❤❡ ♦✉t❝♦♠❡ |C| ✐s n✱ ❛♥❞ T+ ∼ τN ✇❤❡♥ t❤❡ s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡ ❛❜♦rts ✇✐t❤ ❛ ❝❡rt✐✜❝❛t✐♦♥ t❤❛t |C| ✇✐❧❧ ❜❡ ❧❛r❣❡r t❤❛♥ N✳ ✭t❤✐s ✐s ❛❝❤✐❡✈❡❞ t❤r♦✉❣❤ ❛♥t✐❝✐♣❛t❡❞ r❡❥❡❝t✐♦♥✮ ❚❤❡♥ t❤❡ ❝♦♠♣❧❡①✐t② ✐s ♦❢ ♦r❞❡r T ∼ τN αN

• n

αn min(n, N) ❊✈❡♥ ✐♥ t❤❡ ✏❜❡st r❡❛❧✐st✐❝ ❝❛s❡✑ ♦❢ αn ∼ znnγ✱ ✇✐t❤ γ > −✶ ❛♥❞ z < ✶ t❤❛t ❝❛♥ ❜❡ t✉♥❡❞ ❛t ②♦✉r ✇✐❧❧✱ t❤✐s ❣✐✈❡s T ∼ N✷ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✐♥✈❡♥t ❛ ❜❡tt❡r ❛❧❣♦r✐t❤♠ ✐♥ ♦r❞❡r t♦ ❦✐❧❧ t❤❡ ❡①tr❛ ❢❛❝t♦r N ❛s ♠✉❝❤ ❛s ✇❡ ❝❛♥✳ ❚❤✐s ✇✐❧❧ ✏❛❧♠♦st✑ ❜❡ ❞♦♥❡✱ ❜✉t ♦♥❧② ✉♣ t♦ r❡❛❝❤ ❝♦♠♣❧❡①✐t② N

✸ ✷ ✱

t❤r♦✉❣❤ t❤❡ tr✐❝❦ ♦❢ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✱ ✇❤✐❝❤ ✐s ✐♥ ♠② ✏●❆❙❈♦♠✷✵✶✻✑ ♣❛♣❡r✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P❧❛♥ ♦❢ t❤❡ t❛❧❦

■t✬s ♥♦✇ ❝❧❡❛r✿ ■✬♠ tr②✐♥❣ t♦ ❵s❡❧❧✬ ②♦✉ t✇♦ r❡❝❡♥t t♦♦❧s ❢♦r ✐♠♣r♦✈✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ✏❇♦❧t③♠❛♥♥✕❧✐❦❡✑ ❛❧❣♦r✐t❤♠s✱ t❤❛t ✐s✱ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠s ✇❤✐❝❤ ✇♦✉❧❞ ❜❡ ❧✐♥❡❛r ✐❢ ✐t ✇❡r❡♥✬t ❢♦r s♦♠❡ s✐③❡ ❝♦♥str❛✐♥t✱ ❛♥❞ ❛r❡ ✐♥st❡❛❞ T ∼ N✶+γ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♠❛♥② r❡♣❡t✐t✐♦♥s ♥❡❝❡ss❛r② t♦ ❣❡t t❤❡ ❞❡s✐r❡❞ s✐③❡✳ ■ ♣r♦♣♦s❡ ②♦✉ t✇♦ ♠❛✐♥ t♦♦❧s✿ ❇♦❧t③♠❛♥♥ ❝❛s❡✿ ✉s❡ t❤❡ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦ ✐♥ ✇✐❞❡ ❣❡♥❡r❛❧✐t②✱ ❜✉t t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ♦♥❧② ❞❡❝r❡❛s❡s t♦ γ

✷

❉✐s❝❧❛✐♠❡r✱ t❤✐s ✐s ✐♥ ♣❛rt ✏✉♥❞❡r ❝♦♥str✉❝t✐♦♥✑✦ ❇r✐❞❣❡ ❝❛s❡✿ ✐♥ s✉✐t❛❜❧❡ ❝✐r❝✉♠st❛♥❝❡s✱ ②♦✉ ❝❛♥ ✉s❡ t❤❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ tr✐❝❦ ❛♥❞ r❡♠♦✈❡ t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ❇❡❢♦r❡ t❤✐s✱ ■ ✇❛♥t t♦ ❞✐s❝✉ss t❤❡ t❤❡♦r❡t✐❝❛❧ ❧✐♠✐t t♦ ♣♦ss✐❜❧❡ ✐♠♣r♦✈❡♠❡♥ts✱ ❛♥❞ ❛ ♥✐❝❡ ❛❧❣♦r✐t❤♠✱ ❞✉❡ t♦ ❇❛❝❤❡r✱ ❇♦❞✐♥✐✱ ❍♦❧❧❡♥❞❡r ❛♥❞ ▲✉♠❜r♦s♦✱ t❤❛t ❝❛♥ r❡❛❝❤ t❤✐s ❧✐♠✐t ✐♥ ❛ s♣❡❝✐❛❧ ❝❛s❡✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞

❲❡ ❤❛✈❡ s❡❡♥ t❤❛t t❤❡ ♥❛ï✈❡ ❛❧❣♦r✐t❤♠ ❤❛s ❝♦♠♣❧❡①✐t② n✶+ d

✷ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡✱ ❛♥❞ n✷ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡✳

❚❤✐s s❡❡♠s ❜❛❞✳ ❇✉t ❤♦✇ ❜❛❞ ❡①❛❝t❧②❄ ❍♦✇ ❣♦♦❞ ❝❛♥ ✇❡ ♣♦ss✐❜❧② ❞♦❄ ▲❡t ✉s tr② t♦ ✉♥❞❡rst❛♥❞ t❤❡ ✐♥tr✐♥s✐❝ ♠✐♥✐♠❛❧ ❝♦♠♣❧❡①✐t② ♦❢ ❛ ♣r♦❜❧❡♠✳ ❚❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐s ❞❡✜♥❡❞ ♦♥❧② ✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✱ ❛♥❞ ✇✐t❤ s♦♠❡ ❞❡❣r❡❡ ♦❢ ❛r❜✐tr❛r✐♥❡ss✳ ■♥st❡❛❞✱ ❢♦r t❤❡ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✱ t❤❛t ✐s✱ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ r❛♥❞♦♠ ❜✐ts ✉s❡❞ ❢♦r s❛♠♣❧✐♥❣ ❛♥ ♦❜❥❡❝t ♦❢ s✐③❡ N✱ ❛❧s♦ t❤❡ ♦✈❡r❛❧❧ ❝♦♥st❛♥ts ❞♦ ♠❛tt❡r✳ ❚❤❡ ✐♥tr✐♥s✐❝ ♠✐♥✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t② ♦❢ ❛♥ ❡①❛❝t s❛♠♣❧✐♥❣ ♣r♦❜❧❡♠ ✐s ❣✐✈❡♥ ❜② t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ♠❡❛s✉r❡✿ SN = −

• C∈CN

µN(C) ln µN(C)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

fi(x)

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

CN = {①} ① = (x✶, . . . , xN) ∈ NN, |①| :=

i xi, r❛♥❞♦♠ ✈❡❝t♦r ♦❢ ✐♥t❡❣❡rs

µN,M(①) = ✶

Z

N

i=✶ f i (xi) × δ|①|,M ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦ ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t ✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

fi(x)

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

CN = {①} ① = (x✶, . . . , xN) ∈ NN, |①| :=

i xi, ←

− r❛♥❞♦♠ ✈❡❝t♦r

♦❢ ✐♥t❡❣❡rs

µN,M(①) = ✶

Z

N

i=✶ f i (xi) × δ|①|,M ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦

← − ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t

✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮

↑ ✈❛r✐❛❜❧❡s ❛r❡ ◆❖❚

✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

fi(x)

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

CN = {①} ① = (x✶, . . . , xN) ∈ NN, |①| :=

i xi, ←

− r❛♥❞♦♠ ✈❡❝t♦r

♦❢ ✐♥t❡❣❡rs

µN,M(①) = ✶

Z

N

i=✶ f i (xi) × δ|①|,M

← − ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿

t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦ ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t ✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮

↑ ✈❛r✐❛❜❧❡s ❛r❡ ◆❖❚

✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

fi(x)

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

CN = {①} ① = (x✶, . . . , xN) ∈ NN, |①| :=

i xi, ←

− r❛♥❞♦♠ ✈❡❝t♦r

♦❢ ✐♥t❡❣❡rs

µN,M(①) = ✶

Z

N

i=✶ f i (xi) × δ|①|,M ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦

← − ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t

✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮

↑

✈❛r✐❛❜❧❡s ❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✿ ❞♦❛❜❧❡ ❜② ✉s✐♥❣ ♣❡r♠✉t❛t✐♦♥ s②♠♠❡tr② ❬▲✳ ❉❡✈r♦②❡✱ ✷✵✶✷❪

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

■♥ t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ♦♥❧② ❝♦♥s✐❞❡r ♣r♦❜❧❡♠s ✐♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♦❢ ❛ s♣❡❝✐❛❧ ❢♦r♠✿

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

fi(x)

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

CN = {①} ① = (x✶, . . . , xN) ∈ NN, |①| :=

i xi, ←

− r❛♥❞♦♠ ✈❡❝t♦r

♦❢ ✐♥t❡❣❡rs

µN,M(①) = ✶

Z

N

i=✶ f i (xi) × δ|①|,M ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t✿ t❤❡ ♣r♦❜❧❡♠ tr✐✈✐❛❧✐s❡s✦

← − ◆❖❚ ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t

✭❛ s✐♥❣❧❡ ❧✐♥❡❛r ❝♦♥str❛✐♥t✮

↑ ✈❛r✐❛❜❧❡s ❛r❡ ◆❖❚

✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞

❲❡ ❛ss✉♠❡ E(

i xi) = M✱ s♦ t❤❛t µN,[α](①) :=

• M

µN,M αN,M =

• i

fi(xi)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

❙♦✱ ✇❤❛t✬s t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ ❛ ♠❡❛s✉r❡ µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M❄

❙✐♠♣❧❡ ❢❛❝t ✶✿ S[µN,[α]] =

i S[fi] = Θ(N)

❙✐♠♣❧❡ ❢❛❝t ✷✿ S[µN,[α]] = −

• m
• ①:|①|=m

αN,m µN,m(①) ln(αN,m µN,m(①)) = EαN

• S(µN,m)
• + S(αN)

◆♦t❡ t❤❛t✱ ❜② ❈▲❚✱ S(αN) = Θ(ln N)✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

❙♦✱ ✇❤❛t✬s t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ ❛ ♠❡❛s✉r❡ µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M❄

❙✐♠♣❧❡ ❢❛❝t ✶✿ S[µN,[α]] =

i S[fi] = Θ(N)

❙✐♠♣❧❡ ❢❛❝t ✷✿ S[µN,[α]] = −

• m
• ①:|①|=m

αN,m µN,m(①) ln(αN,m µN,m(①)) = EαN

• S(µN,m)
• + S(αN)

◆♦t❡ t❤❛t✱ ❜② ❈▲❚✱ S(αN) = Θ(ln N)✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

❙♦✱ ✇❤❛t✬s t❤❡ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ♦❢ ❛ ♠❡❛s✉r❡ µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M❄

❙✐♠♣❧❡ ❢❛❝t ✶✿ S[µN,[α]] =

i S[fi] = Θ(N)

❙✐♠♣❧❡ ❢❛❝t ✷✿ S[µN,[α]] = −

• m
• ①:|①|=m

αN,m µN,m(①) ln(αN,m µN,m(①)) = EαN

• S(µN,m)
• + S(αN)

◆♦t❡ t❤❛t✱ ❜② ❈▲❚✱ S(αN) = Θ(ln N)✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

❆ ❜✐t ♠♦r❡ s✉❜t❧❡✿ ❇② ❈▲❚✱ ❛♥❞ ❝♦♥s✐❞❡r✐♥❣ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❧✐st (f✶, . . . , fN) ✐♥t♦ t✇♦ ❧✐sts ♦❢ s✐③❡ N✶ ❛♥❞ N✷ = N − N✶✱ ✇❡ ❤❛✈❡ αN,mS[µN,m] = −

• m✶

αN✶,m✶αN−N✶,m−m✶ ×

• ①✶:|①✶|=m✶

①✷:|①✷|=m−m✶

fN✶,m✶(①✶)fN−N✶,m−m✶(①✷) ln

• fN✶,m✶(①✶)fN−N✶,m−m✶(①✷)
• =
• m✶

αN✶,m✶αN−N✶,m−m✶

• S[µN✶,m✶] + S[µN−N✶,m−m✶]
• s♦ t❤❛t t❤❡ ❛♥s❛t③ S[fN,M+δ] − S[fN,M] = O(δ✷/N)

❝❛♥ ❜❡ ♣r♦✈❡♥ s❡❧❢✲❝♦♥s✐st❡♥t✳ ❚❤❡s❡ r❡❛s♦♥✐♥❣s ✐♠♣❧② S[fN,M] =

i

S[fi] ✶ + O ln N

N

• ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦

❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

❆ ❜✐t ♠♦r❡ s✉❜t❧❡✿ ❇② ❈▲❚✱ ❛♥❞ ❝♦♥s✐❞❡r✐♥❣ ❛ ♣❛rt✐t✐♦♥ ♦❢ t❤❡ ❧✐st (f✶, . . . , fN) ✐♥t♦ t✇♦ ❧✐sts ♦❢ s✐③❡ N✶ ❛♥❞ N✷ = N − N✶✱ ✇❡ ❤❛✈❡ αN,mS[µN,m] = −

• m✶

αN✶,m✶αN−N✶,m−m✶ ×

• ①✶:|①✶|=m✶

①✷:|①✷|=m−m✶

fN✶,m✶(①✶)fN−N✶,m−m✶(①✷) ln

• fN✶,m✶(①✶)fN−N✶,m−m✶(①✷)
• =
• m✶

αN✶,m✶αN−N✶,m−m✶

• S[µN✶,m✶] + S[µN−N✶,m−m✶]
• s♦ t❤❛t t❤❡ ❛♥s❛t③ S[fN,M+δ] − S[fN,M] = O(δ✷/N)

❝❛♥ ❜❡ ♣r♦✈❡♥ s❡❧❢✲❝♦♥s✐st❡♥t✳ ❚❤❡s❡ r❡❛s♦♥✐♥❣s ✐♠♣❧② S[fN,M] =

i

S[fi] ✶ + O ln N

N

• ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦

❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

■♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡✱ t❤❡ ♠❡❛s✉r❡ µN,α ✐s t❤❡ ♦♥❡ ✐♥❞✉❝❡❞ ❜② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ A✶(z)✱ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ ❵❝♦♠❜✐♥❛t♦r✐❛❧ s♣❡❝✐✜❝❛t✐♦♥✬ s②st❡♠      A✶ = F✶(A✶, . . . , Ak, z) ✳ ✳ ✳ Ak = Fk(A✶, . . . , Ak, z) ❋r♦♠ t❤❡ ♣♦s✐t✐✈✐t② ♦❢ t❤❡ Fi✬s ❝♦❡✣❝✐❡♥ts✱ ✇❡ ❝❛♥ ✐♥t❡r♣r❡t A✶(z) ❛s ❛ s✉♠ ♦✈❡r s✉✐t❛❜❧❡ ✭❝♦❧♦✉r❡❞✱ ✇❡✐❣❤t❡❞✮ tr❡❡s✱ ✇❤❡r❡ t❤❡ ❜r❛♥❝❤✐♥❣ r✉❧❡s ❛r❡ ❞❡s❝r✐❜❡❞ ❜② t❤❡ s♣❡❝✐✜❝❛t✐♦♥✱ ❛♥❞ ❞❡♣❡♥❞ ♦♥ z✳ ❊❛❝❤ tr❡❡ ❝♦rr❡s♣♦♥❞s t♦ ♦♥❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ❡①❛❝t❧② s❛♠♣❧❡❞ ❢r♦♠ µN,α ❆s ❛ r❡s✉❧t✱ ❛♥② s✉❝❤ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❜❛r❡❧② s✉❜✲❝r✐t✐❝❛❧ ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t A st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✵

A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t A st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✵

A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t Az st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✶

A A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t Az st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✶

A A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BBz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✶

A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BBz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✶

A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BAAAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✶

A A B B A A A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BAAAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✶

A A B B A A A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BAzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✷

A A B B A A A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BAzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✷

A A B B A A A

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BBBzAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✷

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BBBzAz st❛❝❦ s✐③❡✿ ✹ ♦❜❥✳ s✐③❡✿ ✷

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BzzBzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✹

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BzzBzAz st❛❝❦ s✐③❡✿ ✸ ♦❜❥✳ s✐③❡✿ ✹

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BzzBzzz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✺

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t BzzBzzz st❛❝❦ s✐③❡✿ ✷ ♦❜❥✳ s✐③❡✿ ✺

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t zzzzBzzz st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✼

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t zzzzBzzz st❛❝❦ s✐③❡✿ ✶ ♦❜❥✳ s✐③❡✿ ✼

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

❇② r❛♥❞♦♠✐s✐♥❣ ♦♥ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦s✐t✐♦♥✱ ❛♥② ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ r❡✇r✐t✐♥❣ s②st❡♠✱ ❊①❛♠♣❧❡✿ ❢♦r A = A z + B✷ + z B = A✸ + z✷ ✇❡ ❝♦✉❧❞ ❣❡t zzzzzzzzz st❛❝❦ s✐③❡✿ ✵ ♦❜❥✳ s✐③❡✿ ✾

A A B B A A A B B

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

■♥ t❤❡ ❧✐♠✐t✱ ♦♥ ❛ s✉❝❝❡ss❢✉❧ r✉♥✱ t❤❡ st❛❝❦ s✐③❡ ♣r♦✜❧❡ ✐s ❛♥ ❡①❝✉rs✐♦♥✱ ✇❤✐❧❡ t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡ ✐s ❛ str❛✐❣❤t ❧✐♥❡✱ ❜♦t❤ ♦❢ ❧❡♥❣t❤ L ❈❛❧❧ n✶(t), . . . , nk(t), s(t) t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts A✶✱ ✳ ✳ ✳ Ak ✐♥ t❤❡ st❛❝❦✱ ❛♥❞ ♦❢ t❤❡ ♦❜❥❡❝t s✐③❡✳ ∼ √ N N L = Θ(N) ❆t ❧✐♥❡❛r ♦r❞❡r✱ t❤❡s❡ q✉❛♥t✐t✐❡s s❛t✐s❢② ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠

✶

❤❛s ❛ ✉♥✐q✉❡ ❡✐❣❡♥✈❡❝t♦r

✶

✇✐t❤ ❛❧❧ ✵✱ ✶✱ ❛♥❞ ❡✐❣❡♥✈❛❧✉❡ ③❡r♦✱ ❛❧❧ ♦t❤❡r ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ♥❡❣❛t✐✈❡✳ ❚❤✐s ✐s t❤❡ ❡✐❣❡♥✈❛❧✉❡✴✈❡❝t♦r ❛ss♦❝✐❛t❡❞ t♦ t❤❡ st❛❝❦ s✐③❡ ❡①❝✉rs✐♦♥✱ ✇❤✐❝❤ ❤❛s ❤❡✐❣❤t ✱ s♦ ✐t ✈❛♥✐s❤❡s ❛t ❧✐♥❡❛r ♦r❞❡r✳ ❋♦r t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡✱ ✇❡ ❤❛✈❡

✶

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

■♥ t❤❡ ❧✐♠✐t✱ ♦♥ ❛ s✉❝❝❡ss❢✉❧ r✉♥✱ t❤❡ st❛❝❦ s✐③❡ ♣r♦✜❧❡ ✐s ❛♥ ❡①❝✉rs✐♦♥✱ ✇❤✐❧❡ t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡ ✐s ❛ str❛✐❣❤t ❧✐♥❡✱ ❜♦t❤ ♦❢ ❧❡♥❣t❤ L ❈❛❧❧ n✶(t), . . . , nk(t), s(t) t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts A✶✱ ✳ ✳ ✳ Ak ✐♥ t❤❡ st❛❝❦✱ ❛♥❞ ♦❢ t❤❡ ♦❜❥❡❝t s✐③❡✳ ❆t ❧✐♥❡❛r ♦r❞❡r✱ t❤❡s❡ q✉❛♥t✐t✐❡s s❛t✐s❢② ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ˙ ni = t−✶

j nj(−δij + Ai Aj ∂ ∂Ai Fj)| z→z∗ Aj→A∗

j

=:

j Mijnj/t

M ❤❛s ❛ ✉♥✐q✉❡ ❡✐❣❡♥✈❡❝t♦r (p✶, . . . , pk) ✇✐t❤ ❛❧❧ pi > ✵✱

i pi = ✶✱

❛♥❞ ❡✐❣❡♥✈❛❧✉❡ ③❡r♦✱ ❛❧❧ ♦t❤❡r ❡✐❣❡♥✈❛❧✉❡s ❛r❡ ♥❡❣❛t✐✈❡✳ ❚❤✐s ✐s t❤❡ ❡✐❣❡♥✈❛❧✉❡✴✈❡❝t♦r ❛ss♦❝✐❛t❡❞ t♦ t❤❡ st❛❝❦ s✐③❡ ❡①❝✉rs✐♦♥✱ ✇❤✐❝❤ ❤❛s ❤❡✐❣❤t ∼ √ N✱ s♦ ✐t ✈❛♥✐s❤❡s ❛t ❧✐♥❡❛r ♦r❞❡r✳ ❋♦r t❤❡ ♦❜❥❡❝t s✐③❡ ♣r♦✜❧❡✱ ✇❡ ❤❛✈❡ ˙ s = t−✶

j nj( z Aj ∂ ∂z Fj)| z→z∗ Aj→A∗

j

≃

j pj( z Aj ∂ ∂z Fj)| z→z∗ Aj→A∗

j ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❜♦✉♥❞ ✐♥ t❤❡ ❇♦❧t③♠❛♥♥ ❝❛s❡

◆♦t❡ ❤♦✇ t❤❡ r❛♥❞♦♠✐s❛t✐♦♥ ♦❢ t❤❡ ❣r♦✇t❤ ♣♦s✐t✐♦♥ ❤❛s ♠❛❞❡ t❤❡ ❞②♥❛♠✐❝s ❛s②♠♣t♦t✐❝❛❧❧② ❤♦♠♦❣❡♥❡♦✉s ✐♥ t✐♠❡ ✭t❤❛t ✐s✱ ❤♦♠♦❣❡♥❡♦✉s ✉♣ t♦ O(N− ✶

✹ ) ✢✉❝t✉❛t✐♦♥s✱

✇❤❡♥ t❤❡ st❛❝❦ s✐③❡ ✐s ❢❛r ❢r♦♠ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s✮ ❆s ❛ r❡s✉❧t✱ ❢r♦♠ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ s♣❡❝✐✜❝❛t✐♦♥✱ ❛♥❞ t❤❡ s❡❧❡❝t✐♦♥ ♦❢ t❤❡ ✏❣♦♦❞✑ ❝r✐t✐❝❛❧ ♣♦✐♥t z∗✱ ✇❡ ❝❛♥ r❡❛❞ t❤❡ ❧✐♠✐t ♣❛r❛♠❡t❡rs pi ❛♥❞ ˙ s ❆♥② ●❛❧t♦♥✕❲❛ts♦♥ ❜r❛♥❝❤✐♥❣ ♦♥ ❛ t②♣❡✲Ai ♦❜❥❡❝t ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝♦♥str✉❝t✐♦♥ ✇✐t❤ ❛♥ ✐♥tr✐♥s✐❝ ❙❤❛♥♥♦♥ ❡♥tr♦♣② Si ❏✉st ❛s ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡✱ ❤❛✈✐♥❣ ❛♥ ❡①❝✉rs✐♦♥ ✐♥st❡❛❞ ♦❢ ❛ ❣❡♥❡r✐❝ ✇❛❧❦ ♦♥❧② r❡❞✉❝❡s t❤❡ ❡♥tr♦♣② ❜② ❛ ❢❛❝t♦r ∼ (ln N)/N ❚❤✐s ❣✐✈❡s L = N/ ˙ s + O( √ N)✱ ❛♥❞ S ≃ N

• ✶

˙ s

k

i=✶ piSi

• (✶ + O(N− ✶

✹ ))✱

✇❤✐❝❤ ✐s ♦✉r ✜♥❛❧ r❡s✉❧t

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞✿ s✉♠♠❛r②

■♥ ❝♦♥❝❧✉s✐♦♥✱ ✐♥ ♦✉r ❝❛s❡ ♦❢ st✉❞②✱ ②♦✉ ❤❛✈❡ ❛♥ ♦♣t✐♠❛❧ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✐❢ t❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐s ❧✐♥❡❛r✱ ❛♥❞ t❤❡ r❛♥❞♦♠ ❜✐t ❝♦♠♣❧❡①✐t② Tr❛♥❞(N) ✐s✱ ✉♣ t♦ ❝♦rr❡❝t✐♦♥s✱ ❇r✐❞❣❡ ❝❛s❡✿ µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M

Tr❛♥❞(N, M) =

i

S[fi]

• (✶ + o (✶))

❇♦❧t③♠❛♥♥ ❝❛s❡✿ {Ai = Fi(A✶, . . . , Ak, z)}i=✶,...,k Tr❛♥❞(N) = N ˙ s

k

• i=✶

piSi

• (✶ + o(✶))

■ ✇✐❧❧ ♥♦t s❤♦✇ ❛♥② ❛❧❣♦r✐t❤♠ ♦❢ ♠✐♥❡ t❤❛t r❡❛❝❤❡s ♦♣t✐♠❛❧✐t②✳ ❇✉t ■ ✇✐❧❧ s❤♦✇ ②♦✉ t❤❛t ♦♣t✐♠❛❧✐t② ❡①✐sts✦ ❇② ♣r❡s❡♥t✐♥❣ ②♦✉ t❤❡ ✏♠♦t❤❡r ♦❢ ❛❧❧ ✭❇r✐❞❣❡✲❝❛s❡ ❡①❛❝t s❛♠♣❧✐♥❣✮ ❛❧❣♦r✐t❤♠s✑

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❤❛♥♥♦♥ ❝♦♠♣❧❡①✐t② ❜♦✉♥❞✿ s✉♠♠❛r②

■♥ ❝♦♥❝❧✉s✐♦♥✱ ✐♥ ♦✉r ❝❛s❡ ♦❢ st✉❞②✱ ②♦✉ ❤❛✈❡ ❛♥ ♦♣t✐♠❛❧ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✐❢ t❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐s ❧✐♥❡❛r✱ ❛♥❞ t❤❡ r❛♥❞♦♠ ❜✐t ❝♦♠♣❧❡①✐t② Tr❛♥❞(N) ✐s✱ ✉♣ t♦ ❝♦rr❡❝t✐♦♥s✱ ❇r✐❞❣❡ ❝❛s❡✿ µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M

Tr❛♥❞(N, M) =

i

S[fi]

• (✶ + o (✶))

❇♦❧t③♠❛♥♥ ❝❛s❡✿ {Ai = Fi(A✶, . . . , Ak, z)}i=✶,...,k Tr❛♥❞(N) = N ˙ s

k

• i=✶

piSi

• (✶ + o(✶))

■ ✇✐❧❧ ♥♦t s❤♦✇ ❛♥② ❛❧❣♦r✐t❤♠ ♦❢ ♠✐♥❡ t❤❛t r❡❛❝❤❡s ♦♣t✐♠❛❧✐t②✳ ❇✉t ■ ✇✐❧❧ s❤♦✇ ②♦✉ t❤❛t ♦♣t✐♠❛❧✐t② ❡①✐sts✦ ❇② ♣r❡s❡♥t✐♥❣ ②♦✉ t❤❡ ✏♠♦t❤❡r ♦❢ ❛❧❧ ✭❇r✐❞❣❡✲❝❛s❡ ❡①❛❝t s❛♠♣❧✐♥❣✮ ❛❧❣♦r✐t❤♠s✑

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ❤✐❞❞❡♥ ✐♥ ❛ s♠❛❧❧ ❝♦r♥❡r ♦❢ t❤❡ ♣❛♣❡r ❇❛❝❤❡r✱ ❇♦❞✐♥✐✱ ❍♦❧❧❡♥❞❡r ❛♥❞ ▲✉♠❜r♦s♦✱ ▼❡r❣❡❙❤✉✤❡✿ ❆ ❱❡r② ❋❛st✱ P❛r❛❧❧❡❧ ❘❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❆❧❣♦r✐t❤♠ ❤tt♣s✿✴✴❛r①✐✈✳♦r❣✴♣❞❢✴✶✺✵✽✳✵✸✶✻✼ ❚❤❡ ♣r♦❜❧❡♠✿ ❡①❛❝t s❛♠♣❧✐♥❣ ♦❢ str✐♥❣s ✐♥ {•, ◦}n ✇✐t❤ #{•} = k ❇❇❍▲ s♦❧✈❡s ✐t ✐♥ ❧✐♥❡❛r t✐♠❡ ❛♥❞ ♦♣t✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✳ ✭✇❤✐❝❤ ✐s Tr❛♥❞(n) = n(−p ln p − (✶ − p) ln(✶ − p)) + o(n)✱ ✇✐t❤ p = k

n✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬

❋✐rst ♥❛ï✈❡ ✐❞❡❛✿ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑ ♥❛ï✈❡ ❛♣♣r♦❛❝❤✳ ❙❛♠♣❧❡ n ✈❛r✐❛❜❧❡s ① = (x✶, . . . , xn) ∈ {✵, ✶}n✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥p✱ r❡st❛rt ✐❢ |①| = k✳ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ ∼ n

✸ ✷ ✱

❜❡❝❛✉s❡ |①| ✐s ❞✐str✐❜✉t❡❞ r♦✉❣❤❧② ❛s ❛ ●❛✉ss✐❛♥ ♦❢ ✈❛r✐❛♥❝❡ θ(n) ❛♥❞ ♠❡❛♥ k✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬

❙❡❝♦♥❞ ♥❛ï✈❡ ✐❞❡❛✿ ♣r♦❥❡❝t ❞♦✇♥ ❢r♦♠ ❋✐s❤❡r✕❨❛t❡s ❚❤❡ ❋✐s❤❡r✕❨❛t❡s ❛❧❣♦r✐t❤♠ s❛♠♣❧❡s ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ σ ∈ Sn ✇✐t❤ ♦♣t✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✿ Tr❛♥❞(n) ≃ ln n! ≃ n(ln n − ✶) ■t ✇♦r❦s ❜② s❛♠♣❧✐♥❣ ② ∈ {✶} × {✶, ✷} × {✶, ✷, ✸} × · · · × {✶, . . . , n}✱ ❛♥❞ ❞♦✐♥❣ ❛s ❢♦❧❧♦✇s✿ ❚❤❡♥✱ ❵♣r♦❥❡❝t✐♥❣ ❞♦✇♥✬ ♠❡❛♥s xi = ✶ ✐✛ σ−✶(i) ≤ k ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ ∼ n ln n✱ ❜❡❝❛✉s❡✱ ❡✈❡♥ ✐❢ ❋✐s❤❡r✕❨❛t❡s ✐s ♦♣t✐♠❛❧✱ t❤❡ ♣r♦❥❡❝t✐♦♥ t❤r♦✇s ❛✇❛② ♠♦st ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥

1 2 3 4 5 6 7 8 9 1 3 7 5 1 8 1 0 9 2 4 6 1 1 1 2 3 6 2 5 7 6

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬

❙❡❝♦♥❞ ♥❛ï✈❡ ✐❞❡❛✿ ♣r♦❥❡❝t ❞♦✇♥ ❢r♦♠ ❋✐s❤❡r✕❨❛t❡s ❚❤❡ ❋✐s❤❡r✕❨❛t❡s ❛❧❣♦r✐t❤♠ s❛♠♣❧❡s ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ σ ∈ Sn ✇✐t❤ ♦♣t✐♠❛❧ r❛♥❞♦♠✲❜✐t ❝♦♠♣❧❡①✐t②✿ Tr❛♥❞(n) ≃ ln n! ≃ n(ln n − ✶) ■t ✇♦r❦s ❜② s❛♠♣❧✐♥❣ ② ∈ {✶} × {✶, ✷} × {✶, ✷, ✸} × · · · × {✶, . . . , n}✱ ❛♥❞ ❞♦✐♥❣ ❛s ❢♦❧❧♦✇s✿ ❚❤❡♥✱ ❵♣r♦❥❡❝t✐♥❣ ❞♦✇♥✬ ♠❡❛♥s xi = ✶ ✐✛ σ−✶(i) ≤ k ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ ∼ n ln n✱ ❜❡❝❛✉s❡✱ ❡✈❡♥ ✐❢ ❋✐s❤❡r✕❨❛t❡s ✐s ♦♣t✐♠❛❧✱ t❤❡ ♣r♦❥❡❝t✐♦♥ t❤r♦✇s ❛✇❛② ♠♦st ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥

1 2 3 4 5 6 7 8 9 1 3 7 5 1 8 1 0 9 2 4 6 1 1 1 2 3 6 2 5 7 6

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬

❚❤❡ ❣♦♦❞ ✐❞❡❛✿ ❙❛♠♣❧❡ t❤❡ n ✈❛r✐❛❜❧❡s ① = (x✶, . . . , xn) ∈ {✵, ✶}n✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥p✱ ♦♥❡ ❜② ♦♥❡ ✉♣ t♦ ✇❤❡♥ ②♦✉ ❤❛✈❡ k ❡♥tr✐❡s xi = ✶✱ ♦r n − k ❡♥tr✐❡s xi = ✵✳ ❚❤❡♥ ❝♦♠♣❧❡t❡ ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ✇✐t❤ ✇❤❛t ✐s ♥❡❡❞❡❞✱ ❋✐♥❛❧❧②✱ ♣❡r❢♦r♠ ❋✐s❤❡r✕❨❛t❡s s❤✉✤✐♥❣s ♦♥ t❤❡s❡ ❧❛st ❛❞❞❡❞ st❡♣s✳ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ Tr❛♥❞(n) = S[µ] + O(√n ln n) ❜❡❝❛✉s❡ t❤❡ ✜♥❛❧ s❤✉✤❡s ❛r❡ ❥✉st ❛ ❢❡✇✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬

❚❤❡ ❣♦♦❞ ✐❞❡❛✿ ❙❛♠♣❧❡ t❤❡ n ✈❛r✐❛❜❧❡s ① = (x✶, . . . , xn) ∈ {✵, ✶}n✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥p✱ ♦♥❡ ❜② ♦♥❡ ✉♣ t♦ ✇❤❡♥ ②♦✉ ❤❛✈❡ k ❡♥tr✐❡s xi = ✶✱ ♦r n − k ❡♥tr✐❡s xi = ✵✳ ❚❤❡♥ ❝♦♠♣❧❡t❡ ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ✇✐t❤ ✇❤❛t ✐s ♥❡❡❞❡❞✱ ❋✐♥❛❧❧②✱ ♣❡r❢♦r♠ ❋✐s❤❡r✕❨❛t❡s s❤✉✤✐♥❣s ♦♥ t❤❡s❡ ❧❛st ❛❞❞❡❞ st❡♣s✳ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ Tr❛♥❞(n) = S[µ] + O(√n ln n) ❜❡❝❛✉s❡ t❤❡ ✜♥❛❧ s❤✉✤❡s ❛r❡ ❥✉st ❛ ❢❡✇✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ ❵t❤❡ ♠♦t❤❡r ♦❢ ❛❧❧ ❛❧❣♦r✐t❤♠s✬

❚❤❡ ❣♦♦❞ ✐❞❡❛✿ ❙❛♠♣❧❡ t❤❡ n ✈❛r✐❛❜❧❡s ① = (x✶, . . . , xn) ∈ {✵, ✶}n✱ ✐✳✐✳❞✳ ✇✐t❤ ❇❡r♥p✱ ♦♥❡ ❜② ♦♥❡ ✉♣ t♦ ✇❤❡♥ ②♦✉ ❤❛✈❡ k ❡♥tr✐❡s xi = ✶✱ ♦r n − k ❡♥tr✐❡s xi = ✵✳ ❚❤❡♥ ❝♦♠♣❧❡t❡ ❞❡t❡r♠✐♥✐st✐❝❛❧❧② ✇✐t❤ ✇❤❛t ✐s ♥❡❡❞❡❞✱ ❋✐♥❛❧❧②✱ ♣❡r❢♦r♠ ❋✐s❤❡r✕❨❛t❡s s❤✉✤✐♥❣s ♦♥ t❤❡s❡ ❧❛st ❛❞❞❡❞ st❡♣s✳ ❆✈❡r❛❣❡ ❝♦♠♣❧❡①✐t②✿ Tr❛♥❞(n) = S[µ] + O(√n ln n) ❜❡❝❛✉s❡ t❤❡ ✜♥❛❧ s❤✉✤❡s ❛r❡ ❥✉st ❛ ❢❡✇✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇❇❍▲ ❛❧❣♦r✐t❤♠✿ t❤❡ ❝♦❞❡

❆❧❣♦r✐t❤♠ ✿ ❇❇❍▲ s❤✉✤✐♥❣ ❛❧❣♦r✐t❤♠ ❜❡❣✐♥ a = k, b = n − k, i = ✵❀ r❡♣❡❛t i ✰✰❀ νi ← − ❇❡r♥β❀ ✐❢ νi = ✶ t❤❡♥ a ✲✲ ❡❧s❡ b ✲✲ ✉♥t✐❧ a < ✵ ♦r b < ✵ ❝♦♠♣❧❡①✐t② ∼ n❀ ✐❢ a < ✵ t❤❡♥ ¯ ν = ✵ ❡❧s❡ ¯ ν = ✶❀ ❢♦r j ← i t♦ n ❞♦ νj = ¯ ν❀ h ← − ❘♥❞■♥tj❀ s✇❛♣ νj ❛♥❞ νh ❝♦♠♣❧❡①✐t② ∼ √n ln n❀ r❡t✉r♥ ν ❡♥❞

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P❧❛♥ ♦❢ t❤❡ t❛❧❦

■ r❡❝❛❧❧ ②♦✉ t❤❛t ■✬♠ tr②✐♥❣ t♦ ❵s❡❧❧✬ ②♦✉ t✇♦ r❡❝❡♥t t♦♦❧s ❢♦r ✐♠♣r♦✈✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ✏❇♦❧t③♠❛♥♥✕❧✐❦❡✑ ❛❧❣♦r✐t❤♠s✱ t❤❛t ✐s✱ ❡①❛❝t s❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠s ✇✐t❤ ❝♦♠♣❧❡①✐t② T ∼ N✶+γ ❜❡❝❛✉s❡ ♦❢ t❤❡ ♠❛♥② r❡♣❡t✐t✐♦♥s ♥❡❝❡ss❛r② t♦ ❣❡t t❤❡ ❞❡s✐r❡❞ s✐③❡✳ ■ ♣r♦♣♦s❡ ②♦✉ t✇♦ ♠❛✐♥ t♦♦❧s✿ ❇♦❧t③♠❛♥♥ ❝❛s❡✿ ✉s❡ t❤❡ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦ ✐♥ ✇✐❞❡ ❣❡♥❡r❛❧✐t②✱ ❜✉t t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ♦♥❧② ❞❡❝r❡❛s❡s t♦ γ

✷

❇r✐❞❣❡ ❝❛s❡✿ ✐♥ s✉✐t❛❜❧❡ ❝✐r❝✉♠st❛♥❝❡s✱ ②♦✉ ❝❛♥ ✉s❡ t❤❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ tr✐❝❦ ❛♥❞ r❡♠♦✈❡ t❤❡ ❡①tr❛ ❡①♣♦♥❡♥t ❲❡ ❛r❡ r❡❛❞② t♦ ❣♦✦ ▲❡t ✉s st❛rt ✇✐t❤ t❤❡ ❇r✐❞❣❡ ❝❛s❡ ❛♥❞ t❤❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ tr✐❝❦

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ♦♥❡ s❧✐❞❡

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

fi(x)

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

Cn = {①}, ① = (x✶, . . . , xN) ∈ NN, |①| :=

i xi

µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M

Pr♦❜❧❡♠✿ ❆ss✉♠❡ t❤❛t s❛♠♣❧✐♥❣ ❢r♦♠ ❡❛❝❤ ❞✐str✐❜✳ fi ❝♦sts O(✶)✳ ❋✐♥❞ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t s❛♠♣❧❡s ❢r♦♠ t❤❡ ❞✐str✐❜✉t✐♦♥ µN,M ✐♥ ❛✈❡r❛❣❡ ❧✐♥❡❛r t✐♠❡✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ ♦♥❡ s❧✐❞❡

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

fi(x)

0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5 0 1 2 3 4 0.5

01234 0.5 01234 0.5 01234 0.5 01234 0.5 01234 0.5 01234 0.5

qi(s)

01234 0.5 01234 0.5 01234 0.5 01234 0.5 01234 0.5 01234 0.5

0 1 0.2 0.4 0.6 0.8 1

g(x) Cn = {①}, ① = (x✶, . . . , xN) ∈ NN, |①| :=

i xi

µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M

❖✉r s♦❧✉t✐♦♥✿ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts g(x) ∈ {❇❡r♥b, P♦✐ss, ●❡♦♠b}✱ ❛♥❞ {qi(s)}✶≤i≤n;s∈N r❡❛❧ ♣♦s✐t✐✈❡✱ s✉❝❤ t❤❛t fi(x) =

s qi(s)g∗s(x)✳ ❚❤❡♥ ♦✉r ♥❡✇ ❛❧❣♦r✐t❤♠ ❞♦❡s ✐t✦

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t✇♦ s❧✐❞❡s

❖✉r ♥❡✇ tr✐❝❦ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡❛s✿ ◮ ❘❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠s ❤❛✈❡ ❛♥ ❡①tr❛ ❢❛❝t♦r ✐♥ t❤❡✐r ❝♦♠♣❧❡①✐t②✱ ♦♥ t❤❡ s❝❛❧❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡✳ ■♥ ♦r❞❡r t♦ ❤❛✈❡ t❤❡ ♦♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② s❝❛❧✐♥❣✱ ②♦✉ ♥❡❡❞ t❤❡ ❛✈❡r❛❣❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ t♦ ❜❡ Θ(✶)✱ ✐✳❡✳ ♥♦t t♦ s❝❛❧❡ ✇✐t❤ t❤❡ s✐③❡ n✳ ◮ P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❣✐✈❡s fi(x) =

s qi(s)g∗s(x)✳

❆s ❛ r❡s✉❧t t❤❡ ♠❡❛s✉r❡ µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M

✐s ❛ ♠❛r❣✐♥❛❧ ♦❢ ❛ ♠❡❛s✉r❡ ✐♥ t✇♦ s❡ts ♦❢ ✈❛r✐❛❜❧❡s✿ µN,M(①, s) = ✶

Z

N

i=✶

• qi(si) g∗si(xi)
• × δ|①|,M✳

◮ ❨♦✉ ❝❛♥ ✜rst s❛♠♣❧❡ s✱ ✇✐t❤ ♠❡❛s✉r❡ µ✶(s) = N

i=✶ qi(si)✱

t❤❡♥ ❛❝❝❡♣t t❤✐s ✈❡❝t♦r s ✇✐t❤ r❛t❡ a(s) ∝ g∗|s|(M)✱ ❛♥❞ ✜♥❛❧❧② s❛♠♣❧❡ ① ✇✐t❤ ♠❡❛s✉r❡ µ✷(① | s) = N

i=✶ g∗si(xi)✳

◮ ❚❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ ✐s ❤✐❣❤ ❜❡❝❛✉s❡✱ ❛❧t❤♦✉❣❤ g∗|s|(M) = Θ(N− ✶

✷ )✱ ✇❡ ❤❛✈❡ g∗|s|(M)/ maxN(g∗N(M)) = Θ(✶)✳ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ r❡♠✐♥❞❡r ♦❢ t❤❡ ♥❛ï✈❡ ❛❧❣♦r✐t❤♠ ✐♥ t❤❡ ❇r✐❞❣❡ ❝❛s❡

❯♣ t♦ r❡❞❡✜♥✐♥❣ t❤❡ fi✬s✱ ✇❡ ❝❛♥ ❛ss✉♠❡ ✇✳❧✳♦✳❣✳ t❤❛t E(|①|) =

i E[fi] = M✳

❆ss✉♠❡ t❤❛t ❜♦t❤ M ❛♥❞ σ✷ :=

i V❛r[fi] ❛r❡ Θ(N)✳

❚❤❡ ❵❇r✐❞❣❡ ❝❛s❡✬ r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✇♦✉❧❞ ❣✐✈❡✿ ❆❧❣♦r✐t❤♠ ✿ ◆❛ï✈❡ r❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ ❝♦♠♣❧❡①✐t② ∼ N✸/✷ ❜❡❣✐♥ r❡♣❡❛t |①| = ✵❀ ❢♦r i ← ✶ t♦ N ← ❝♦♠♣❧❡①✐t② ∼ N❀ ❞♦ xi ⇐ fi |①|✰❂xi❀ ✉♥t✐❧ |①| = M ← ❝♦♠♣❧❡①✐t② ∼ √ N❀ r❡t✉r♥ (x✶, . . . , xN) ❡♥❞

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ r❡❥❡❝t✐♦♥ ♣❛r❛❞✐❣♠

▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐♥ ❛♥② r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠✱ ②♦✉ ✇❛♥t t♦ ❞♦ ❡①❛❝t s❛♠♣❧✐♥❣ ❢♦r µ(①)✱ ✇❤❡♥ µ(①) ∝ µ✵(①)a(①)✱ ✇✐t❤ a(①) ∈ [✵, ✶]✱ s✉♣♣♦s✐♥❣ t❤❛t ②♦✉ ❦♥♦✇ ❤♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ µ✵ ❆❧❣♦r✐t❤♠ ✿ ❘❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ T[µ] ∼ T[µ✵] E(a(①))−✶ ❜❡❣✐♥ r❡♣❡❛t ① ⇐ µ✵ ; ← ❝♦♠♣❧❡①✐t② T[µ✵] α ⇐ ❇❡r♥a(①)❀ ✉♥t✐❧ α = ✶ ← ❝♦♠♣❧❡①✐t② E(a(①))−✶❀ r❡t✉r♥ ① ❡♥❞

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ r❡❥❡❝t✐♦♥ ♣❛r❛❞✐❣♠ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s

◆♦✇ ❛ss✉♠❡ µ(①) ∝

② µ✶(②) µ✷(① | ②) a(②)✱ ✇✐t❤ a(②) ∈ [✵, ✶]✱

s✉♣♣♦s✐♥❣ t❤❛t ②♦✉ ❦♥♦✇ ❤♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ µ✶✱ ❛♥❞ µ✷(· | ②) ❆❧❣♦r✐t❤♠ ✿ ❘❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s ❜❡❣✐♥ r❡♣❡❛t ② ⇐ µ✶ ; ← s❛♠♣❧❡ ❛ t❡♥t❛t✐✈❡ ② ✇✐t❤ µ✶ α ⇐ ❇❡r♥a(②) ✉♥t✐❧ α = ✶ ← ❛❝❝❡♣t ② ✇✐t❤ r❛t❡ a(②)❀ ① ⇐ µ✷(· | ②) ; ← s❛♠♣❧❡ ① ✇✐t❤ µ✷( · |②) r❡t✉r♥ ① ❡♥❞ T =

• ② µ✶(②)
• T✶(②) + a(②)T✷(②)
• ② µ✶(②)a(②)

= E(T✶ + aT✷) E(a) ≤ T ♠❛①

✶

E(a) +T ♠❛①

✷

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ r❡❥❡❝t✐♦♥ ♣❛r❛❞✐❣♠ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s

◆♦✇ ❛ss✉♠❡ µ(①) ∝

② µ✶(②) µ✷(① | ②) a(②)✱ ✇✐t❤ a(②) ∈ [✵, ✶]✱

s✉♣♣♦s✐♥❣ t❤❛t ②♦✉ ❦♥♦✇ ❤♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ µ✶✱ ❛♥❞ µ✷(· | ②) ❆❧❣♦r✐t❤♠ ✿ ❘❡❥❡❝t✐♦♥ s❛♠♣❧✐♥❣ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s ❜❡❣✐♥ r❡♣❡❛t ② ⇐ µ✶ ; ← ❝♦♠♣❧❡①✐t② T✶(②) α ⇐ ❇❡r♥a(②) ✉♥t✐❧ α = ✶ ← ❝♦♠♣❧❡①✐t② a(②)−✶❀ ① ⇐ µ✷(· | ②) ; ← ❝♦♠♣❧❡①✐t② T✷(②) r❡t✉r♥ ① ❡♥❞ T =

• ② µ✶(②)
• T✶(②) + a(②)T✷(②)
• ② µ✶(②)a(②)

= E(T✶ + aT✷) E(a) ≤ T ♠❛①

✶

E(a) +T ♠❛①

✷

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♣r♦✈✐❞❡s ❛ ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡

P♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ t❡❧❧s t❤❛t✱ ❢♦r ❛❧❧ i✱ fi(x) =

s qi(s)g∗s(x)✱ ✇✐t❤ qi(s) ≥ ✵✳

❋r♦♠ t❤❡ ♥♦r♠❛❧✐s❛t✐♦♥ ♦❢ t❤❡ fi✬s ❛♥❞ ♦❢ g✱ ✐t ❢♦❧❧♦✇s t❤❛t ❛❧s♦ t❤❡ qi(s) ❛r❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✳ ❆s ❛ r❡s✉❧t t❤❡ ♠❡❛s✉r❡ µN,M(①) = ✶

Z

N

i=✶ fi(xi) × δ|①|,M

✐s ❛ ♠❛r❣✐♥❛❧ ♦❢ ❛ ♠❡❛s✉r❡ ✐♥ t✇♦ s❡ts ♦❢ ✈❛r✐❛❜❧❡s✿ µN,M(①, s) = ✶

Z

N

i=✶

• qi(si) g∗si(xi)
• × δ|①|,M✳

❚❤✐s ✐s ❡①❛❝t❧② ❛s ✐♥ ❛ ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡✱ ✇✐t❤ ❝♦rr❡s♣♦♥❞❡♥❝❡ s❛♠♣❧❡ s ✇✐t❤ ♠❡❛s✉r❡ µ✶(s) µ✶(s) = n

i=✶ qi(si)

❛❝❝❡♣t s ✇✐t❤ r❛t❡ a(s) a(s) ∝ g∗|s|(m) s❛♠♣❧❡ ① ✇✐t❤ ♠❡❛s✉r❡ µ✷(① | s) µ✷(① | s) = n

i=✶ g∗si(xi)

◆♦t❡✿ ❛❧t❤♦✉❣❤ µ✶ ❛♥❞ µ✷ ❞❡♣❡♥❞ ♦♥ t❤❡ ✈❡❝t♦r s✱ t❤❡ r❛t❡ a ♦♥❧② ❞❡♣❡♥❞s ♦♥ |s| =

i si✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

■♥❝r❡❛s✐♥❣ t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡

❚❤❡ ❝r✉❝✐❛❧ ♣♦✐♥t ✐s t❤❛t t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛❧❧♦✇s t♦ ✐♥❝r❡❛s❡ t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡✦ ■♥ t❤❡ ❵♦r❞✐♥❛r②✬ r❡❥❡❝t✐♦♥ s❝❤❡♠❡✱ ②♦✉ ❛❝❝❡♣t ① ✐✛ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ❡✈❡♥t ♦❝❝✉rs ✭✐♥ ♦✉r ❝❛s❡✱ |①| = M✮✳ ■❢ t❤✐s ♣r♦❜❛❜✐❧✐t② ✐s ✐♥tr✐♥s✐❝❛❧❧② s♠❛❧❧ ✭✐♥ ♦✉r ❝❛s❡✱ Θ(N−✶/✷)✮✱ t❤❡r❡ ✐s ♥♦t❤✐♥❣ ②♦✉ ❝❛♥ ❞♦✳ ■♥ t❤❡ r❡❥❡❝t✐♦♥ s❝❤❡♠❡ ❢♦r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡s✱ t❤❡ r❛t❡ a(s) ✐s ❞❡✜♥❡❞ ✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❢❛❝t♦r✱ ❛s ❧♦♥❣ ❛s maxs a(s) ≤ ✶✳ ❍❡r❡✱ t❤❡ ♦❜✈✐♦✉s ❝❤♦✐❝❡ ❢♦r a(s) ✐s a(s) = g∗|s|(M)✱ ✇❤✐❝❤ ✐s Θ(N− ✶

✷ )✳

❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♣✉s❤ ✐t ✉♣ t♦ a(s) = g∗|s|(M) maxn(g∗n(M))✳ ❆s ✇❡ ✇✐❧❧ s❡❡✱ ✇✐t❤ t❤✐s ❝❤♦✐❝❡ E(a(s)) = Θ(✶)✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❍♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ ❇❡r♥a(s)

❚❤✐s ✐❞❡❛ ✐s ♥♦t s✉✣❝✐❡♥t ❜② ✐ts❡❧❢✳ ❊✈❡♥ ✐❢ ②♦✉ ❦♥♦✇ ✐♥ ❛❞✈❛♥❝❡ t❤❛t✱ ❛❢t❡r ♠❛①✐♠✐s❛t✐♦♥✱ E(a(s)) = Θ(✶)✱ ②♦✉ st✐❧❧ ❤❛✈❡ ❛ ♣r♦❜❧❡♠✿ s❛♠♣❧✐♥❣ ❛ ❇❡r♥♦✉❧❧✐ r♥❞ ✈❛r ✇✐t❤ ♣❛r❛♠❡t❡r a(s) ✐s ❞✐✣❝✉❧t ✐❢ ②♦✉ ❞♦ ♥♦t ❤❛✈❡ ❛♥ ❛♥❛❧②t✐❝ ❡①♣r❡ss✐♦♥ ❢♦r a(s)✳ ■t ✐s ♥♦t ❝♦♠♣✉❧s♦r② t♦ ❤❛✈❡ ❛♥ ❛♥❛❧②t✐❝ ❡①♣r❡ss✐♦♥ ❢♦r a(s) ✭❥✉st t❤✐♥❦ t♦ ❤♦✇ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ❛❧❣♦r✐t❤♠✿ x ⇐ ❘♥❞[✵, ✶]❀ y ⇐ ❘♥❞[✵, ✶]❀ r❡t✉r♥ s✐❣♥(✶ − x✷ − y✷) s❛♠♣❧❡s ❇❡r♥π/✹ ✇✐t❤♦✉t ❦♥♦✇✐♥❣ π✳ ✳ ✳ ✮ ❤♦✇❡✈❡r✱ ✐t ♠❛❦❡s ❧✐❢❡ ❡❛s✐❡r✱ ❛♥❞ ✐♥ ♦✉r ❝❛s❡ ✇❡ ❤❛✈❡ ✐t ❢♦r ❢r❡❡ ✐❢ ✇❡ ❝❤♦♦s❡ t❤❡ ❜❛s❡ ❢✉♥❝t✐♦♥ g(x) ❢♦r ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❤❡ ❧✐st g(x) ∈ {❇❡r♥b, P♦✐ss, ●❡♦♠b}

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❍♦✇ t♦ s❛♠♣❧❡ ❢r♦♠ ❇❡r♥a(s)

❊①❛♠♣❧❡ ✇✐t❤ ❇❡r♥♦✉❧❧✐ ✭t❤❡ ♦t❤❡r ❝❛s❡s ❛r❡ s✐♠✐❧❛r✮ ✭❥✉st ✇r✐t❡ a(s) ❢♦r a(s)✱ ✇✐t❤ s = |s|✮ a(s) = g∗s(M) maxn(g∗n(M)) = bM(✶ − b)s−M s

M

• maxn
• bM(✶ − b)n−M n

M

• ❚❤❡ ♠❛① ✐s r❡❛❧✐s❡❞ ❢♦r n = ¯

n := ⌊M/b⌋✱ t❤✉s a(s) = (✶ − b)s−¯

n s!(¯

n − M)! ¯ n!(s − M)!

• ♦♦❞ ♥❡✇s ✶✿ ❚❤✐s ✐s ❡❛s✐❧② ❡✈❛❧✉❛t❡❞ t♦ ❤✐❣❤ ♣r❡❝✐s✐♦♥

✭✐✳❡✳✱ ❝❛❧❝✉❧❛t✐♥❣ d ❜✐♥❛r② ❞✐❣✐ts ❤❛s ❝♦♠♣❧❡①✐t② ≪ ✷d✮✱ s♦ t❤❛t t❤❡ ❛✈❡r❛❣❡ ❝♦st ♦❢ ❇❡r♥a(s) ✐s Θ(✶)✳

• ♦♦❞ ♥❡✇s ✷✿ ❋♦r ❧❛r❣❡ M✱ ❛♥❞ b = Θ(✶)✱ a(s) ❝♦♥✈❡r❣❡s t♦ ❛♥

✉♥✲♥♦r♠❛❧✐s❡❞ ●❛✉ss✐❛♥ ❝❡♥t❡r❡❞ ❛r♦✉♥❞ ¯ n✱ ❛♥❞ ♦❢ ✈❛r✐❛♥❝❡ Θ(M)✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ r♦✉❣❤ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣❧❡①✐t②

❘❡❝❛❧❧ t❤❡ ❜❛s✐❝ st❡♣s ✐♥ t❤❡ r❡❥❡❝t✐♦♥ ❛❧❣♦ ❢♦r ♦✉r ❞❡❝♦♠♣♦s❡❞ ♠❡❛s✉r❡✿ s❛♠♣❧❡ s ✇✐t❤ ♠❡❛s✉r❡ µ✶(s) µ✶(s) = n

i=✶ qi(si)

❛❝❝❡♣t s ✇✐t❤ r❛t❡ a(s) a(s) = g∗s(M)/g∗¯

n(M)

s❛♠♣❧❡ ① ✇✐t❤ ♠❡❛s✉r❡ µ✷(① | s) µ✷(① | s) = n

i=✶ g∗si(xi)

❛♥❞ t❤❛t t❤✐s ❛❧❣♦r✐t❤♠ ❤❛s ❝♦♠♣❧❡①✐t② T ≤ T ♠❛①

✶

Eµ✶(a(s)) + T ♠❛①

✷

✇❤❡r❡ T ♠❛①

✶

, T ♠❛①

✷

= Θ(n). ❯♥❞❡r ♠✐❧❞ ❈▲❚ ❤②♣♦t❤❡s❡s✱ t❤❡ ♠❡❛s✉r❡ ♦♥ s = |s| ✐♥❞✉❝❡❞ ❜② µ✶(s) ✐s ❛ ✭♥♦r♠❛❧✐s❡❞✮ ●❛✉ss✐❛♥ ❝❡♥t❡r❡❞ ✐♥ ¯ n✱ ✇✐t❤ ✈❛r✐❛♥❝❡ σ✷

✶N✱

✇❤✐❧❡ a(s) ✐s ❛♥ ✉♥✲♥♦r♠❛❧✐s❡❞ ●❛✉ss✐❛♥✱ ❝❡♥t❡r❡❞ ✐♥ ¯ n✱ ✇✐t❤ ✈❛r✐❛♥❝❡ σ✷

✷N✿

E(a) ≃

• ❞x

✶

√

✷πσ✷

✶N exp

• − x✷

✷N

• ✶

σ✷

✶ + ✶

σ✷

✷

=

σ✷

√

σ✷

✶+σ✷ ✷

T T ♠❛①

✶

• ✶ + (σ✶/σ✷)✷ + T ♠❛①

✷

= Θ(N)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♣r❡❝✐s❡ r❡s✉❧t

❚❤❡ t❤r❡❡ ❢✉♥❞❛♠❡♥t❛❧ ❞✐str✐❜✉t✐♦♥s g∗s

β (r) =

   ❇❡r♥∗s

β (r) = βr(✶ − β)s−rs r

• β ∈ ]✵, ✶[

P♦✐sss(r) = e−s sr

r!

β = ✵

• ❡♦♠∗s

−β(r) = |β|r(✶ + |β|)−s−rs+r−✶ r

• β ∈ ] − ∞, ✵[

❛r❡ s✉❝❤ t❤❛t g∗s

α ❤❛s ❛ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ gβ ✐✛ α ≤ β✳

s s

−∞ ✵ ✶ β δx,✶ ❇✐♥♦β P♦✐ss

• ❡♦♠−β

❋♦r t❤❡ ❧✐st ♦❢ ❢✉♥❝t✐♦♥s F = {f✶, . . . , fn} ✐♥ ♦✉r ♠❡❛s✉r❡✱ ❝❛❧❧ β♠✐♥(F) t❤❡ s♠❛❧❧❡st ✈❛❧✉❡ ♦❢ β s✉❝❤ t❤❛t ❛❧❧ t❤❡ fi✬s ❤❛✈❡ ❛ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ gβ✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♣r❡❝✐s❡ r❡s✉❧t

❚❤❡♥✱ t❤❡ ❧❛r❣❡st ✈❛❧✉❡ ❢♦r E(a) t❤❛t ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✇✐t❤✐♥ ♦✉r ❢r❛♠❡✇♦r❦ ✐s a♠❛①(F) :=

• ✶ − β♠✐♥(F) ·

i E[fi]

• i V❛r[fi]

s s

−∞ ✵ ✶ β δx,✶ ❇✐♥♦β P♦✐ss

• ❡♦♠−β

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ❜♦♥✉s s✉r♣r✐s❡

■❢ ②♦✉ ❤❛✈❡ t❤❛t g(x) ✐s ❇❡r♥b ♦r ●❡♦♠b✱ t❤❡♥ t❤❡ ❇❇❍▲ ❛❧❣♦r✐t❤♠ ❛t ♣❛r❛♠❡t❡rs [n, m] ❝❛♥ ❜❡ ✉s❡❞ ✇❤❡♥ s❛♠♣❧✐♥❣ ❢r♦♠

i g∗si(xi) δ|x|,M✱ ❜② r❡✇r✐t✐♥❣ xi = y(✶) i

+ · · · + y(si)

i

❈❛❧❧✐♥❣ s =

i si✱

◮ ❇❇❍▲[s, M] ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ g = ❇❡r♥b ❝❛s❡✱ ❜② ✐❞❡♥t✐❢②✐♥❣ t❤❡ ♦✉t❝♦♠❡ str✐♥❣ ♦❢ ❇❇❍▲ ✇✐t❤ t❤❡ ❧✐st ♦❢ y(j)

i

✬s✳ ◮ ❇❇❍▲[s + M − ✶, M] ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ g = ●❡♦♠b ❝❛s❡✱ ❜② ✐❞❡♥t✐❢②✐♥❣ t❤❡ ❧❡♥❣t❤s ♦❢ r✉♥s ♦❢ ✵s ✐♥ t❤❡ ♦✉t❝♦♠❡ str✐♥❣ ♦❢ ❇❇❍▲ ✇✐t❤ t❤❡ ❧✐st ♦❢ y(j)

i

✬s✳ ❚❤❡ P♦✐ss♦♥✐❛♥ ❝❛s❡ ✭♠♦r❡ s❡❧❞♦♠❧② ♥❡❡❞❡❞✮ ❝❛♥ ❜❡ ❞❡❛❧❡❞ ✇✐t❤ ❛ s♠❛❧❧ ❛❧❣♦r✐t❤♠ t❤❛t ■ ✐♥✈❡♥t❡❞✱ s✐♠✐❧❛r ✐♥ s♣✐r✐t t♦ ❇❇❍▲

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❊①❛♠♣❧❡s ♦❢ ❛♣♣❧✐❝❛t✐♦♥

❙♦✱ ✇❡ ❤❛✈❡ ❝♦♥str✉❝t❡❞ ♦✉r ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❧✐♥❡❛r✲t✐♠❡ ❡①❛❝t s❛♠♣❧✐♥❣ ♦❢ s✉♠✲❝♦♥str❛✐♥❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ✐♥ t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡② ❛r❡ ♥♦t ❡q✉❛❧❧② ❞✐str✐❜✉t❡❞✳ ❍♦✇❡✈❡r✱ ②♦✉ ❝♦✉❧❞ ❥✉st t❤✐♥❦✿ ✓✇❤♦ ❝❛r❡s ❛❜♦✉t ♥♦t✲❡q✉❛❧❧②✲❞✐str✐❜✉t❡❞ ✈❛r✐❛❜❧❡s❄ ❆❢t❡r ❛❧❧✱ ❡✈❡r② t✐♠❡ ■ ✇❛♥t❡❞ t♦ ❣❡♥❡r❛t❡ ✇❛❧❦s✱ tr❡❡s✱ ❡t❝✳✱ ■ ❛❧✇❛②s ✇❛♥t❡❞ ❡q✉❛❧❧②✲❞✐str✐❜✉t❡❞ ✈❛r✐❛❜❧❡s✳ ✳ ✳ ✔ ❚❤❡ ♣♦✐♥t ✐s✿ ❡①❛♠♣❧❡s ♦❢ t❤✐s s♦rt ♠❛② ❜❡ ❤✐❞❞❡♥ ❜❡②♦♥❞ s♦♠❡ s♠❛rt ❜✐❥❡❝t✐♦♥✱ st❛rt✐♥❣ ❢r♦♠ ♠♦r❡ ❝✉st♦♠❛r② ✭❛♥❞ s②♠♠❡tr✐❝✮ ♣r♦❜❧❡♠s✳ ❚❤✐s ✐s ✇❡❧❧ ✐❧❧✉str❛t❡❞ ❜② t✇♦ ❝❧❛ss✐❝❛❧ ❡①❛♠♣❧❡s✿

• ❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞
• P❡r♠✉t❛t✐♦♥s ✇✐t❤ m ❝②❝❧❡s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✜rst ❦✐♥❞

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞

❈❛❧❧ Ss❡t

n,m t❤❡ ❡♥s❡♠❜❧❡ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ ❛ s❡t ✇✐t❤ n ✭❧❛❜❡❧❧❡❞✮

❡❧❡♠❡♥ts ✐♥t♦ m ✭✉♥❧❛❜❡❧❡❞✮ ♥♦♥✲❡♠♣t② s✉❜s❡ts✳ ❲✳❧✳♦✳❣✳ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t t❤❡ s❡t ❤❛s ❛ t♦t❛❧ ♦r❞❡r✐♥❣✳ ❊①❛♠♣❧❡✱ ❢♦r (n, m) = (✷✽, ✾)✱ ❛♥❞ t❤❡ s❡t {a, b, c, d, e, f , g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, α, β} ❝♦♥s✐❞❡r t❤❡ ♣❛rt✐t✐♦♥

• {a, g, t}, {b, d, m, o, α}, {c, j, s, y}, {e, h, v}, {f , k, q}, {i, l, z},

{n, p, u}, {r, β}, {w, x}

• ❆❧t❤♦✉❣❤ t❤❡ s❡ts ❛r❡ ♥♦t ❧❛❜❡❧❡❞✱ t❤❡② ❛r❡ ❝❛♥♦♥✐❝❛❧❧② ♦r❞❡r❡❞✱ ❡✳❣✳

❜② t❤❡✐r s♠❛❧❧❡st ❡❧❡♠❡♥t✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ❤❛✈❡ ❛ ❝❛♥♦♥✐❝❛❧ ✐♥❝✐❞❡♥❝❡ ♠❛tr✐① ✱ ✇✐t❤ ✶ ✐❢ t❤❡ ❡❧❡♠❡♥t ✐s ✐♥ s✉❜s❡t ✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞

❈❛❧❧ Ss❡t

n,m t❤❡ ❡♥s❡♠❜❧❡ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ ❛ s❡t ✇✐t❤ n ✭❧❛❜❡❧❧❡❞✮

❡❧❡♠❡♥ts ✐♥t♦ m ✭✉♥❧❛❜❡❧❡❞✮ ♥♦♥✲❡♠♣t② s✉❜s❡ts✳ ❲✳❧✳♦✳❣✳ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t t❤❡ s❡t ❤❛s ❛ t♦t❛❧ ♦r❞❡r✐♥❣✳ ❊①❛♠♣❧❡✱ ❢♦r (n, m) = (✷✽, ✾)✱ ❛♥❞ t❤❡ s❡t {a, b, c, d, e, f , g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, α, β} ❝♦♥s✐❞❡r t❤❡ ♣❛rt✐t✐♦♥

• {a, g, t}, {b, d, m, o, α}, {c, j, s, y}, {e, h, v}, {f , k, q}, {i, l, z},

{n, p, u}, {r, β}, {w, x}

• ❆❧t❤♦✉❣❤ t❤❡ s❡ts ❛r❡ ♥♦t ❧❛❜❡❧❡❞✱ t❤❡② ❛r❡ ❝❛♥♦♥✐❝❛❧❧② ♦r❞❡r❡❞✱ ❡✳❣✳

❜② t❤❡✐r s♠❛❧❧❡st ❡❧❡♠❡♥t✳ ❆s ❛ r❡s✉❧t✱ ✇❡ ❤❛✈❡ ❛ ❝❛♥♦♥✐❝❛❧ ✐♥❝✐❞❡♥❝❡ ♠❛tr✐① T✱ ✇✐t❤ Tij = ✶ ✐❢ t❤❡ ❡❧❡♠❡♥t j ✐s ✐♥ s✉❜s❡t i✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞

• {a, g, t}, {b, d, m, o, α}, {c, j, s, y}, {e, h, v}, {f , k, q}, {i, l, z},

{n, p, u}, {r, β}, {w, x}

• a b c d e f g h i j k l mn o p q r s t u v w x y z α β

✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❈❛❧❧ ❜❛❝❦❜♦♥❡ B(T) t❤❡ ❧✐st ♦❢ s♠❛❧❧❡st ❡❧❡♠❡♥ts ✐♥ t❤❡ s✉❜s❡ts✱ ❤❡r❡ B = {a, b, c, e, f , i, n, r, w}✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❛rt✐t✐♦♥s ✇✐t❤ ✐s t❤❡ tr✐✈✐❛❧ ♣r♦❞✉❝t✿

✶

✱ ❜✉t t❤❡ q✉❛♥t✐t✐❡s ❛r❡ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞✿ ✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞

• {a, g, t}, {b, d, m, o, α}, {c, j, s, y}, {e, h, v}, {f , k, q}, {i, l, z},

{n, p, u}, {r, β}, {w, x}

• a b c d e f g h i j k l mn o p q r s t u v w x y z α β

✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ cy

• ❈❛❧❧ ❜❛❝❦❜♦♥❡ B(T) t❤❡ ❧✐st ♦❢ s♠❛❧❧❡st ❡❧❡♠❡♥ts ✐♥ t❤❡ s✉❜s❡ts✱

❤❡r❡ B = {a, b, c, e, f , i, n, r, w}✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❛rt✐t✐♦♥s T ✇✐t❤ B(T) = B ✐s t❤❡ tr✐✈✐❛❧ ♣r♦❞✉❝t✿ m

y=✶ ycy ✱ ❜✉t t❤❡ q✉❛♥t✐t✐❡s cy

❛r❡ ❧✐♥❡❛r❧② ❝♦♥str❛✐♥❡❞✿

y cy = n − m✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❡t ♣❛rt✐t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ s❡❝♦♥❞ ❦✐♥❞

❆s ❛ r❡s✉❧t✱ s❛♠♣❧✐♥❣ ✉♥✐❢♦r♠❧② s❡t ♣❛rt✐t✐♦♥s ✐♥ Sn,m✱ ✇❤✐❝❤ ❜✐❥❡❝t✐✈❡❧② ❝♦✐♥❝✐❞❡s t♦ s❛♠♣❧✐♥❣ ✉♥✐❢♦r♠❧② t❤❡ t❛❜❧❡❛✉① T✱ ❜♦✐❧s ❞♦✇♥ t♦ s❛♠♣❧✐♥❣ t❤❡ ❜❛❝❦❜♦♥❡ B ✇✐t❤ t❤❡ ♥♦♥✲✉♥✐❢♦r♠ ♠❡❛s✉r❡ µn,m(c✶, . . . , cm) ∝ m

y=✶ ycy × δ|c|,n−m

❚❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ■♥tr♦❞✉❝❡ ❛♥ ❛♣♣r♦♣r✐❛t❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ω

• y cy ✱ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ E(|c|) = n − m

✭t❤❡ ❣♦♦❞ ❝❤♦✐❝❡ ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❡q✉❛t✐♦♥ n

m = − ln(✶−ω) ω

✮ ❚❤❡ ❢✉♥❝t✐♦♥s fy(cy) ❛r❡ ●❡♦♠by (cy)✱ ✇✐t❤ by =

ωy n−ωy

◆♦✇✱ ●❡♦♠a ❤❛s ❛ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥ t❡r♠s ♦❢ ❇❡r♥b

• ❡♦♠a(x) =

s ●❡♦♠

a a+b (s) ❇❡r♥∗s

b (x)

❈❤♦♦s✐♥❣ ❢♦r s✐♠♣❧✐❝✐t② b = ✶

✷✱ ♦✉r ❛❧❣♦r✐t❤♠ ✇♦r❦s✱ ✇✐t❤ ❛♥

❛✈❡r❛❣❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ E(a) =

• e−θ−✶+θ

✷(eθ−✶−θ)

✭ω = ✶ − e−θ✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶)(✷)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶)(✷✸)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶)(✷✸)(✹)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶✺)(✷✸)(✹)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶✺)(✷✻✸)(✹)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶✺)(✷✻✸)(✹)(✼)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶✺)(✷✻✸✽)(✹)(✼)
• ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦

❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶✺)(✷✻✸✽)(✹)(✼)
• ❈❛❧❧ B(σ) = {✵, ✵, ✶, ✵, ✶, ✶, ✵, ✶}✱

t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ ✏❜❧❛❝❦ r♦✇s✑ ♦❢ T(σ)✱ t❤❡ ❜❛❝❦❜♦♥❡ ♦❢ σ✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ✬s ✇✐t❤ ❜❛❝❦❜♦♥❡

✶

✐s ✶ ✱ ❛♥❞ ✇❡ ♠✉st ❤❛✈❡ ① ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

m✲❝②❝❧❡ ♣❡r♠✉t❛t✐♦♥s✱ ❛♥❞ ❙t✐r❧✐♥❣ ♥✉♠❜❡rs ♦❢ t❤❡ ✶st ❦✐♥❞

❈❛❧❧ S❝②❝

n,m t❤❡ s❡t ♦❢ ♣❡r♠✉t❛t✐♦♥s σ ∈ Sn ✇✐t❤ m ❝②❝❧❡s✳

❉❡s❝r✐❜❡ σ t❤r♦✉❣❤ t❤❡ ✐♥s❡rt✐♦♥ t❛❜❧❡ ❛ss♦❝✐❛t❡❞ t♦ ✐ts ❣r♦✇t❤✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r σ =

• (✶✺)(✷✻✸✽)(✹)(✼)
• σ =
• (✶✺)(✷✻✸✽)(✹)(✼)
• ❈❛❧❧ B(σ) = {✵, ✵, ✶, ✵, ✶, ✶, ✵, ✶}✱

t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ ✏❜❧❛❝❦ r♦✇s✑ ♦❢ T(σ)✱ t❤❡ ❜❛❝❦❜♦♥❡ ♦❢ σ✳ ❚❤❡ ♥✉♠❜❡r ♦❢ σ✬s ✇✐t❤ ❜❛❝❦❜♦♥❡ B = (x✶, . . . , xn) ✐s

y(y − ✶)xy ✱

❛♥❞ ✇❡ ♠✉st ❤❛✈❡ |①| = m ❆❣❛✐♥✱ t❤✐s ✐s ❡①❛❝t❧② ♦✉r ❢r❛♠❡✇♦r❦✦ ❥✉st ✇✐t❤ ✐♥❤♦♠♦❣❡♥❡♦✉s ❇❡r♥♦✉❧❧✐ ✈❛r✐❛❜❧❡s✱ ✐♥st❡❛❞ ♦❢ ✐♥❤♦♠♦❣❡♥❡♦✉s ●❡♦♠❡tr✐❝ ✈❛r✐❛❜❧❡s✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P❧❛♥ ♦❢ t❤❡ t❛❧❦

❙♦✱ ■✬♠ tr②✐♥❣ t♦ ❵s❡❧❧✬ ②♦✉ t✇♦ r❡❝❡♥t t♦♦❧s ❢♦r ✐♠♣r♦✈✐♥❣ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✏❇♦❧t③♠❛♥♥✕❧✐❦❡✑ ♣r♦❜❧❡♠s ■ ✐♥tr♦❞✉❝❡❞✳ ■♥ t❤❡ ✏❇r✐❞❣❡ ❝❛s❡✑✱ ✐❢ ✇❡ ❤❛✈❡ ♣♦s✐t✐✈❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ fi✬s ✐♥ t❡r♠s ♦❢ ❛ ❢✉♥❝t✐♦♥ g ❜❡✐♥❣ ❇❡r♥ ♦r ●❡♦♠✱ ■ s❤♦✉❧❞ ❤❛✈❡ ❝♦♥✈✐♥❝❡❞ ②♦✉ t❤❛t ♠② tr✐❝❦ ✐s ❵♦♣t✐♠❛❧ ✉♣ t♦ ❛ ❢❛❝t♦r✬✳ ❨♦✉ s❤❛❧❧ ❜❡ ❤❛♣♣② ✇✐t❤ t❤✐s✱ ✉♥❧❡ss ②♦✉ r❡❛❧❧② s❡❛r❝❤ ❢♦r r❛♥❞✲❜✐t ♦♣t✐♠❛❧✐t② ✭❧✐❦❡ ✐♥ t❤❡ ♥✐❝❡ ❇❇❍▲ ❛❧❣♦r✐t❤♠✮✳ ❍♦✇❡✈❡r✱ ✇❡ ❛r❡ ♥♦t ❛❧✇❛②s s♦ ❧✉❝❦②✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ✇❡ ❛r❡ ✐♥ t❤❡ ✏❇♦❧t③♠❛♥♥ ❝❛s❡✑✱ ✇❡ ❛❧r❡❛❞② st❛rt ❢r♦♠ ❛ ✇♦rs❡ ❝♦♠♣❧❡①✐t② ✭∼ N✷ ❜❡❝❛✉s❡ ♦❢ ❢❛t t❛✐❧s✮✱ ❛♥❞ ✇❡ ❤❛✈❡ ♠✉❝❤ ❧❡ss t♦♦❧s ✐♥ ♦✉r ❤❛♥❞s✳ ✳ ✳ ■t✬s t✐♠❡ ❢♦r ♠❡ ❢♦r tr②✐♥❣ t♦ s❡❧❧ ②♦✉ t❤❡ s❡❝♦♥❞ t♦♦❧✿ t❤❡ ✐♠♣r♦✈❡❞ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✿ t❤❡ ♣r♦❜❧❡♠

❲❡ ❤❛✈❡ t✇♦ ♠❡❛s✉r❡s ♦♥ Z✱ p(x) ❛♥❞ q(x)✳ ▲❡t p(x) q(x)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✿ t❤❡ ♣r♦❜❧❡♠

❲❡ ❤❛✈❡ t✇♦ ♠❡❛s✉r❡s ♦♥ Z✱ p(x) ❛♥❞ q(x)✳ ▲❡t f (x) = p(x)q(x)

• y p(y)q(y)

p(x) q(x) f (x)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✿ t❤❡ ♣r♦❜❧❡♠

❲❡ ❤❛✈❡ t✇♦ ♠❡❛s✉r❡s ♦♥ Z✱ p(x) ❛♥❞ q(x)✳ ▲❡t f (x) = p(x)q(x)

• y p(y)q(y)

p(x) q(x) f (x) ❲❡ ❤❛✈❡ t✇♦ ❜❧❛❝❦✲❜♦① ❛❧❣♦r✐t❤♠s t❤❛t s❛♠♣❧❡ ❢r♦♠ p ❛♥❞ ❢r♦♠ q✱ ❛♥❞ ✇❡ ✇❛♥t t♦ s❛♠♣❧❡ ❢r♦♠ f

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠

❚❤❡ r✉❧❡s ♦❢ t❤❡ ❣❛♠❡ ❛r❡ ❝❧❡❛r✳ ❲❡ ❤❛✈❡ ♥♦ ❡①♣❧♦✐t❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ✇❤❛ts♦❡✈❡r ♦♥ p ❛♥❞ q✳ ❲❡ ♦♥❧② ❤❛✈❡ t❤❡ ❜❧❛❝❦ ❜♦①❡s ❚❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ s❡❡♠s t♦ ❜❡ t❤❡ ♦♥❧② ❝❛♥❞✐❞❛t❡✿ ❆❧❣♦r✐t❤♠ ✿ ❖❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❜❡❣✐♥ r❡♣❡❛t x ⇐ p ; ②⇐ q ✉♥t✐❧ x = y❀ r❡t✉r♥ x ❡♥❞ ❉❡✜♥❡ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t (f , g) =

x f (x)g(x)✳

❚❤❡ ❵r❡♣❡❛t✬ ❧♦♦♣ ✐s r❡♣❡❛t❡❞ ♦♥ ❛✈❡r❛❣❡ ✶/(p, q) t✐♠❡s✳ ■❢ ✇❡ ❤❛✈❡ ❛ s✐③❡ ♣❛r❛♠❡t❡r n✱ t❤✐s ♠❛② ❜❡ ❧❛r❣❡✱ ❛♥❞ ✇❡ ✇❛♥t t♦ ♠❛❦❡ ❜❡tt❡r

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❝❛❧✐♥❣ ♦❢ t❤❡ s✐③❡ ♣❛r❛♠❡t❡r

❙❛② t❤❛t t❤❡r❡ ❡①✐sts ❛ κ ≥ ✸ s✉❝❤ t❤❛t

x(p(x)κ + q(x)κ) ≪ (p, q)

✭❜② ❈❛✉❝❤②✕❙❝❤✇❛r③✱ ✐t ❝❛♥✬t ❜❡ κ = ✷✮✳ ❚❤❡♥ ✇❡ ✇❛♥t t♦ ❜r✐♥❣ ❞♦✇♥ t❤❡ ❝♦♠♣❧❡①✐t② ❢r♦♠ ∼ ✶/(p, q)✳ t♦ ∼ ✶/

• (p, q)

✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡ t②♣✐❝❛❧ ❝❛s❡ ✐s κ = ✸ ❛❜♦✈❡✱ ❛♥❞ (p, p)✱ (p, q)✱ (q, q) ❛r❡ ❛❧❧ Θ(n−✷α) ✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s Θ(n✷α) ❛♥❞ ✇❡ ✇❛♥t t♦ ❣♦ ❞♦✇♥ t♦ Θ(nα ln n) ♦r Θ(nα) ✲ ✛ Θ(n✷α)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❙❝❛❧✐♥❣ ♦❢ t❤❡ s✐③❡ ♣❛r❛♠❡t❡r

❙❛② t❤❛t t❤❡r❡ ❡①✐sts ❛ κ ≥ ✸ s✉❝❤ t❤❛t

x(p(x)κ + q(x)κ) ≪ (p, q)

✭❜② ❈❛✉❝❤②✕❙❝❤✇❛r③✱ ✐t ❝❛♥✬t ❜❡ κ = ✷✮✳ ❚❤❡♥ ✇❡ ✇❛♥t t♦ ❜r✐♥❣ ❞♦✇♥ t❤❡ ❝♦♠♣❧❡①✐t② ❢r♦♠ ∼ ✶/(p, q)✳ t♦ ∼ ✶/

• (p, q)

✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡ t②♣✐❝❛❧ ❝❛s❡ ✐s κ = ✸ ❛❜♦✈❡✱ ❛♥❞ (p, p)✱ (p, q)✱ (q, q) ❛r❡ ❛❧❧ Θ(n−✷α) ✭♠❛②❜❡ ✉♣ t♦ ❧♦❣s✮ ❚❤❡♥✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s Θ(n✷α) ❛♥❞ ✇❡ ✇❛♥t t♦ ❣♦ ❞♦✇♥ t♦ Θ(nα ln n) ♦r Θ(nα) ✲ ✛ Θ(n✷α)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ❝♦♠♣❧❡①✐t② ♣❛r❛❞✐❣♠

✭②♦✉ ♥❡❡❞ t❤✐s ♦♥❧② ✐❢ ②♦✉ ❝❛r❡ ❛❜♦✉t ❧♦❣s✮

■♥ ❛❧❣♦r✐t❤♠ ❝♦♠♣❧❡①✐t② ②♦✉ ♥♦r♠❛❧❧② ❥✉st ❝♦✉♥t ♦♣❡r❛t✐♦♥s✳ ◆♦♥❡t❤❡❧❡ss✱ ✇❤❡♥ ②♦✉ ❤❛✈❡ ❜❧❛❝❦ ❜♦①❡s✱ ✐t ✐s ✇✐s❡ t♦ ❝♦✉♥t s❡♣❛r❛t❡❧② ♦♣❡r❛t✐♦♥s ❛♥❞ ❜❧❛❝❦✲❜♦① q✉❡r✐❡s✱ ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛❧❧② ♠✉❝❤ ♠♦r❡ ❡①♣❡♥s✐✈❡ t❤❛♥ ❛ s✐♥❣❧❡ ♦♣❡r❛t✐♦♥ ❍❡r❡ ✇❡ ❤❛✈❡ t✇♦ ❜❧❛❝❦ ❜♦①❡s✱ ❢♦r s✐♠♣❧✐❝✐t② ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❡② ❤❛✈❡ s✐♠✐❧❛r ❝♦♠♣❧❡①✐t② ◆✐❝❡ ♥♦t❛t✐♦♥✿

• ❢♦r ❜❧❛❝❦✲❜♦① q✉❡r✐❡s

❢♦r ♦♣❡r❛t✐♦♥s ❚❤❡♥✱ ❡✳❣✳✱ Θ(nx + ny ) s✐♠♣❧✐✜❡s t♦ Θ(nx ) ✐❢ x ≥ y✱ ❜✉t st❛②s ❛s ✐s ✐❢ x < y✱ ❛s ✇❡ ♦♥❧② ❦♥♦✇ t❤❛t / > ✶ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♦❜✈✐♦✉s r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s Θ(n✷α ) ❲❡ ✇❛♥t t♦ ❣♦ ❞♦✇♥ t♦ Θ(nα + nα ln(n) )

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❚❤❡ ♥❛ï✈❡ ❵❜✐rt❤❞❛② ♣❛r❛❞♦①✬ ❛❧❣♦r✐t❤♠

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠ ❆❧❣♦r✐t❤♠ ✿ ❇✐rt❤❞❛② ♣❛r❛❞♦①✱ ✜rst tr② ❜❡❣✐♥ r❡♣❡❛t (x✶, . . . , xk) ⇐ p ; ✭②✶, . . . , yk) ⇐ q ✉♥t✐❧ ∃! (i, j) | xi = yj❀ r❡t✉r♥ xi ❡♥❞ ❇❡st ❤♦♣❡✿ ✐♥ ❡❛❝❤ ❵r❡♣❡❛t✬ ❝②❝❧❡ t❤❡r❡ ❛r❡ ∼ P♦✐ssk✷(p,q) ♣❛✐rs (i, j) s✉❝❤ t❤❛t xi = yj✱ s♦ ✐❢ ✇❡ t✉♥❡ k ∼ ✶/

• (p, q)

t❤❡ ❝②❝❧❡ ❝♦sts Θ(k )✱ ❛♥❞ ✐s r❡♣❡❛t❡❞ ♦♥ ❛✈❡r❛❣❡ Θ(✶) t✐♠❡s ❚❤❡ ❝♦♠♣❧❡①✐t② ❞r♦♣s ❞♦✇♥ t♦ Θ(nα )

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆♥ ❡❛s② ✇✐♥❄

❆♥ ❡❛s② ✇✐♥❄ ✳ ✳ ✳ ◆♦✦ ❜❡s✐❞❡s ❛ ❜✉♥❝❤ ♦❢ s♦❧✈❛❜❧❡ ♠✐♥♦r ✐ss✉❡s

✭✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs r❡❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ P♦✐ss♦♥✐❛♥❄✮ ✭t❤❡ ❵✇r♦♥❣✬ ♥❛ï✈❡ s❡❛r❝❤ ❢♦r ❛ ❣♦♦❞ ♣❛✐r t❛❦❡s t✐♠❡

✷✮

t❤❡r❡ ✐s t❤❡ ♦♥❡ ❜✐❣ ♣r♦❜❧❡♠✿ ❚❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐s ❜✐❛s❡❞✦ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✇✐t❤ ✐s ✐♥ ❢❛❝t ♣r♦♣♦rt✐♦♥❛❧ t♦ ✱ ❛♥❞ ❜♦♦st❡❞ ❜② ❛ ❢❛❝t♦r

✷✱

✇❤✐❝❤ ✐s ❣♦♦❞✳ ✳ ✳ ✳ ✳ ✳ ❜✉t ❦♥♦✇✐♥❣ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣♦♦❞ ♣❛✐r ❣✐✈❡s ❛ ❜✐❛s✦ ❍♦✇❡✈❡r t❤❡ ✇❤♦❧❡ ✐❞❡❛ r❡♠❛✐♥s ✈❛❧✉❛❜❧❡✱ ❜❡❝❛✉s❡✱ ✐❢ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✐s s♠❛❧❧✱ ❤❛✈✐♥❣ ♥♦ ❢✉rt❤❡r ♣❛✐rs ✐s ❵♥♦r♠❛❧✬✱ s♦ t❤❡ ❜✐❛s ✐s s♠❛❧❧✱ ❛♥❞ ♠❛②❜❡ ❝❛♥ ❜❡ ❝♦rr❡❝t❡❞ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✲❝❤❡❛♣ tr✐❝❦✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆♥ ❡❛s② ✇✐♥❄

❆♥ ❡❛s② ✇✐♥❄ ✳ ✳ ✳ ◆♦✦ ❜❡s✐❞❡s ❛ ❜✉♥❝❤ ♦❢ s♦❧✈❛❜❧❡ ♠✐♥♦r ✐ss✉❡s

✭✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs r❡❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ P♦✐ss♦♥✐❛♥❄✮ ✭t❤❡ ❵✇r♦♥❣✬ ♥❛ï✈❡ s❡❛r❝❤ ❢♦r ❛ ❣♦♦❞ ♣❛✐r t❛❦❡s t✐♠❡

✷✮

t❤❡r❡ ✐s t❤❡ ♦♥❡ ❜✐❣ ♣r♦❜❧❡♠✿ ❚❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐s ❜✐❛s❡❞✦ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✇✐t❤ ✐s ✐♥ ❢❛❝t ♣r♦♣♦rt✐♦♥❛❧ t♦ ✱ ❛♥❞ ❜♦♦st❡❞ ❜② ❛ ❢❛❝t♦r

✷✱

✇❤✐❝❤ ✐s ❣♦♦❞✳ ✳ ✳ ✳ ✳ ✳ ❜✉t ❦♥♦✇✐♥❣ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣♦♦❞ ♣❛✐r ❣✐✈❡s ❛ ❜✐❛s✦ ❍♦✇❡✈❡r t❤❡ ✇❤♦❧❡ ✐❞❡❛ r❡♠❛✐♥s ✈❛❧✉❛❜❧❡✱ ❜❡❝❛✉s❡✱ ✐❢ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✐s s♠❛❧❧✱ ❤❛✈✐♥❣ ♥♦ ❢✉rt❤❡r ♣❛✐rs ✐s ❵♥♦r♠❛❧✬✱ s♦ t❤❡ ❜✐❛s ✐s s♠❛❧❧✱ ❛♥❞ ♠❛②❜❡ ❝❛♥ ❜❡ ❝♦rr❡❝t❡❞ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✲❝❤❡❛♣ tr✐❝❦✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆♥ ❡❛s② ✇✐♥❄

❆♥ ❡❛s② ✇✐♥❄ ✳ ✳ ✳ ◆♦✦ ❜❡s✐❞❡s ❛ ❜✉♥❝❤ ♦❢ s♦❧✈❛❜❧❡ ♠✐♥♦r ✐ss✉❡s

✭✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs r❡❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ P♦✐ss♦♥✐❛♥❄✮ ✭t❤❡ ❵✇r♦♥❣✬ ♥❛ï✈❡ s❡❛r❝❤ ❢♦r ❛ ❣♦♦❞ ♣❛✐r t❛❦❡s t✐♠❡ k✷✮

t❤❡r❡ ✐s t❤❡ ♦♥❡ ❜✐❣ ♣r♦❜❧❡♠✿ ❚❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐s ❜✐❛s❡❞✦ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✇✐t❤ ✐s ✐♥ ❢❛❝t ♣r♦♣♦rt✐♦♥❛❧ t♦ ✱ ❛♥❞ ❜♦♦st❡❞ ❜② ❛ ❢❛❝t♦r

✷✱

✇❤✐❝❤ ✐s ❣♦♦❞✳ ✳ ✳ ✳ ✳ ✳ ❜✉t ❦♥♦✇✐♥❣ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣♦♦❞ ♣❛✐r ❣✐✈❡s ❛ ❜✐❛s✦ ❍♦✇❡✈❡r t❤❡ ✇❤♦❧❡ ✐❞❡❛ r❡♠❛✐♥s ✈❛❧✉❛❜❧❡✱ ❜❡❝❛✉s❡✱ ✐❢ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✐s s♠❛❧❧✱ ❤❛✈✐♥❣ ♥♦ ❢✉rt❤❡r ♣❛✐rs ✐s ❵♥♦r♠❛❧✬✱ s♦ t❤❡ ❜✐❛s ✐s s♠❛❧❧✱ ❛♥❞ ♠❛②❜❡ ❝❛♥ ❜❡ ❝♦rr❡❝t❡❞ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✲❝❤❡❛♣ tr✐❝❦✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆♥ ❡❛s② ✇✐♥❄

❆♥ ❡❛s② ✇✐♥❄ ✳ ✳ ✳ ◆♦✦ ❜❡s✐❞❡s ❛ ❜✉♥❝❤ ♦❢ s♦❧✈❛❜❧❡ ♠✐♥♦r ✐ss✉❡s

✭✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs r❡❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ P♦✐ss♦♥✐❛♥❄✮ ✭t❤❡ ❵✇r♦♥❣✬ ♥❛ï✈❡ s❡❛r❝❤ ❢♦r ❛ ❣♦♦❞ ♣❛✐r t❛❦❡s t✐♠❡ k✷✮

t❤❡r❡ ✐s t❤❡ ♦♥❡ ❜✐❣ ♣r♦❜❧❡♠✿ ❚❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐s ❜✐❛s❡❞✦ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✇✐t❤ ✐s ✐♥ ❢❛❝t ♣r♦♣♦rt✐♦♥❛❧ t♦ ✱ ❛♥❞ ❜♦♦st❡❞ ❜② ❛ ❢❛❝t♦r

✷✱

✇❤✐❝❤ ✐s ❣♦♦❞✳ ✳ ✳ ✳ ✳ ✳ ❜✉t ❦♥♦✇✐♥❣ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣♦♦❞ ♣❛✐r ❣✐✈❡s ❛ ❜✐❛s✦ ❍♦✇❡✈❡r t❤❡ ✇❤♦❧❡ ✐❞❡❛ r❡♠❛✐♥s ✈❛❧✉❛❜❧❡✱ ❜❡❝❛✉s❡✱ ✐❢ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✐s s♠❛❧❧✱ ❤❛✈✐♥❣ ♥♦ ❢✉rt❤❡r ♣❛✐rs ✐s ❵♥♦r♠❛❧✬✱ s♦ t❤❡ ❜✐❛s ✐s s♠❛❧❧✱ ❛♥❞ ♠❛②❜❡ ❝❛♥ ❜❡ ❝♦rr❡❝t❡❞ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✲❝❤❡❛♣ tr✐❝❦✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆♥ ❡❛s② ✇✐♥❄

❆♥ ❡❛s② ✇✐♥❄ ✳ ✳ ✳ ◆♦✦ ❜❡s✐❞❡s ❛ ❜✉♥❝❤ ♦❢ s♦❧✈❛❜❧❡ ♠✐♥♦r ✐ss✉❡s

✭✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs r❡❛❧❧② ❞✐str✐❜✉t❡❞ ❛s ❛ P♦✐ss♦♥✐❛♥❄✮ ✭t❤❡ ❵✇r♦♥❣✬ ♥❛ï✈❡ s❡❛r❝❤ ❢♦r ❛ ❣♦♦❞ ♣❛✐r t❛❦❡s t✐♠❡ k✷✮

t❤❡r❡ ✐s t❤❡ ♦♥❡ ❜✐❣ ♣r♦❜❧❡♠✿ ❚❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐s ❜✐❛s❡❞✦ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs (i, j) ✇✐t❤ xi = yj = z ✐s ✐♥ ❢❛❝t ♣r♦♣♦rt✐♦♥❛❧ t♦ f (z)✱ ❛♥❞ ❜♦♦st❡❞ ❜② ❛ ❢❛❝t♦r k✷✱ ✇❤✐❝❤ ✐s ❣♦♦❞✳ ✳ ✳ ✳ ✳ ✳ ❜✉t ❦♥♦✇✐♥❣ t❤❛t ②♦✉ ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣♦♦❞ ♣❛✐r ❣✐✈❡s ❛ ❜✐❛s✦ ❍♦✇❡✈❡r t❤❡ ✇❤♦❧❡ ✐❞❡❛ r❡♠❛✐♥s ✈❛❧✉❛❜❧❡✱ ❜❡❝❛✉s❡✱ ✐❢ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❣♦♦❞ ♣❛✐rs ✐s s♠❛❧❧✱ ❤❛✈✐♥❣ ♥♦ ❢✉rt❤❡r ♣❛✐rs ✐s ❵♥♦r♠❛❧✬✱ s♦ t❤❡ ❜✐❛s ✐s s♠❛❧❧✱ ❛♥❞ ♠❛②❜❡ ❝❛♥ ❜❡ ❝♦rr❡❝t❡❞ ✇✐t❤ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✲❝❤❡❛♣ tr✐❝❦✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

P♦✐ss♦♥✐s❛t✐♦♥ ♦❢ t❤❡ ♥❛ï✈❡ ❛❧❣♦r✐t❤♠

❆♥❛❧②s✐♥❣ t❤❡ ❜✐❛s ♦❢ t❤❡ ♣r❡✈✐♦✉s ❛❧❣♦r✐t❤♠ ✐s ❝♦♠♣❧✐❝❛t❡❞✳ ■t ❣❡ts ❡❛s✐❡r ✐❢ ✇❡ ❵P♦✐ss♦♥✐s❡✬ k✱ ✐✳❡✳ ✇❡ r❛t❤❡r ❝♦♥s✐❞❡r✿ ❆❧❣♦r✐t❤♠ ✿ ❇✐rt❤❞❛② ♣❛r❛❞♦①✱ P♦✐ss♦♥✐s❡❞ ❜❡❣✐♥ r❡♣❡❛t s❛♠♣❧❡ kp ❛♥❞ kq ✇✐t❤ P♦✐ssk❀ (x✶, . . . , xkp) ⇐ p ; ✭②✶, . . . , ykq) ⇐ q ✉♥t✐❧ ∃! (i, j) | xi = yj❀ r❡t✉r♥ xi ❡♥❞ ❈❛❧❧ νz = (az, bz)✱ ✇✐t❤ az = #{i | xi = z} ❛♥❞ bz = #{j | yj = z} ❚❤❡ ❣♦♦❞ ❢❛❝t ♦❢ P♦✐ss♦♥✐s❛t✐♦♥ ✐s t❤❛t t❤❡ νz✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s s♦ t❤❛t ✇❡ ❝❛♥ ❡❛s✐❧② ❛♥❛❧②s❡ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ r❡t✉r♥✐♥❣ z

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❜✐❛s

❋❛❝t✿ ♣r♦❜(az, bz) = P♦✐sskp(z)(az)P♦✐sskq(z)(bz) t❤✉s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡r❡ ✐s ❛ s✐♥❣❧❡ ♣❛✐r ❢♦r z ✐s P♦✐sskp(z)(✶)P♦✐sskq(z)(✶) = e−k(p(z)+q(z))k✷p(z)q(z) ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡r❡ ❛r❡ ♥♦ ♣❛✐rs ❢♦r ✈❛❧✉❡s z′ = z ✐s ✶ −

• ✶ − P♦✐sskp(z′)(✵)
• ✶ − P♦✐sskq(z′)(✵)
• =

e−k(p(z′)+q(z′)) ekp(z′) + ekq(z′) − ✶

• ❛s ❛ r❡s✉❧t✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ r❡t✉r♥✐♥❣ z ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦†

f (z) ekp(z) + ekq(z) − ✶ ❛♥❞ ✇❡ ✇♦✉❧❞ ❜❡ ✜♥❡ ✐❢ ✇❡ ❝♦✉❧❞ ❞❡✈✐s❡ ❛ ✜♥❛❧ r❡❥❡❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ s♣✉r✐♦✉s ❢❛❝t♦r ❇✐❛s(z) := ✶/(ekp(z) + ekq(z) − ✶)

† ■✳❡✳✱ ✉♣ t♦ ❛ ❢❛❝t♦r✱ ❧✐❦❡ z′ ψ(z′)✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ z✱

❛❧t❤♦✉❣❤ ♣♦ss✐❜❧② ❝♦♠♣❧✐❝❛t❡❞ t♦ ❝❛❧❝✉❧❛t❡

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ♣❤✐❧♦s♦♣❤✐❝❛❧ ❞✐❣r❡ss✐♦♥

❙♦✱ ✇❡ ✇❛♥t t♦ ❛❞❞ ❛ ✜♥❛❧ ♣r♦❝❡❞✉r❡ ❢♦r ❝♦rr❡❝t✐♥❣ t❤❡ s♣✉r✐♦✉s ❢❛❝t♦r ❇✐❛s(z) := ✶/(ekp(z) + ekq(z) − ✶)✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r s♦♠❡ C > maxz(ekp(z) + ekq(z) − ✶)✱ ❜✉t st✐❧❧ ✇✐t❤

z f (z) ekp(z)+ekq(z)−✶ C

= Θ(✶)✱ ■ ✇❛♥t t♦ s❛♠♣❧❡ ❛ ❇❡r♥♦✉❧❧✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦❢ ♣❛r❛♠❡t❡r ekp(z)+ekq(z)−✶

C

❍♦✇❡✈❡r✱ r❡❝❛❧❧ t❤❛t ✇❡ ❤❛✈❡ ♥♦ ❛♥❛❧②t✐❝ ✐♥❢♦r♠❛t✐♦♥ ♦♥ p ❛♥❞ q ✭✇❡ ♦♥❧② ❤❛✈❡ t❤❡ ❜❧❛❝❦ ❜♦①❡s✦✮ ❈❛♥ ✇❡ ❡✈❡r s❛♠♣❧❡ ❇❡r♥ξ ✇✐t❤♦✉t ❡✈❛❧✉❛t✐♥❣ ξ❄ ❚❤✐s ✐s ♥♦t ✐♠♣♦ss✐❜❧❡ ❛ ♣r✐♦r✐✱ ❥✉st r❡♠❡♠❜❡r t❤❡ ❢❛♠♦✉s ❛❧❣♦r✐t❤♠ ❢♦r ❇❡r♥π/✹ t❤❛t ♠❛❦❡s ♥♦ ✉s❡ ♦❢ π✳ ✳ ✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ✜rst s♦❧✉t✐♦♥ ✇✐t❤ st❛t✐❝ ❧✐sts

❆❧❣♦r✐t❤♠ ✿ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✱ ✇✐t❤ st❛t✐❝ ❧✐sts r❡♣❡❛t s❛♠♣❧❡ kp ❛♥❞ kq ✇✐t❤ P♦✐ssk❀ (x✶, . . . , xkp) ⇐ p ; ✭②✶, . . . , ykq) ⇐ q ; s♦rt t❤❡ ❧✐sts ❛❜♦✈❡✱ ♣r♦❞✉❝❡ t❤❡ ❧✐st ♦❢ {(z, az, bz)}❀ ✐❢ N• = ✵ ❛♥❞ N• = ✶ t❤❡♥ ν ← az + bz − ✶ ❢♦r t❤❡ ♦♥❧② z ∈ Ω•❀ ✇✐♥ ← ❇❡r♥ν/✸❀ ✉♥t✐❧ ✇✐♥=tr✉❡❀ r❡t✉r♥ z❀ ✵ ✶ ✷ ✸ ✹ ✵ ✶ ✷ ✸ ✹

✶ ✸ ✷ ✸ ✷ ✸

✶ ✶

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❲❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ✉♥❜✐❛s❡❞

▲❡t ✉s ✜rst s❡❡ ✇❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ❝♦rr❡❝t ❈❛❧❧ ❢♦r s❤♦rt φ(z) = e−k(p(z)+q(z)) ❚❤❡ ❛❧❣♦r✐t❤♠ ♠❛② r❡t✉r♥ z ♦♥❧② ✐❢ ♥♦ z′ = z ❛rr✐✈❡s ♦✉t ♦❢ t❤❡ Ω• r❡❣✐♦♥✱ ❡❛❝❤ z′ ♠❛❦❡s ❛ ❢❛❝t♦r φ(z′)

• ✶ + kp(z′) + (kp(z′))✷

✷

+ kq(z′) + (kq(z′))✷

✷

• ✵

✶ ✷ ✸ ✹ ✵ ✶ ✷ ✸ ✹

✶ ✸ ✷ ✸ ✷ ✸

✶ ✶

❚❤❡♥✱ ✐t s❤❛❧❧ ❛❧s♦ ❤❛♣♣❡♥ t❤❛t z ❢❛❧❧s ✇✐t❤✐♥ t❤❡ Ω• r❡❣✐♦♥ t❤✐s ♠❛❦❡s ❛ ❢❛❝t♦r k✷p(z)q(z) φ(z)

• ✶

✸

✶ +

✷ ✸ kp(z) ✷

+ ✶

(kp(z))✷ ✻

+

✷ ✸ kq(z) ✷

+ ✶

(kq(z))✷ ✻

• ❲❡ ✇♦✉❧❞ ❤❛✈❡ ✇♦♥ ✐❢ t❤❡ t✇♦ ❢❛❝t♦rs ✐♥ ♣❛r❡♥t❤❡s✐s

✇❡r❡ ♣r♦♣♦rt✐♦♥❛❧✳ ❇✉t t❤❡② ❛r❡ ♥♦t✳ ❋♦r t❤✐s r❡❛s♦♥ ✇❡ ♣✉t ❛ ✜♥❛❧ ❇❡r♥♦✉❧❧✐ r❡❥❡❝t✐♦♥✱ t❤❛t ❝♦rr❡❝ts ❢♦r t❤❡ ❢❛❝t♦r✐❛❧s

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❲❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ✉♥❜✐❛s❡❞

▲❡t ✉s ✜rst s❡❡ ✇❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ❝♦rr❡❝t ❈❛❧❧ ❢♦r s❤♦rt φ(z) = e−k(p(z)+q(z)) ❚❤❡ ❛❧❣♦r✐t❤♠ ♠❛② r❡t✉r♥ z ♦♥❧② ✐❢ ♥♦ z′ = z ❛rr✐✈❡s ♦✉t ♦❢ t❤❡ Ω• r❡❣✐♦♥✱ ❡❛❝❤ z′ ♠❛❦❡s ❛ ❢❛❝t♦r φ(z′)

• ✶ + kp(z′) + (kp(z′))✷

✷

+ kq(z′) + (kq(z′))✷

✷

• ✵

✶ ✷ ✸ ✹ ✵ ✶ ✷ ✸ ✹

✶ ✸ ✷ ✸ ✷ ✸

✶ ✶

❚❤❡♥✱ ✐t s❤❛❧❧ ❛❧s♦ ❤❛♣♣❡♥ t❤❛t z ❢❛❧❧s ✇✐t❤✐♥ t❤❡ Ω• r❡❣✐♦♥ t❤✐s ♠❛❦❡s ❛ ❢❛❝t♦r k✷p(z)q(z) φ(z)

• ✶

✸·✶ + ✷ ✸· kp(z) ✷

+ ✶· (kp(z))✷

✻

+ ✷

✸· kq(z) ✷

+ ✶· (kq(z))✷

✻

• ❲❡ ✇♦✉❧❞ ❤❛✈❡ ✇♦♥ ✐❢ t❤❡ t✇♦ ❢❛❝t♦rs ✐♥ ♣❛r❡♥t❤❡s✐s

✇❡r❡ ♣r♦♣♦rt✐♦♥❛❧✳ ❇✉t t❤❡② ❛r❡ ♥♦t✳ ❋♦r t❤✐s r❡❛s♦♥ ✇❡ ♣✉t ❛ ✜♥❛❧ ❇❡r♥♦✉❧❧✐ r❡❥❡❝t✐♦♥✱ t❤❛t ❝♦rr❡❝ts ❢♦r t❤❡ ❢❛❝t♦r✐❛❧s

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❲❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ❢❛st

❲❡ ❤❛✈❡ t❛❦❡♥ t❤❡ ❛♣♣❛r❡♥t❧② r✐s❦② ❝❤♦✐❝❡ ♦❢ ♠❛❦✐♥❣ ❛ ✈❡r② ❧❛r❣❡ r❡❥❡❝t✐♦♥ r❡❣✐♦♥ Ω•✳ ■❢ ❛♥② z ❢❛❧❧s ❤❡r❡✱ ✇❡ r❡st❛rt ♥♦ ♠❛tt❡r ✇❤❛t✳ ❉♦❡s t❤✐s ❛✛❡❝t t❤❡ ❝♦♠♣❧❡①✐t② ✐♥ ❛♥ ✐♠♣♦rt❛♥t ✇❛②✱ ♦r t❤❡ ❡st✐♠❛t❡s ❢r♦♠ t❤❡ ♥❛ï✈❡ ❜✐rt❤❞❛② ❤❡✉r✐st✐❝s ❛r❡ st✐❧❧ ✈❛❧✐❞❄ ❖♥❡ r✉♥ ❝♦sts ♦♥ ❛✈❡r❛❣❡ ✷k + k ln(k) ✭❢♦r s♦rt✐♥❣✮✱ ✐✳❡✳ ❛❧r❡❛❞② t❤❡ s❡❡❦❡❞ Θ(nα + nα ln(n) )✱ s♦ ✇❡ ✇✐♥ ✐✛ t❤❡ ❛❝❝❡♣t❛♥❝❡ r❛t❡ ♦❢ ♦♥❡ r✉♥ ✐s Θ(✶) ❚❤✐s r❛t❡ ✐s ❡❛s✐❧② ❞❡❞✉❝❡❞ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s ❝❛❧❝✉❧❛t✐♦♥✱ t♦ ❜❡

z

✶ ✸k✷p(z)q(z)

w

φ(w)

• ✶ + kp(w) + (kp(w))✷

✷

+ kq(w) + (kq(w))✷

✷

• =: ✶

✸k✷ (p, q)

• w

ψ(w) ✳ ✳ ✳ ✇❡ ❝❧❛✐♠ t❤❛t

• w

ψ(w) ≃ e−k✷(p,q)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❲❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ❢❛st

■❢ t❤✐s ✐s tr✉❡✱ t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣❧❡①✐t② ❝❛♥ ❜❡ ♠❛❞❡ ❛s s♠❛❧❧ ❛s ❝♦♠♣❧❡①✐t② + Θ(ln n) = min

k

✷k

✶ ✸k✷(p, q)e−k✷(p,q)

= ✶

• (p, q)

min

x

✷√x

x ✸ exp(−x) =

✶

• (p, q)

min

x

✻ex √x = ✻ √ ✷e

• (p, q)

❚❤❡ r❡❛s♦♥ ✇❤② t❤✐s ✐s tr✉❡ ✐s t❤❛t ✶ ψ(w) = ✶ + k✷p(w)q(w)+ k✸

✸! (p(w)✸ + q(w)✸) + · · · ♥❡❣❧✐❣✐❜❧❡ ❜② ♦✉r ❛ss✉♠♣t✐♦♥ ♦♥

✸ ✸

✶ + kp(w) + (kp(w))✷

✷

+ kq(w) + (kq(w))✷

✷

t❤✉s

w ψ(w)−✶ ≃ w(✶ + k✷p(w)q(w)) ≃

• w exp(k✷p(w)q(w)) = exp(k✷(p, q))

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❲❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ❢❛st

■❢ t❤✐s ✐s tr✉❡✱ t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣❧❡①✐t② ❝❛♥ ❜❡ ♠❛❞❡ ❛s s♠❛❧❧ ❛s ❝♦♠♣❧❡①✐t② + Θ(ln n) = min

k

✷k

✶ ✸k✷(p, q)e−k✷(p,q)

= ✶

• (p, q)

min

x

✷√x

x ✸ exp(−x) =

✶

• (p, q)

min

x

✻ex √x = ✻ √ ✷e

• (p, q)

❚❤❡ r❡❛s♦♥ ✇❤② t❤✐s ✐s tr✉❡ ✐s t❤❛t ✶ ψ(w) = ✶ + k✷p(w)q(w)+ k✸

✸! (p(w)✸ + q(w)✸) + · · · ♥❡❣❧✐❣✐❜❧❡ ❜② ♦✉r ❛ss✉♠♣t✐♦♥ ♦♥

✸ ✸

✶ + kp(w) + (kp(w))✷

✷

+ kq(w) + (kq(w))✷

✷

t❤✉s

w ψ(w)−✶ ≃ w(✶ + k✷p(w)q(w)) ≃

• w exp(k✷p(w)q(w)) = exp(k✷(p, q))

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❲❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ❢❛st

■❢ t❤✐s ✐s tr✉❡✱ t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣❧❡①✐t② ❝❛♥ ❜❡ ♠❛❞❡ ❛s s♠❛❧❧ ❛s ❝♦♠♣❧❡①✐t② + Θ(ln n) = min

k

✷k

✶ ✸k✷(p, q)e−k✷(p,q)

= ✶

• (p, q)

min

x

✷√x

x ✸ exp(−x) =

✶

• (p, q)

min

x

✻ex √x = ✻ √ ✷e

• (p, q)

❚❤❡ r❡❛s♦♥ ✇❤② t❤✐s ✐s tr✉❡ ✐s t❤❛t ✶ ψ(w) = ✶ + k✷p(w)q(w)+ k✸

✸! (p(w)✸ + q(w)✸) + · · · ♥❡❣❧✐❣✐❜❧❡ ❜② ♦✉r ❛ss✉♠♣t✐♦♥ ♦♥

✸ ✸

✶ + kp(w) + (kp(w))✷

✷

+ kq(w) + (kq(w))✷

✷

t❤✉s

w ψ(w)−✶ ≃ w(✶ + k✷p(w)q(w)) ≃

• w exp(k✷p(w)q(w)) = exp(k✷(p, q))

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❲❤② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ❢❛st

■❢ t❤✐s ✐s tr✉❡✱ t❤❡ ❛✈❡r❛❣❡ ❝♦♠♣❧❡①✐t② ❝❛♥ ❜❡ ♠❛❞❡ ❛s s♠❛❧❧ ❛s ❝♦♠♣❧❡①✐t② + Θ(ln n) = min

k

✷k

✶ ✸k✷(p, q)e−k✷(p,q)

= ✶

• (p, q)

min

x

✷√x

x ✸ exp(−x) =

✶

• (p, q)

min

x

✻ex √x = ✻ √ ✷e

• (p, q)

❚❤❡ r❡❛s♦♥ ✇❤② t❤✐s ✐s tr✉❡ ✐s t❤❛t ✶ ψ(w) = ✶ + k✷p(w)q(w)+ k✸

✸! (p(w)✸ + q(w)✸) + · · · ♥❡❣❧✐❣✐❜❧❡ ❜② ♦✉r ❛ss✉♠♣t✐♦♥ ♦♥

• z(p(z)✸ + q(z)✸)

✶ + kp(w) + (kp(w))✷

✷

+ kq(w) + (kq(w))✷

✷

t❤✉s

w ψ(w)−✶ ≃ w(✶ + k✷p(w)q(w)) ≃

• w exp(k✷p(w)q(w)) = exp(k✷(p, q))

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❖♥❡ ❛♣♣❧✐❝❛t✐♦♥✿ r❛♥❞♦♠ ❜r✐❞❣❡s ✐♥ r❛♥❞♦♠ ♠❡❞✐❛

❈♦♥s✐❞❡r ❛ ❜✐st♦❝❤❛st✐❝ ❞✐❣r❛♣❤ ✭❛ ❣r❛♣❤ ✇✐t❤ ✉♥✐❢♦r♠ ✐♥✲ ❛♥❞ ♦✉t✲❞❡❣r❡❡✮ ✇❤✐❝❤ ❤❛s s♦♠❡ ❵tr❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ ♦♥ ❛✈❡r❛❣❡✬ ✭■ ❝❤♦♦s❡ ❤❡r❡ ❛ ♠♦❞❡❧ ♦❢ ♣❧❛q✉❡tt❡s ✐✳✐✳❞✳ ♦r✐❡♥t❡❞ ❝❧♦❝❦✇✐s❡ ♦r ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❖♥❡ ❛♣♣❧✐❝❛t✐♦♥✿ r❛♥❞♦♠ ❜r✐❞❣❡s ✐♥ r❛♥❞♦♠ ♠❡❞✐❛

❈♦♥s✐❞❡r ❛ ❜✐st♦❝❤❛st✐❝ ❞✐❣r❛♣❤ ✭❛ ❣r❛♣❤ ✇✐t❤ ✉♥✐❢♦r♠ ✐♥✲ ❛♥❞ ♦✉t✲❞❡❣r❡❡✮ ✇❤✐❝❤ ❤❛s s♦♠❡ ❵tr❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ ♦♥ ❛✈❡r❛❣❡✬ ✭■ ❝❤♦♦s❡ ❤❡r❡ ❛ ♠♦❞❡❧ ♦❢ ♣❧❛q✉❡tt❡s ✐✳✐✳❞✳ ♦r✐❡♥t❡❞ ❝❧♦❝❦✇✐s❡ ♦r ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❖♥❡ ❛♣♣❧✐❝❛t✐♦♥✿ r❛♥❞♦♠ ❜r✐❞❣❡s ✐♥ r❛♥❞♦♠ ♠❡❞✐❛

❈♦♥s✐❞❡r ❛ ❜✐st♦❝❤❛st✐❝ ❞✐❣r❛♣❤ ✭❛ ❣r❛♣❤ ✇✐t❤ ✉♥✐❢♦r♠ ✐♥✲ ❛♥❞ ♦✉t✲❞❡❣r❡❡✮ ✇❤✐❝❤ ❤❛s s♦♠❡ ❵tr❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ ♦♥ ❛✈❡r❛❣❡✬ ✭■ ❝❤♦♦s❡ ❤❡r❡ ❛ ♠♦❞❡❧ ♦❢ ♣❧❛q✉❡tt❡s ✐✳✐✳❞✳ ♦r✐❡♥t❡❞ ❝❧♦❝❦✇✐s❡ ♦r ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❖♥❡ ❛♣♣❧✐❝❛t✐♦♥✿ r❛♥❞♦♠ ❜r✐❞❣❡s ✐♥ r❛♥❞♦♠ ♠❡❞✐❛

❈♦♥s✐❞❡r ❛ ❜✐st♦❝❤❛st✐❝ ❞✐❣r❛♣❤ ✭❛ ❣r❛♣❤ ✇✐t❤ ✉♥✐❢♦r♠ ✐♥✲ ❛♥❞ ♦✉t✲❞❡❣r❡❡✮ ✇❤✐❝❤ ❤❛s s♦♠❡ ❵tr❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ ♦♥ ❛✈❡r❛❣❡✬ ✭■ ❝❤♦♦s❡ ❤❡r❡ ❛ ♠♦❞❡❧ ♦❢ ♣❧❛q✉❡tt❡s ✐✳✐✳❞✳ ♦r✐❡♥t❡❞ ❝❧♦❝❦✇✐s❡ ♦r ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮ ◆♦✇ ✇❡ ❤❛✈❡ ❛ r❛♥❞♦♠ ✐♥st❛♥❝❡ ♦❢ ❛ ❣r❛♣❤✳ ❖♥ t♦♣ ♦❢ t❤✐s✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r r❛♥❞♦♠ ✇❛❧❦s st❛rt✐♥❣ ❛t t❤❡ ♦r✐❣✐♥ ❉♦ ②♦✉ ❤❛✈❡ ❛ ❣♦♦❞ ❛❧❣♦r✐t❤♠ ❢♦r ✉♥✐❢♦r♠❧② s❛♠♣❧✐♥❣ ✇❛❧❦s

✵ ✵✱ ♦❢ ❧❡♥❣t❤ ✷ ✱ st❛rt✐♥❣

❛♥❞ ❛rr✐✈✐♥❣ ❛t t❤❡ ♦r✐❣✐♥❄

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❖♥❡ ❛♣♣❧✐❝❛t✐♦♥✿ r❛♥❞♦♠ ❜r✐❞❣❡s ✐♥ r❛♥❞♦♠ ♠❡❞✐❛

❈♦♥s✐❞❡r ❛ ❜✐st♦❝❤❛st✐❝ ❞✐❣r❛♣❤ ✭❛ ❣r❛♣❤ ✇✐t❤ ✉♥✐❢♦r♠ ✐♥✲ ❛♥❞ ♦✉t✲❞❡❣r❡❡✮ ✇❤✐❝❤ ❤❛s s♦♠❡ ❵tr❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ ♦♥ ❛✈❡r❛❣❡✬ ✭■ ❝❤♦♦s❡ ❤❡r❡ ❛ ♠♦❞❡❧ ♦❢ ♣❧❛q✉❡tt❡s ✐✳✐✳❞✳ ♦r✐❡♥t❡❞ ❝❧♦❝❦✇✐s❡ ♦r ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮ ◆♦✇ ✇❡ ❤❛✈❡ ❛ r❛♥❞♦♠ ✐♥st❛♥❝❡ ♦❢ ❛ ❣r❛♣❤✳ ❖♥ t♦♣ ♦❢ t❤✐s✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r r❛♥❞♦♠ ✇❛❧❦s st❛rt✐♥❣ ❛t t❤❡ ♦r✐❣✐♥ ❉♦ ②♦✉ ❤❛✈❡ ❛ ❣♦♦❞ ❛❧❣♦r✐t❤♠ ❢♦r ✉♥✐❢♦r♠❧② s❛♠♣❧✐♥❣ ✇❛❧❦s

✵ ✵✱ ♦❢ ❧❡♥❣t❤ ✷ ✱ st❛rt✐♥❣

❛♥❞ ❛rr✐✈✐♥❣ ❛t t❤❡ ♦r✐❣✐♥❄

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❖♥❡ ❛♣♣❧✐❝❛t✐♦♥✿ r❛♥❞♦♠ ❜r✐❞❣❡s ✐♥ r❛♥❞♦♠ ♠❡❞✐❛

❈♦♥s✐❞❡r ❛ ❜✐st♦❝❤❛st✐❝ ❞✐❣r❛♣❤ ✭❛ ❣r❛♣❤ ✇✐t❤ ✉♥✐❢♦r♠ ✐♥✲ ❛♥❞ ♦✉t✲❞❡❣r❡❡✮ ✇❤✐❝❤ ❤❛s s♦♠❡ ❵tr❛♥s❧❛t✐♦♥❛❧ ✐♥✈❛r✐❛♥❝❡ ♦♥ ❛✈❡r❛❣❡✬ ✭■ ❝❤♦♦s❡ ❤❡r❡ ❛ ♠♦❞❡❧ ♦❢ ♣❧❛q✉❡tt❡s ✐✳✐✳❞✳ ♦r✐❡♥t❡❞ ❝❧♦❝❦✇✐s❡ ♦r ❝♦✉♥t❡r❝❧♦❝❦✇✐s❡✮ ◆♦✇ ✇❡ ❤❛✈❡ ❛ r❛♥❞♦♠ ✐♥st❛♥❝❡ ♦❢ ❛ ❣r❛♣❤✳ ❖♥ t♦♣ ♦❢ t❤✐s✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r r❛♥❞♦♠ ✇❛❧❦s st❛rt✐♥❣ ❛t t❤❡ ♦r✐❣✐♥ ❉♦ ②♦✉ ❤❛✈❡ ❛ ❣♦♦❞ ❛❧❣♦r✐t❤♠ ❢♦r ✉♥✐❢♦r♠❧② s❛♠♣❧✐♥❣ ✇❛❧❦s W (n)

✵→✵✱ ♦❢ ❧❡♥❣t❤ ✷n✱ st❛rt✐♥❣

❛♥❞ ❛rr✐✈✐♥❣ ❛t t❤❡ ♦r✐❣✐♥❄

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❖♥❡ ❛♣♣❧✐❝❛t✐♦♥✿ r❛♥❞♦♠ ✇❛❧❦s ✐♥ r❛♥❞♦♠ ♠❡❞✐❛

❙❛♠♣❧✐♥❣ r❛♥❞♦♠ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ✷n✱ ✇✐t❤ ♥♦ ♣r❡s❝r✐❜❡❞ ❡♥❞♣♦✐♥t✱ ✐s tr✐✈✐❛❧❧② ❧✐♥❡❛r✳ ✳ ✳ ❜✉t t❤❡ ❡♥❞♣♦✐♥t ① ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✱ ♣r♦❜❛❜❧② ❛t ❞✐st❛♥❝❡ ∼ √n✱ ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ① = ✵ ✐s r♦✉❣❤❧② n− D

✷

◆❛ï✈❡ r❡❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠✿ ❝♦♠♣❧❡①✐t② n✶+ D

✷

◆♦✇✱ ❣r♦✇ t❤❡ ✜rst ❤❛❧❢ ♦❢ t❤❡ ♣❛t❤✱ ♦❢ ❧❡♥❣t❤ n✱ ✉♣ t♦ ✐ts ❡♥❞♣♦✐♥t ① ✭✇❤✐❝❤ ❤❛s ❞✐str✐❜✉t✐♦♥ p(①)✮✱ ❛♥❞ ❣r♦✇ t❤❡ ❧❛st ❤❛❧❢ ♦❢ t❤❡ ♣❛t❤✱ ❛❣❛✐♥ st❛rt✐♥❣ ❢r♦♠ t❤❡ ♦r✐❣✐♥✱ ❛♥❞ ✇❛❧❦✐♥❣ ♦♥ t❤❡ ❞✐❣r❛♣❤ ✇✐t❤ r❡✈❡rs❡❞ ♦r✐❡♥t❛t✐♦♥ ✭t❤✐s ❤❛s ❞✐str✐❜✉t✐♦♥ q(①)✮ ■❢ ②♦✉ ✉s❡ t❤❡ ❛❧❣♦r✐t❤♠ ❞❡s❝r✐❜❡❞ ❤❡r❡✱ ✇✐t❤ t❤❡s❡ p ❛♥❞ q✱ t❤❡ ❝♦♠♣❧❡①✐t② ❣♦❡s ❢r♦♠ n✶+ D

✷ t♦ n✶+ D ✹ ❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♣r♦❜❧❡♠

◆♦✇ s✉♣♣♦s❡ t❤❛t ②♦✉ ❤❛✈❡ ❛ ❢❛♠✐❧② ♦❢ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t ♣r♦❜❧❡♠s✱ t❤❛t ✐s✱ ❢♦r s♦♠❡ s❡t C✱ ②♦✉ ✇❛♥t t♦ s❛♠♣❧❡ x ❢r♦♠ f (x) =

• c∈C gc pc(x) qc(x)
• c,y gc pc(y) qc(y)

✭✇✐t❤

c gc = ✶✱ ❛♥❞ y pc(y) = y pc(y) = ✶ ❢♦r ❛❧❧ c✮

❖✉r ✇✐♥♥✐♥❣ str❛t❡❣② ✇❛s t♦ s❛♠♣❧❡ x ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❢♦r♠ k✷p(x)q(x)

• y

ψ(y) ✇❤❡r❡ t❤❡ k✷ ❢❛❝t♦r ✐s t❤❡ ✏❜✐rt❤❞❛② ♣❛r❛❞♦①✑ ❡♥❤❛♥❝❡♠❡♥t✱ ❛♥❞ t❤❡

• y ψ(y) ✐s ❛♥ ✐rr❡❧❡✈❛♥t ❢❛❝t♦r✱ ♦❢ ♦r❞❡r ✶ ✇❤❡♥ k✷(p, q) ✐s ♦❢ ♦r❞❡r ✶✳

❍♦✇❡✈❡r✱ ✐❢ ✇❡ ❥✉st ❞♦ t❤❡ s❛♠❡ ✐♥ t❤✐s ♠♦r❡ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦✱ t❤❡

y ψ(y) ❢❛❝t♦r ❞❡♣❡♥❞s ♦♥ C✱ ❛♥❞ ✐s ♥♦t ✐rr❡❧❡✈❛♥t ❛♥②♠♦r❡✳ ✳ ✳

❈❛♥ ✇❡ ❝♦rr❡❝t ✐ts ❝♦♥tr✐❜✉t✐♦♥❄

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♣r♦❜❧❡♠

■❢ ✇❡ ❥✉st ❛♣♣❧② t❤❡ str❛t❡❣② ❛❜♦✈❡✱ ✇❡ s❛♠♣❧❡ ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ❧✐❦❡ f❛❧❣(x) ∝

• c∈C

gc ✶

✸k✷pc(x) qc(x) exp(−k✷(pc, qc) + · · · )

❲❡ ❤❛✈❡ ❛ ♥✐❝❡ ✈❛r✐❛♥t ♦❢ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t ❛❧❣♦r✐t❤♠✱ ✇❤✐❝❤ ✉s❡s ✏✐♥❝r❡♠❡♥t❛❧ ❧✐sts✑ ✐♥st❡❛❞ ♦❢ ✏st❛t✐❝✑ ♦♥❡s✳ ❚❤✐s ✇♦r❦s ✇✐t❤ ❝♦♥t✐♥✉♦✉s t✐♠❡ ❛♥❞ P♦✐ss♦♥ ❝❧♦❝❦s✱ ❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❡r♠✐♥❛t✐♥❣ ❛t t✐♠❡ ✐♥ [t, t + dt] ✐s f❛❧❣(x, t)dt ∝

• c∈C

gc ✶

✸t pc(x) qc(x) exp(−t✷(pc, qc) + t✸ · · · )dt

t❤❛t ✐s✱ ✐♥t❡❣r❛t✐♥❣ ♦✈❡r t✐♠❡✱ f❛❧❣(x) ∼ ✶ ✸

• c∈C

gc pc(x)qc(x) (pc, qc) + · · · t❤✐s ♦t❤❡r ❜✐❛s ❢❛❝t♦r ✐s ❛ ❜✐t ❡❛s✐❡r t♦ ❝♦rr❡❝t✳ ✳ ✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❆ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♣r♦❜❧❡♠

❙✉♠♠❛r②✿ t❤❡ ✏✐♥❝r❡♠❡♥t❛❧ ❧✐sts✑ ✈❛r✐❛♥t ♦❢ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t ❛❧❣♦r✐t❤♠ ❣✐✈❡s ❛ ♠❡❛s✉r❡ f❛❧❣(x) ∼ ✶ ✸

• c∈C

gc pc(x)qc(x) (pc, qc) + · · · ✐♥st❡❛❞ ♦❢ t❤❡ ❞❡s✐r❡❞ f (x) ∝

c∈C gc pc(x)qc(x)✳

❲❡ ❝❛♥ ❝♦rr❡❝t t❤✐s ❜✐❛s✱ ✉♣ t♦ ❡rr♦r t❡r♠s t❤❛t ✇❡ st✐❧❧ ❤❛✈❡ t♦ ✜①✱ ❜② ✜rst ❞♦✐♥❣ t❤❡ ❇✐rt❤❞❛② ♣❛r❛❞♦① str❛t❡❣② ✭❛t ❣✐✈❡♥ c ✇✐t❤ ❛ st❛t✐❝ ♣❛r❛♠❡t❡r k s✉❝❤ t❤❛t ✇❡ ❦♥♦✇ t❤❛t k✷ maxc(pc, qc) ✶✮✱ ❛♥❞ t❤❡♥ ♣❡r❢♦r♠ t❤❡ ✐♥❝r❡♠❡♥t❛❧ ❧✐st r✉♥ ❛t c ♦♥❧② ✐❢ ✇❡ ❤❛✈❡ ❢♦✉♥❞ s♦♠❡ ✏❣♦♦❞ ♣❛✐r✑✳ ❚❤✐s ❣✐✈❡s ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❢♦r♠ f❛❧❣(x) ∼ ✶

✸

• c∈C

gc pc(x)qc(x) (pc, qc) + · · ·

• ✶−
• y
• e−kpc(y)+e−kqc(y)−e−k(pc(y)+qc(y))

∼ ✶

✸

• c∈C

gc pc(x)qc(x)k✷(pc, qc) + · · · (pc, qc) + · · · ∝

• c∈C

gc pc(x)qc(x)(✶ + · · · )

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣ ♦❢ s♣❡❝✐✜❝❛❜❧❡ str✉❝t✉r❡s

▲❡t ✉s ❥✉st ♣r❡t❡♥❞ t❤❛t ✇❡ ❝❛♥ s♦❧✈❡ t❤✐s ❵s♠❛❧❧✬ (✶ + · · · ) ♣r♦❜❧❡♠✱ ❛♥❞ s❡❡ ✇❤❛t t❤✐s ✇♦✉❧❞ ✐♠♣❧② ❛t t❤❡ ❧❡✈❡❧ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✳✳✳ ❘❡❝❛❧❧ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❵❝♦♠❜✐♥❛t♦r✐❛❧ s♣❡❝✐✜❝❛t✐♦♥✬ s②st❡♠s✿      A✶ = F✶(A✶, . . . , Ak, z) ✳ ✳ ✳ Ak = Fk(A✶, . . . , Ak, z) ❇② t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ♠❛r❦❡❞ ♦❜❥❡❝ts ✭❛ ✉s❡❢✉❧ tr✐❝❦ ✐♥ ♦r❞❡r t♦ ♠♦❞✐❢② t❤❡ t❛✐❧ ❡①♣♦♥❡♥t ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✮✱ ②♦✉ ❣❡t ❛ss♦❝✐❛t❡❞ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ♠❛r❦❡❞ ❝❧❛ss❡s      A′

✶ = i A′ iF✶,i(A✶, . . . , Ak, z) + F✶,✵(A✶, . . . , Ak, z)

✳ ✳ ✳ A′

k = i A′ iFk,i(A✶, . . . , Ak, z) + Fk,✵(A✶, . . . , Ak, z)

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣ ♦❢ s♣❡❝✐✜❝❛❜❧❡ str✉❝t✉r❡s

◆♦✇✱ r❡❞❡✜♥❡ Ai = ✶

✷(Ar❡❞ i

+ A❜❧✉❡

i

) ✐♥ t❤❡ A′

i ❡q✉❛t✐♦♥s✱ ❛♥❞

♣❡r❢♦r♠ t❤❡ ●❛❧t♦♥✕❲❛ts♦♥ ❡①♣❧♦r❛t✐♦♥ ♦❢ t❤❡ A′

i ❵s♣✐♥❡✬ ♦♥❧②✱

✉♣ t♦ ✇❤❡♥ ②♦✉ ❤✐t t❤❡ ♠❛r❦❡❞ ♦❜❥❡❝t✳ ❆t t❤✐s ♣♦✐♥t ②♦✉ ❤❛✈❡ t②♣✐❝❛❧❧② ♣r♦❞✉❝❡❞✿ ◮ νr❡❞

i

∼ √ N r❡❞ ❜r❛♥❝❤❡s Ar❡❞

i

✭❢♦r t❤❡ ✈❛r✐♦✉s ✶ ≤ i ≤ k✮✱ ◮ ν❜❧✉❡

i

∼ √ N ❜❧✉❡ ❜r❛♥❝❤❡s A❜❧✉❡

i

✱ ◮ n✵ ∼ √ N ❡❧❡♠❡♥t❛r② ✉♥✐ts✱ ◮ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❢❛❝t♦r g✳ ❈❛❧❧ c t❤❡ ❞❛t✉♠ ♦❢ {νr❡❞

i

, ν❜❧✉❡

i

, n✵, g} ❛♥❞ pc(x) = [zx]

• i

(Ar❡❞

i

(z))νr❡❞

i

qc(x) = [zN−n✵(c)−x]

• i

(A❜❧✉❡

i

(z))ν❜❧✉❡

i

❚❤❡♥ ✇❡ ❛r❡ ❡①❛❝t❧② ✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ❛❜♦✈❡✦

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

▲❡t✬s ❝❤♦♦s❡ t❤❡ ❡①❛♠♣❧❡ ♦❢ W✹✲❢r❡❡ ❣r❛♣❤s✱ t❤❛t ✐s✱ ❣r❛♣❤s t❤❛t ❞♦ ♥♦t ❝♦♥t❛✐♥ ❛s ❛ ♠✐♥♦r✳ ❚❤❡s❡ ❣r❛♣❤s✱ r♦♦t❡❞ ❛t ♦♥❡ ❡❞❣❡✱ ❛r❡ ✐♥ ❜✐❥❡❝t✐♦♥ ✇✐t❤ ✷✲t❡r♠✐♥❛❧ ❣r❛♣❤s G[uv] t❤❛t ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ♦♥❡ ❡❞❣❡ ❜② s❡r✐❡s✱ ♣❛r❛❧❧❡❧ ❛♥❞ ❵✇❤❡❛tst♦♥❡ ❜r✐❞❣❡✬ r❡❞✉❝t✐♦♥s✿ G[uv] G u v

• w

− →

• p

− →

• s

− → ❈r✉❝✐❛❧❧②✱ ❢♦r ❛♥② s✉❝❤ ❣r❛♣❤✱ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ tr❡❡ ✐s ✉♥✐q✉❡✦ ❚❤✉s ❝♦✉♥t✐♥❣ t❤❡ ❣r❛♣❤s ✐s ❧✐❦❡ ❝♦✉♥t✐♥❣ t❤❡s❡ tr❡❡s✱ ✇❤✐❝❤ ❣✐✈❡s r✐s❡ t♦ ❛ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✭❥✉st ❧✐❦❡ s❡r✐❡s✲♣❛r❛❧❧❡❧ ❣r❛♣❤s✱ ❜✉t ❛ ❜✐t ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

■♥tr♦❞✉❝❡ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s✿ ◮ z ✐s t❤❡ ♦♥❡✲❡❞❣❡ tr✐✈✐❛❧ ❝❧❛ss❀ ◮ A ✐s ❛❧❧ t❤❡ ✭✷✲t❡r♠✐♥❛❧✮ ❣r❛♣❤s ✇❡ ✇❛♥t❀ ◮ S ❣r❛♣❤s ❝♦♥s✐st ♦❢ ✷ ♦r ♠♦r❡ ❜❧♦❝❦s ✐♥ s❡r✐❡s ✐♥ t❤❡ ♦✉t❡r♠♦st ❧❛②❡r❀ ◮ P ❣r❛♣❤s ❝♦♥s✐st ♦❢ ✷ ♦r ♠♦r❡ ❜❧♦❝❦s❀ ◮ W ❣r❛♣❤s ❝♦♥s✐st ♦❢ ♦♥❡ ❲❤❡❛tst♦♥❡ ❜r✐❞❣❡✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ ●❛❧t♦♥✕❲❛ts♦♥ tr❡❡s ✐♥ ✇❤✐❝❤✿ ◮ ❛❧❧ ❛♥❞ ♦♥❧② t❤❡ ❧❡❛✈❡s ❛r❡ z❀ ◮ A ✐s ♦♥❧② ❛t t❤❡ r♦♦t✱ ❤❛s ❛ s✐♥❣❧❡ ❝❤✐❧❞ ✐♥ t❤❡ ❧✐st {S, P, W , z}❀ ◮ S ❛♥❞ P ❞♦ ♥♦t ❤❛✈❡ ❝❤✐❧❞r❡♥ ✇✐t❤ s❛♠❡ ❧❛❜❡❧✱ ❛♥❞ ❤❛✈❡ ❞❡❣r❡❡ ❛t ❧❡❛st ✷✳ ❆ P ♥♦❞❡ ♦❢ ❞❡❣r❡❡ d ❝♦♠❡s ✇✐t❤ ❛ s②♠♠❡tr② ❢❛❝t♦r ✶/d!❀ ◮ W ❤❛s ❞❡❣r❡❡ ✺ ❛♥❞ s②♠♠❡tr② ❢❛❝t♦r ✶/✷✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

■♥tr♦❞✉❝❡ t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s✿ ◮ z ✐s t❤❡ ♦♥❡✲❡❞❣❡ tr✐✈✐❛❧ ❝❧❛ss❀ ◮ A ✐s ❛❧❧ t❤❡ ✭✷✲t❡r♠✐♥❛❧✮ ❣r❛♣❤s ✇❡ ✇❛♥t❀ ◮ S ❣r❛♣❤s ❝♦♥s✐st ♦❢ ✷ ♦r ♠♦r❡ ❜❧♦❝❦s ✐♥ s❡r✐❡s ✐♥ t❤❡ ♦✉t❡r♠♦st ❧❛②❡r❀ ◮ P ❣r❛♣❤s ❝♦♥s✐st ♦❢ ✷ ♦r ♠♦r❡ ❜❧♦❝❦s❀ ◮ W ❣r❛♣❤s ❝♦♥s✐st ♦❢ ♦♥❡ ❲❤❡❛tst♦♥❡ ❜r✐❞❣❡✳ ❚❤✐s ❣✐✈❡s r✐s❡ t♦ ●❛❧t♦♥✕❲❛ts♦♥ tr❡❡s ✐♥ ✇❤✐❝❤✿ ◮ ❛❧❧ ❛♥❞ ♦♥❧② t❤❡ ❧❡❛✈❡s ❛r❡ z❀ ◮ A ✐s ♦♥❧② ❛t t❤❡ r♦♦t✱ ❤❛s ❛ s✐♥❣❧❡ ❝❤✐❧❞ ✐♥ t❤❡ ❧✐st {S, P, W , z}❀ ◮ S ❛♥❞ P ❞♦ ♥♦t ❤❛✈❡ ❝❤✐❧❞r❡♥ ✇✐t❤ s❛♠❡ ❧❛❜❡❧✱ ❛♥❞ ❤❛✈❡ ❞❡❣r❡❡ ❛t ❧❡❛st ✷✳ ❆ P ♥♦❞❡ ♦❢ ❞❡❣r❡❡ d ❝♦♠❡s ✇✐t❤ ❛ s②♠♠❡tr② ❢❛❝t♦r ✶/d!❀ ◮ W ❤❛s ❞❡❣r❡❡ ✺ ❛♥❞ s②♠♠❡tr② ❢❛❝t♦r ✶/✷✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

       A = S + P + W + z S = (A − S)✷/(✶ − (A − S)) P = exp(A − P) − (✶ + (A − P)) W = A✺/✷ ✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ✜rst ♣♦✐♥t ♦❢ s❧♦♣❡ ③❡r♦✱ ♦♥ t❤❡ ❜r❛♥❝❤ st❛rt✐♥❣ ✐♥ ✵ ✵ ✇✐t❤ s❧♦♣❡ ✶ ✭t❤✐s ❤♦❧❞s ❢♦r ❛♥② ♣♦❧②♥♦♠✐❛❧ ✱ ✐✳❡✳ ❛♥② r❡❝✉rs✐✈❡ ❢❛♠✐❧② ♦❢ ❣r❛♣❤s✮✳ ❚❤✉s ✐s ❛ r♦♦t ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✶

✷

✶

✷

✵✳ ❙♦ ✇❡ ❵❦♥♦✇✬ t❤❡ ❜r❛♥❝❤✐♥❣ r❛t❡s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ❝r✐t✐❝❛❧ ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

   A = S + P + W(A) + z W(A) = A✺/✷ S = (A − S)✷/(✶ − (A − S)) P = exp(A − P) − (✶ + (A − P)) ✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ✜rst ♣♦✐♥t ♦❢ s❧♦♣❡ ③❡r♦✱ ♦♥ t❤❡ ❜r❛♥❝❤ st❛rt✐♥❣ ✐♥ ✵ ✵ ✇✐t❤ s❧♦♣❡ ✶ ✭t❤✐s ❤♦❧❞s ❢♦r ❛♥② ♣♦❧②♥♦♠✐❛❧ ✱ ✐✳❡✳ ❛♥② r❡❝✉rs✐✈❡ ❢❛♠✐❧② ♦❢ ❣r❛♣❤s✮✳ ❚❤✉s ✐s ❛ r♦♦t ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✶

✷

✶

✷

✵✳ ❙♦ ✇❡ ❵❦♥♦✇✬ t❤❡ ❜r❛♥❝❤✐♥❣ r❛t❡s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ❝r✐t✐❝❛❧ ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

   A = S + P + W(A) + z W(A) = A✺/✷ S = A✷/(✶ + A) P = A − ln(✶ + A) z = ln(✶ + A) − A✷ ✶ + A − W(A) (Ac, zc) ✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ✜rst ♣♦✐♥t ♦❢ s❧♦♣❡ ③❡r♦✱ ♦♥ t❤❡ ❜r❛♥❝❤ st❛rt✐♥❣ ✐♥ (✵, ✵) ✇✐t❤ s❧♦♣❡ ✶ ✭t❤✐s ❤♦❧❞s ❢♦r ❛♥② ♣♦❧②♥♦♠✐❛❧ W(A)✱ ✐✳❡✳ ❛♥② r❡❝✉rs✐✈❡ ❢❛♠✐❧② ♦❢ ❣r❛♣❤s✮✳ ❚❤✉s Ac ✐s ❛ r♦♦t ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✶ − A − A✷ − (✶ + A)✷W′(A) = ✵✳ ❙♦ ✇❡ ❵❦♥♦✇✬ t❤❡ ❜r❛♥❝❤✐♥❣ r❛t❡s ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ❝r✐t✐❝❛❧ ●❛❧t♦♥✕❲❛ts♦♥ ♣r♦❝❡ss✳

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

▲❡t✬s ❣♦ ❜❛❝❦ t♦ t❤❡ ❵❝♦♠❜✐♥❛t♦r✐❛❧✬ s②st❡♠✳✳✳    A = S + P + W(A) + z S =

k≥✷(A − S)k

P =

k≥✷ ✶ k!(A − P)k

❲✐t❤ ♦♥❡ ♠❛r❦✐♥❣ ✇❡ ❣❡t    A′ = S′ + P′ + A′W′(A) + ✶ S′ = (A′ − S′)

k≥✶(k + ✶)(A − S)k

P′ = (A′ − P′)

k≥✶ ✶ k!(A − P)k

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

❆ t②♣✐❝❛❧ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❵s♣✐♥❡✬ ♦❢ t❤❡ tr❡❡ ❢♦❧❧♦✇✐♥❣ t❤❡ X ′ ❝❧❛ss✿    A = S + P + W(A) + z S =

k≥✷(A − S)k

P =

k≥✷ ✶ k!(A − P)k

   A′ = S′ + P′ + A′W′(A) + ✶ S′ = (A′ − S′)

k≥✶(k + ✶)(A − S)k

P′ = (A′ − P′)

k≥✶ ✶ k!(A − P)k

✭t❤❡ ❝♦❧♦✉rs t❡❧❧ ❤♦✇ t♦ ✐♠♣❧❡♠❡♥t t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦✮

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

❆ t②♣✐❝❛❧ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❵s♣✐♥❡✬ ♦❢ t❤❡ tr❡❡ ❢♦❧❧♦✇✐♥❣ t❤❡ X ′ ❝❧❛ss✿

      

W = ✶

✷(S+P+W +z)✺

S =

k≥✷(P+W +z)k

P =

k≥✷ ✶ k!(S+W +z)k       

W ′ = ✺

✷(S′+P′+W ′+✶)(S+P+W +z)✹

S′ = (P′+W ′+✶)

k≥✶(k+✶)(P+W +z)k

P′ = (S′+W ′+✶)

k≥✶ ✶ k!(S+W +z)k

✭t❤❡ ❝♦❧♦✉rs t❡❧❧ ❤♦✇ t♦ ✐♠♣❧❡♠❡♥t t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦✮

A S P W P P P P S

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣

• r❛♣❤s ✇✐t❤ ♥♦ W✹ ♠✐♥♦r

❆ t②♣✐❝❛❧ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❵s♣✐♥❡✬ ♦❢ t❤❡ tr❡❡ ❢♦❧❧♦✇✐♥❣ t❤❡ X ′ ❝❧❛ss✿

      

W = ✶

✷(S+P+W +z)✺

S =

k≥✷(P+W +z)k

P =

k≥✷ ✶ k!(S+W +z)k       

W ′ = ✺

✷(S′+P′+W ′+✶)(S+P+W +z)✹

S′ = (P′+W ′+✶)

k≥✶(k+✶)(P+W +z)k

P′ = (S′+W ′+✶)

k≥✶ ✶ k!(S+W +z)k

✭t❤❡ ❝♦❧♦✉rs t❡❧❧ ❤♦✇ t♦ ✐♠♣❧❡♠❡♥t t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t tr✐❝❦✮

A S P W P P P P S S P W P S S S P

❆♥❞r❡❛ ❙♣♦rt✐❡❧❧♦ ❚❤❡ ❝❤❛❧❧❡♥❣❡ ♦❢ ❧✐♥❡❛r✲t✐♠❡ ❇♦❧t③♠❛♥♥ s❛♠♣❧✐♥❣