On the Flooding Overhead of Fountain Codes in Wireless Sensor … · 2020-02-14 · Wireless Sensor...
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Submitted on 27 Sep 2010
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On the Flooding Overhead of Fountain Codes inWireless Sensor Networks
Anya Apavatjrut, Katia Jaffrès-Runser, Claire Goursaud, Cédric Lauradoux
To cite this version:Anya Apavatjrut, Katia Jaffrès-Runser, Claire Goursaud, Cédric Lauradoux. On the Flooding Over-head of Fountain Codes in Wireless Sensor Networks. [Research Report] RR-7397, INRIA. 2010,pp.16. �inria-00521326�
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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
On the Flooding Overhead of Fountain Codes in
Wireless Sensor Networks
Anya Apavatjrut — Katia Jaffrès-Runser — Claire Goursaud — Cédric Lauradoux
N° 7397
September 2010
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Centre de recherche INRIA Grenoble – Rhône-Alpes655, avenue de l’Europe, 38334 Montbonnot Saint Ismier
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥
❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s
❆♥②❛ ❆♣❛✈❛t❥r✉t∗✱ ❑❛t✐❛ ❏❛✛rès✲❘✉♥s❡r∗ ✱ ❈❧❛✐r❡ ●♦✉rs❛✉❞∗ ✱
❈é❞r✐❝ ▲❛✉r❛❞♦✉①∗
❚❤❡♠❡ ✿➱q✉✐♣❡s✲Pr♦❥❡ts ❙❲■◆●
❘❛♣♣♦rt ❞❡ r❡❝❤❡r❝❤❡ ♥➦ ✼✸✾✼ ✖ ❙❡♣t❡♠❜❡r ✷✵✶✵ ✖ ✶✸ ♣❛❣❡s
❆❜str❛❝t✿ ❚❤✐s ♣❛♣❡r ❝♦♥❝❡♥tr❛t❡s ♦♥ t❤❡ ♣r♦♣❡r ✉s❡ ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡s ❢♦rt❤❡ tr❛♥s♠✐ss✐♦♥ ♦❢ s♣♦r❛❞✐❝ ❞❛t❛ ✐♥ ❛ ✇✐r❡❧❡ss s❡♥s♦r ♥❡t✇♦r❦ ✭❲❙◆✮✳ ❋♦✉♥t❛✐♥❝♦❞❡s ♦✛❡r ❣r❡❛t ♣❡rs♣❡❝t✐✈❡s ❢♦r t❤❡ s❡❧❢✲♦r❣❛♥✐③❛t✐♦♥ ♦❢ ❲❙◆s✿ t❤❡② s❡❧❢ ❛❞❛♣tt♦ t❤❡ ❝❤❛♥♥❡❧ ❡rr♦r r❛t❡ ✇✐t❤♦✉t ❛♥② ❝♦♥tr♦❧ ❞❛t❛✳ ❲❤❡♥ ❞❡♣❧♦②✐♥❣ ❢♦✉♥t❛✐♥❝♦❞❡s ♦♥ ❛ ❲❙◆✱ t✇♦ ♣r♦❜❧❡♠s ❛r✐s❡✳ ❋✐rst✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❞❛t❛ tr❛♥s♠✐tt❡❞ ❜②❛ s❡♥s♦r ✐s s♠❛❧❧ ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ s✐③❡ ❝♦♥s✐❞❡r❡❞ tr❛❞✐t✐♦♥❛❧❧② ✇✐t❤ ❢♦✉♥t❛✐♥❝♦❞❡s✳ ❚❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❞❡❝♦❞✐♥❣ ♦✈❡r❤❡❛❞ ❢♦r ❢♦✉♥t❛✐♥ ❝♦❞❡s ✐s ♦❢t❡♥ ❞♦♥❡ ❢♦r❧❛r❣❡ ❞❛t❛✳ ❙❡❝♦♥❞✱ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❛r❡ ❞♦♥❡ ✐♥ ❛♥ ❤♦♣✲❜②✲❤♦♣ ❢❛s❤✐♦♥✳ ■t✐♠♣❧✐❡s t❤❛t t❤❡ ❞❡st✐♥❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛ ❝❛♥ ♥♦t ❛❝❦♥♦✇❧❡❞❣❡ ✐♥st❛♥t❛♥❡♦✉s❧② ✐tsr❡❝❡♣t✐♦♥ t♦ t❤❡ s♦✉r❝❡✳ ❚❤❡r❡❢♦r❡✱ t❤❡ tr❛♥s♠✐ss✐♦♥s ♦❢ ✉s❡❧❡ss ♣❛❝❦❡ts ❢♦r t❤❡❞❡st✐♥❛t✐♦♥ ❝❛♥ ♥♦t ❜❡ ♣r❡✈❡♥t❡❞✳ ❚❤❡ ✐♠♣❛❝t ♦❢ t❤✐s ✢♦♦❞✐♥❣ tr❛✣❝ ✐s ❛♥❛❧②③❡❞✳■t ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❛t❛ s✐③❡ k ❛♥❞ ♦♥ ♥✉♠❜❡r ♦❢ ❤♦♣s n ❜❡t✇❡❡♥ t❤❡ s♦✉r❝❡ ❛♥❞t❤❡ ❞❡st✐♥❛t✐♦♥✳ ❖✉r ✇♦r❦ ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ♥❡t✇♦r❦✐♥❣ ❝♦✉♥t❡r♣❛rt ♦❢t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ❜② P❛❦③❛❞ ❛♥❞ ❛❧✳ ❛t ■❙■❚ ✷✵✵✺ ❛♣♣❧✐❡❞ t♦ ❲❙◆s✳ ❚❤❡❝♦♥t❡①t ♦❢ ♦✉r st✉❞② ✐s ❛ ❧✐♥❡ ♥❡t✇♦r❦✱ ✐✳❡✳ ❛ ❝❛s❝❛❞❡ ♦❢ n ❡r❛s✉r❡ ❝❤❛♥♥❡❧s✳❚❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ❤❛s ❜❡❡♥ ❡✈❛❧✉❛t❡❞ ❛s ✇❡❧❧ t❤r♦✉❣❤ r❡❛❧✐st✐❝ s✐♠✉❧❛t✐♦♥s ❢♦rt❤r❡❡ ❞✐✛❡r❡♥t r❡❧❛②✐♥❣ str❛t❡❣✐❡s ✇❤❡r❡ ♣❛❝❦❡ts ❛r❡ ❧♦st ❞✉❡ t♦ ❜♦t❤ s♠❛❧❧ s❝❛❧❡❢❛❞✐♥❣ ❛♥❞ ❝♦❧❧✐s✐♦♥s ❢♦r ❛♥ ✉♥s❧♦tt❡❞ ■❊❊❊ ✽✵✷✳✶✺✳✹ ♠❡❞✐✉♠ ❛❝❝❡ss ❧❛②❡r✳
❑❡②✲✇♦r❞s✿ ✢♦♦❞✐♥❣ ♦✈❡r❤❡❛❞✱ ❢♦✉♥t❛✐♥ ❝♦❞❡s✱ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t✱ ✇✐r❡❧❡sss❡♥s♦r ♥❡t✇♦r❦s
∗ ❈■❚■ ❧❛❜♦r❛t♦r②✱ ■◆❙❆ ❞❡ ▲②♦♥✱ ❋r❛♥❝❡
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❘és✉♠é ✿ ❈❡t ❛rt✐❝❧❡ ♣r♦♣♦s❡ ❧✬♦♣t✐♠✐s❛t✐♦♥ ❞❡ ❧✬✉s❛❣❡ ❞✬✉♥ ❝♦❞❡ ❢♦♥t❛✐♥❡ ♣♦✉r✉♥❡ tr❛♥s♠✐ss✐♦♥ s♣♦r❛❞✐q✉❡ ❞❡ ❞♦♥♥é❡s ❞❛♥s ❧❡s rés❡❛✉① ❞❡ ❝❛♣t❡✉r✳ ❉❛♥s❝❡ ❝♦♥t❡①t❡✱ ❝❡s ❝♦❞❡s ♣rés❡♥t❡♥t ✉♥ ❛✈❛♥t❛❣❡ r❡❧❛t✐❢ à ❧✬❛✉t♦✲♦r❣❛♥✐s❛t✐♦♥ ❞✉rés❡❛✉✿ s♦♥ t❛✉① ❞❡ tr❛♥s♠✐ss✐♦♥ ❞❡ ❞♦♥♥é❡s ❡st ✈❛r✐❛❜❧❡ ❡t s✬❛❞❛♣t❡ ✐♥tr✐♥✲sèq✉❡♠❡♥t à ❧❛ q✉❛❧✐té ❞✉ ❝❛♥❛❧ r❛❞✐♦ s❛♥s q✉✬✐❧ ♥✬② ❛✐t ❜❡s♦✐♥ ❞❡ ♣❛q✉❡ts ❞❡❝♦♥trô❧❡ s✉♣♣❧é♠❡♥t❛✐r❡s✳ ❈❡♣❡♥❞❛♥t✱ ♣♦✉r ♣♦✉✈♦✐r ✉t✐❧✐s❡r ❧❡s ❝♦❞❡s ❢♦♥t❛✐♥❡s❞❛♥s ❧❡s rés❡❛✉① ❞❡ ❝❛♣t❡✉rs✱ ✐❧ ❡st ♥é❝❡ss❛✐r❡ ❞❡ ♣r❡♥❞r❡ ❡♥ ❝♦♠♣t❡ ❧❡s ❝♦♥✲tr❛✐♥t❡s s✉✐✈❛♥t❡s ✿ ♣r❡♠✐èr❡♠❡♥t✱ ❧❡ s✉r❝♦ût ❧✐é ❛✉ ❞é❝♦❞❛❣❡ ♥✬❡st ♣❧✉s ♥é❣✲❧✐❣❡❛❜❧❡ ❧♦rsq✉❡ ❧❛ t❛✐❧❧❡ ❞❡s ❞♦♥♥é❡s ❞❡✈✐❡♥t ♣❡t✐t❡ ❡t ❞❡✉①✐è♠❡♠❡♥t✱ ✐❧ ❡①✐st❡✉♥ ❞é❜♦r❞❡♠❡♥t ❞✉ ✢✉① ❞❡s ♣❛q✉❡ts ❝♦❞és r❡❧❛t✐❢ à ❧❛ tr❛♥s♠✐ss✐♦♥ ❡♥ ♠♦❞❡♠✉❧t✐✲s❛✉ts ❞❡ ❧✬❛❝q✉✐tt❡♠❡♥t✳ ❊♥ ❡✛❡t✱ s✐ ❧✬❛❝q✉✐tt❡♠❡♥t ♠❡ts ♣❧✉s ❞❡ t❡♠♣sà ❛tt❡✐♥❞r❡ ❧❛ s♦✉r❝❡ r❡❧❛t✐✈❡♠❡♥t à ❧❛ ✈✐t❡ss❡ ❞❡ r❡❧❛②❛❣❡ ❞❡s ♣❛q✉❡ts ❝♦❞és♣❛r ❧❛ s♦✉r❝❡✱ ❝❡rt❛✐♥s r❡❧❛✐s ❛✐♥s✐ q✉❡ ❧❛ s♦✉r❝❡ é♠❡tt❡♥t ✉♥ ♥♦♠❜r❡ ❞❡ ♣❛q✉❡ts✐♥✉t✐❧❡s ❛✉ ❞é❝♦❞❛❣❡ ❛✈❛♥t ❞✬❛rrêt❡r ❧❛ tr❛♥s♠✐ss✐♦♥✳ ❈✬❡st ♣r✐♥❝✐♣❛❧❡♠❡♥t ❧❡❝❛s ♣♦✉r ✉♥❡ tr❛♥s♠✐ss✐♦♥ ❡♥ ❝❛s❝❛❞❡ ♠✉❧t✐✲s❛✉ts ♣♦✉r ❧❛q✉❡❧❧❡ ❧❡ ♥♦♠❜r❡ ❞❡r❡❧❛✐s ❡st é❧❡✈é✳ ❉❛♥s ❝❡t ❛rt✐❝❧❡✱ ♥♦✉s ét✉❞✐♦♥s ❝❡ s✉r❝♦ût q✉✐ ✈❛r✐❡ ❛✈❡❝ ❧❛q✉❛♥t✐té ❞❡ ♣❛q✉❡ts à tr❛♥s♠❡ttr❡ ❡t ❧❡ ♥♦♠❜r❡ ❞❡ s❛✉ts ❡♥tr❡ ❧❛ s♦✉r❝❡ ❡t ❧❡❞❡st✐♥❛t❛✐r❡✳ ❈❡ s✉r❝♦ût ❞❡ tr❛♥s♠✐ss✐♦♥ ❡st ❛♥❛❧②sé ♣♦✉r ♣❧✉s✐❡✉rs str❛té❣✐❡s❞❡ r❡❧❛②❛❣❡✱ t♦✉t ❞✬❛❜♦r❞ ❛♥❛❧②t✐q✉❡♠❡♥t ♣✉✐s ♣❛r s✐♠✉❧❛t✐♦♥ ♣♦✉r ✉♥ ♣r♦t♦❝♦❧❡❩✐❣❜❡❡ ■❊❊❊ ✽✵✷✳✶✺✳✹✳
▼♦ts✲❝❧és ✿ s✉r❝♦ût ❧✐é à ❧✬❛❝q✉✐tt❡♠❡♥t ❞✉ ✢✉① ❞❡ tr❛♥s♠✐ss✐♦♥✱ ❝♦❞❡s❢♦♥t❛✐♥❡s✱ rés❡❛✉① ❞❡ ❝❛♣t❡✉rs✱ ■❊❊❊ ✽✵✷✳✶✺✳✹✳
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✸
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤✐s ♣❛♣❡r ✐s ❞❡❞✐❝❛t❡❞ t♦ t❤❡ ❞❡♣❧♦②♠❡♥t ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡ ✐♥ ❛ ✇✐r❡❧❡ss s❡♥s♦r♥❡t✇♦r❦ ✭❲❙◆✮✳ ❆❲❙◆ ✐s ❝♦♠♣♦s❡❞ ♦❢ s❡♥s♦r ♥♦❞❡s ✇✐t❤ r❡str✐❝t❡❞ ❝❛♣❛❜✐❧✐t✐❡s✭♠❡♠♦r②✱ ❡♥❡r❣② ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣♦✇❡r✮ ❛♥❞ ❛ s❡t ♦❢ s✐♥❦ ♥♦❞❡s ✇❤✐❝❤ ❣❛t❤❡rt❤❡ s❡♥s❡❞ ❞❛t❛✳ ■t ✐s ❛ss✉♠❡❞ ❤❡r❡ t❤❛t t❤❡ s✐♥❦ ♥♦❞❡s ❤❛✈❡ ✉♥❧✐♠✐t❡❞ r❡s♦✉r❝❡s❝♦♠♣❛r❡❞ t♦ t❤❡ s❡♥s♦rs✳ ❆ ❞✐r❡❝t ❝♦♠♠✉♥✐❝❛t✐♦♥ ❜❡t✇❡❡♥ ❛ s♦✉r❝❡ S ✭❛♥②s❡♥s♦r✮ ❛♥❞ ❛ ❞❡st✐♥❛t✐♦♥ D ✭❛♥② s✐♥❦✮ ✐s ♥♦t ❛❧✇❛②s ♣♦ss✐❜❧❡✿ r❡❧❛②✐♥❣ ♥♦❞❡sRi ❛r❡ ✉s❡❞ t♦ ❝❛rr② t❤❡ ❞❛t❛ ❢♦❧❧♦✇✐♥❣ ❛ ❤♦♣✲❜②✲❤♦♣ ♠♦❞❡❧✳ ❚❤❡ ❝❛s❝❛❞❡ ♦❢ n
❝❤❛♥♥❡❧s ❬✶✸❪ ❢♦r♠s t❤❡ ♠❛✐♥ ♣❛t❤ ❛♥❞ t❤❡ r❡✈❡rs❡ ♣❛t❤ ❢♦r♠s t❤❡ ❢❡❡❞❜❛❝❦ ♣❛t❤✭❋✐❣✳ ✶✮✳ ❙✉❝❤ ❛ ❝❛s❝❛❞❡ ✐s ❡st❛❜❧✐s❤❡❞ ❜② ❛ r♦✉t✐♥❣ ❛❧❣♦r✐t❤♠ ✇❤♦s❡ st✉❞② ✐s♦✉t ♦❢ t❤❡ s❝♦♣❡ ♦❢ t❤✐s ♣❛♣❡r✳
Rn−1
SDR1
R2
R3
❋✐❣✉r❡ ✶✿ ❆ ♠❛✐♥ ❛♥❞ ❢❡❡❞❜❛❝❦ ♣❛t❤ ✇✐t❤ n ❤♦♣s✳
■♥ ❛ ❲❙◆✱ t❤❡ ❝❛♣❛❝✐t② ♦❢ ❡❛❝❤ ❝❤❛♥♥❡❧ ✐♥ ❛ ♣❛t❤ ✐s ✉♥❦♥♦✇♥ ♣r✐♦r r♦❧❧✲♦✉t❛♥❞ ❡✈❡♥ ❞✉r✐♥❣ t❤❡ ❧✐❢❡t✐♠❡ ♦❢ t❤❡ ♥❡t✇♦r❦✿ t❤❡ ♥♦❞❡s ❛r❡ ✉s✉❛❧❧② s❝❛tt❡r❡❞❛❝r♦ss t❤❡ ♠♦♥✐t♦r❡❞ ❛r❡❛ ✐♥ ❛ r❛♥❞♦♠ ✇❛② ❛♥❞ t❤❡ ❡♥✈✐r♦♥♠❡♥t ♠❛② ✈❛r②✳❋♦✉♥t❛✐♥ ❝♦❞❡s ❛r❡ ❛ ♣r♦♠✐s✐♥❣ s♦❧✉t✐♦♥ t♦ ♣r♦✈✐❞❡ r♦❜✉st♥❡ss ❤❡r❡ ❬✺✱ ✶✹✱ ✶✱ ✻❪✳■♥❞❡❡❞✱ t❤❡② ❛r❡ r❛t❡❧❡ss✱ ✐✳❡✳ ❛ s♦✉r❝❡ S ❝❛♥ ♣♦t❡♥t✐❛❧❧② ❣❡♥❡r❛t❡ ❛ ❧✐♠✐t❧❡ss♥✉♠❜❡r ♦❢ ❡♥❝♦❞❡❞ ♣❛❝❦❡ts ✉♥t✐❧ ✐t r❡❝❡✐✈❡s ❛♥ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t ❢r♦♠ D✳ ❚❤❡②❝❛♥ ❛❞❛♣t t♦ t❤❡ ❝❤❛♥♥❡❧ ♦♥ t❤❡ ✢②✳ ❆♥♦t❤❡r ❛❞✈❛♥t❛❣❡ ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡s ♦✈❡rs❝❤❡♠❡s s✉❝❤ ❛s ❛✉t♦♠❛t✐❝ r❡♣❡❛t r❡q✉❡st ✭❆❘◗✮ ✐s t❤❡ ❧✐♠✐t❡❞ ✉s❡ ♦❢ t❤❡ ❢❡❡❞✲❜❛❝❦ ♣❛t❤✳ D ♦♥❧② ❛❝❦♥♦✇❧❡❞❣❡s ❛ s✉❝❝❡ss❢✉❧ ❞❡❝♦❞✐♥❣ t♦ S✳ ❚❤✐s ✐s ✇❤② ❢♦✉♥t❛✐♥❝♦❞❡s ❤❛✈❡ ❢♦✉♥❞ ✐♠♣♦rt❛♥t ❛♣♣❧✐❝❛t✐♦♥s ✐♥ s❛t❡❧❧✐t❡ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❬✹❪✱ ❝♦♥t❡♥t❞✐str✐❜✉t✐♦♥ ❬✸❪ ♦r ✉♥❞❡r✇❛t❡r ♥❡t✇♦r❦✐♥❣ ❬✺❪✳
■♥ ♦r❞❡r t♦ t❛❦❡ t❤❡ ❢✉❧❧ ❛❞✈❛♥t❛❣❡ ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡s ✐♥ ❛ ❲❙◆✱ ✐t ✐s ♠❛♥❞❛✲t♦r② t♦ ❛❝❝♦✉♥t ❢♦r t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ❞❛t❛ s❡♥t ❜② S ❛♥❞ ♦❢ t❤❡ ❢❡❡❞❜❛❝❦❝❤❛♥♥❡❧✳
❚❤❡ ❞❛t❛ tr❛♥s♠✐tt❡❞ ❜② ❛ s♦✉r❝❡ S ✐♥ ❛ ❲❙◆ ❞❡♣❡♥❞s ♦♥ t❤❡ t❛r❣❡t ❛♣✲♣❧✐❝❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ♠♦st ♦❢ t❤❡ t✐♠❡✱ s❡♥s♦rs ❣❛t❤❡r s♠❛❧❧ ❛♠♦✉♥ts ♦❢ ❞❛t❛✭❡✳❣✳ t❡♠♣❡r❛t✉r❡✱ ❧✐❣❤t ❧❡✈❡❧ ♦r ❛♥② ♣❤②s✐❝❛❧ q✉❛♥t✐t②✳ ✳ ✳ ✮ ❛♥❞ s❡♥❞ t❤❡♠ ❛t ❛r❡❣✉❧❛r ♦r ✐rr❡❣✉❧❛r ♣❛❝❡ t♦ t❤❡ s✐♥❦ D✳ ❍❡♥❝❡✱ ✐t ✐s r❡❛s♦♥❛❜❧❡ t♦ ❝♦♥s✐❞❡r t❤❛t♦♥❧② ❢❡✇ ♣❛❝❦❡ts ❤❛✈❡ t♦ ❜❡ s❡♥t ❛t t❤❡ s❛♠❡ t✐♠❡✱ tr❛✣❝ ✐s s♣♦r❛❞✐❝ ❛♥❞ ❞❛t❛♣❛❝❦❡ts ❛r❡ s❤♦rt✳ ❚❤✐s r❡♣r❡s❡♥ts ❛ ♠❛❥♦r ♠♦❞✐✜❝❛t✐♦♥ ❢♦r t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢❢♦✉♥t❛✐♥ ❝♦❞❡s✳ ■♥ ♣r❡✈✐♦✉s ❛♣♣❧✐❝❛t✐♦♥s ❬✹✱ ✸❪✱ ❧❛r❣❡ ❞❛t❛ ❛r❡ ♦❢t❡♥ ❛ss✉♠❡❞✳❚❤❡ ♠❛✐♥ ✐♠♣❛❝t ♦❢ t❤✐s ❝❤❛♥❣❡ ✐s t❤❡ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞❡❝♦❞✐♥❣ ♦✈❡r❤❡❛❞ ǫ
♦❢ ♠♦st ❢♦✉♥t❛✐♥ ❝♦❞❡s ✭❘❛♣t♦r✱ ▲❚✳ ✳ ✳ ✮✳ ❘❛♥❞♦♠ ❧✐♥❡❛r ❝♦❞❡s ❛r❡ ❧❡ss ❛✛❡❝t❡❞❜② t❤✐s ♠♦❞✐✜❝❛t✐♦♥ ❛♥❞ t❤✐s ✐s ✇❤② t❤❡② ❛r❡ ✉s❡❞ ✐♥ t❤✐s ♣❛♣❡r ❞❡s♣✐t❡ t❤❡✐r❤✐❣❤ ❞❡❝♦❞✐♥❣ ❝♦♠♣❧❡①✐t②✳
❚❤❡ ❢❡❡❞❜❛❝❦ ♣❛t❤ ❢r♦♠ D t♦ S ❞♦❡s ♥♦t ❛❧❧♦✇ t♦ tr❛♥s♠✐t ✐♥st❛♥t❛♥❡♦✉s❧②t❤❡ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t t♦ S ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥✳ ❲❤✐❧❡ t❤❡ ❛❝❦♥♦✇❧✲❡❞❣❡♠❡♥t ✐s ❡♥ r♦✉t❡ t♦ S✱ t❤❡ ❢♦✉♥t❛✐♥ st✐❧❧ ✢♦✇s✱ ✐✳❡✳ S ❛♥❞ t❤❡ n− 1 r❡❧❛②✐♥❣♥♦❞❡s Ri ❝❛♥ st✐❧❧ tr❛♥s♠✐t ❞❛t❛✳ ❚❤❡s❡ ❛❞❞✐t✐♦♥❛❧ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❛r❡ ✇❛st✐♥❣t❤❡ ♥♦❞❡s r❡s♦✉r❝❡s✳ ❲❡ ❝❛❧❧ t❤❡♠ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ❜② ❛♥❛❧♦❣② t♦ t❤❡ ❛♠♦✉♥t
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✹
♦❢ ✇❛t❡r ✇❛st❡❞ ❜② ❛ ❤♦s❡♣✐♣❡ ❜❡❢♦r❡ s♦♠❡❜♦❞② ✐♥ t❤❡ ❣❛r❞❡♥ ✐s ❛❜❧❡ t♦ t✉r♥✲♦✛t❤❡ t❛♣✳
❚❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ❞❡♣❡♥❞s ♦♥ t❤❡ s✐③❡ ♦❢ t❤❡ ❞❛t❛ t♦ ❜❡tr❛♥s♠✐tt❡❞ k ❛♥❞ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❤♦♣s n✳ ■♥❞❡❡❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛❝❦❡tsr❡q✉✐r❡❞ t♦ r❡❛❝❤ ❛ s✉❝❝❡ss❢✉❧ ❞❡❝♦❞✐♥❣ ❛t D ✐s ❧♦✇❡r ❜♦✉♥❞❡❞ ❜② n(k + ǫ)✳ ❆s✐♠♣❧❡ ♠♦❞❡❧ ❜❛s❡❞ ♦♥ t❤❡ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t ♣r♦❣r❡ss✐♦♥ ❣✐✈❡s ✉s ❛ ✢♦♦❞✐♥❣tr❛✣❝ ✐♥ O(n2) ❢♦r ❝❡rt❛✐♥ r❡❧❛②✐♥❣ str❛t❡❣✐❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝❤♦✐❝❡s ♦❢ n ❛♥❞ k
❛r❡ ❝r✐t✐❝❛❧ t♦ ♣r❡✈❡♥t t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ t♦ ❜❡❝♦♠❡ ❞♦♠✐♥❛♥t✳ ❖✉r ❛♥❛❧②s✐s ✐ss✉♣♣♦rt❡❞ ❜② r❡❛❧✐st✐❝ s✐♠✉❧❛t✐♦♥s ✇❤❡r❡ ❜♦t❤ s♠❛❧❧ s❝❛❧❡ ❢❛❞✐♥❣ ❛♥❞ ❝♦❧❧✐s✐♦♥s❢♦r ❛♥ ✉♥s❧♦tt❡❞ ❩✐❣❇❡❡ ■❊❊❊ ✽✵✷✳✶✺✳✹ ♠❡❞✐✉♠ ❛❝❝❡ss ❧❛②❡r ❬✶✼❪ ❛r❡ ❝♦♥s✐❞❡r❡❞✉s✐♥❣ t❤❡ ❲❙◆❡t ♥❡t✇♦r❦ s✐♠✉❧❛t♦r ❬✼❪✳
❚❤❡ r❡st ♦❢ t❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ❡①✐st✐♥❣✇♦r❦s ♦♥ ❢♦✉♥t❛✐♥ ❝♦❞❡s ❛r❡ s✉♠♠❛r✐③❡❞✳ ■♥ ❙❡❝t✐♦♥ ✸✱ t❤❡ ✢♦♦❞✐♥❣ ♦✈❡r❤❡❛❞♣r♦❜❧❡♠ ❞✉❡ t♦ t❤❡ ❤♦♣✲❜②✲❤♦♣ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t tr❛♥s♠✐ss✐♦♥ ✐s ❛♥❛❧②③❡❞ ❛♥❞s❡❝t✐♦♥ ✹ ♣r❡s❡♥ts t❤❡ ✐♠♣❛❝t ♦❢ ❞✐✛❡r❡♥t r❡❧❛②✐♥❣ str❛t❡❣✐❡s ♦♥ t❤✐s ♦✈❡r❤❡❛❞✳❋✐♥❛❧❧②✱ ❙❡❝t✐♦♥ ✺ ❝♦♥❝❧✉❞❡s t❤❡ ♣❛♣❡r✳
✷ ❋♦✉♥t❛✐♥ ❝♦❞❡s
❚❤❡ ❝♦♥❝❡♣t ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡s ✇❛s ✜rst ♣r❡s❡♥t❡❞ ✐♥ ✶✾✾✽ ❜② ❇②❡rs ❛♥❞ ❛❧✳ ❬✹❪✳❙✐♥❝❡ t❤❡♥✱ ♠❛♥② ✇♦r❦s ❤❛✈❡ ❜❡❡♥ ❞♦♥❡ ♦♥ t❤✐s t♦♣✐❝s✳ ❚❤❡ ❞❡s✐❣♥ ♦❢ ❛ ❢♦✉♥t❛✐♥❝♦❞❡ ❢♦r ❛ ❧✐♥❡ ♥❡t✇♦r❦ ✐s ❝♦♠♣♦s❡❞ ♦❢ ❢♦✉r t❛s❦s✿ ✭❛✮ t❤❡ ❡♥❝♦❞✐♥❣ ❛❧❣♦r✐t❤♠❞❡s✐❣♥✱ ✭❜✮ t❤❡ ❞❡❝♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ❞❡s✐❣♥ ✐♥❝❧✉❞✐♥❣ t❤❡ ❞❡❝♦❞✐♥❣ ♦✈❡r❤❡❛❞ ǫ✱ ✭❝✮t❤❡ r❡❧❛②✐♥❣ str❛t❡❣② ❝❤♦✐❝❡ ❛♥❞ ✭❞✮ t❤❡ ✢♦♦❞✐♥❣ ♦✈❡r❤❡❛❞ ❛♥❛❧②s✐s✳ ❚❤❡ ✜rstt❤r❡❡ t❛s❦s ✭❛ t♦ ❝✮ ❛r❡ ❡♥t✇✐♥❡❞ ❛♥❞ ❤❛✈❡ ❜❡❡♥ ❛❞❞r❡ss❡❞ ✐♥ ♠❛♥② ✇♦r❦s ❬✸✱✶✷✱ ✶✻✱ ✷✱ ✾❪✳ ❚❤❡ ❞❡s✐❣♥ ♦❢ ❡✣❝✐❡♥t r❡❧❛②✐♥❣ str❛t❡❣✐❡s ❢♦r ❢♦✉♥t❛✐♥ ❝♦❞❡s ❤❛✈❡r❡❝❡♥t❧② ❞❡s❡r✈❡❞ t❤❡ ❛tt❡♥t✐♦♥ ♦❢ t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦♠♠✉♥✐t② ❞✉❡ t♦ t❤❡❞❡✈❡❧♦♣♠❡♥t ♦❢ ♥❡t✇♦r❦ ❝♦❞✐♥❣ t❡❝❤♥✐q✉❡s ❬✶✺✱ ✶✱ ✻❪✳ ❚♦ t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡❛✉t❤♦rs✱ t❤✐s ♣❛♣❡r ✐s t❤❡ ✜rst ♦♥❡ ❛❞❞r❡ss✐♥❣ t❤❡ ✢♦♦❞✐♥❣ ✐ss✉❡✳ ■♥ ✇❤❛t ❢♦❧❧♦✇s✱✇❡ ❜r✐❡✢② s✉♠♠❛r✐③❡ t❛s❦s ✭❛✮ t♦ ✭❝✮✳
✷✳✶ ❋♦✉♥t❛✐♥ ❡♥❝♦❞✐♥❣
●✐✈❡♥ ❛ s❡t ♦❢ k ✐♥♣✉t s②♠❜♦❧s✱ t❤❡ s♦✉r❝❡ ❣❡♥❡r❛t❡s ❛♥ ✐♥✜♥✐t❡ str❡❛♠ ♦❢ ♦✉t♣✉ts②♠❜♦❧s ✇❤✐❝❤ r❡s✉❧ts ❢r♦♠ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛ s✉❜s❡t ♦❢ ✐♥♣✉t s②♠❜♦❧s✳❋♦r ❡❛❝❤ ❡♥❝♦❞❡❞ s②♠❜♦❧✱ ❛ ❞❡❣r❡❡ d ∈ [1, k] ✐s ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ ❛ ❞❡❣r❡❡❞✐str✐❜✉t✐♦♥✳ ❚❤❡♥✱ d r❛♥❞♦♠❧② ❝❤♦s❡♥ s②♠❜♦❧s ❛r❡ ❧✐♥❡❛r❧② ❝♦♠❜✐♥❡❞ ♦✈❡r F2
t♦ ❢♦r♠ t❤❡ ❡♥❝♦❞❡❞ s②♠❜♦❧✳ ❚❤✐s ❝♦❞✐♥❣ ✐s r❡♣❡❛t❡❞ ✉♥t✐❧ ❛♥ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t✐s r❡❝❡✐✈❡❞✳
❚❤❡ ❞❡❣r❡❡ ❞✐str✐❜✉t✐♦♥ ✐s t❤❡ ❝❡♥tr❛❧ ♣❛r❛♠❡t❡r ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ ❢♦✉♥t❛✐♥❝♦❞❡s t❤❛t ✐s ❧✐♥❦❡❞ t♦ t❤❡ ❞❡❝♦❞✐♥❣ ❛❧❣♦r✐t❤♠✳ ❚❤❡ ♠♦st s✐♠♣❧❡ ❞✐str✐❜✉t✐♦♥ ✐st❤❡ ✉♥✐❢♦r♠ ♦♥❡ ♦❢ r❛♥❞♦♠ ❧✐♥❡❛r ❝♦❞❡s✳ ❆♥♦t❤❡r ♣♦ss✐❜✐❧✐t② ✐s ❘♦❜✉st ❙♦❧✐t♦♥❞✐str✐❜✉t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ ▲❚✲❝♦❞❡ ✐♥tr♦❞✉❝❡❞ ❜② ▲✉❜② ❬✶✷❪✳ ❆ ♠♦❞✐✜❝❛t✐♦♥ ♦❢t❤❡ ❡♥❝♦❞✐♥❣ s❝❤❡♠❡ ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❜② ❙❤♦❦r♦❧❧❛❤✐ ✇✐t❤ ❘❛♣t♦r ❝♦❞❡ ❬✶✻❪✳❆ ♣r❡❝♦❞❡ ✐s ❛♣♣❧✐❡❞ ♦♥ t❤❡ k ✐♥♣✉t s②♠❜♦❧s ♣r✐♦r t♦ t❤❡ ❢♦✉♥t❛✐♥ ❡♥❝♦❞✐♥❣✳❚♦ ❝♦♥❝❧✉❞❡ ♦♥ t❤❡ ❞❡❣r❡❡ ❞✐str✐❜✉t✐♦♥✱ t❤❡ r❡s✉❧ts ❬✾✱ ✶✽❪ ❛r❡ ✇♦rt❤ ♠❡♥t✐♦♥✲✐♥❣✳ ❚❤❡② st✉❞② t❤❡ ♦♣t✐♠❛❧ ❞❡❣r❡❡ ❞✐str✐❜✉t✐♦♥ t❤❛t ♠✉st ❜❡ ❛ss♦❝✐❛t❡❞ t♦ ❛●❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥ ❞❡❝♦❞✐♥❣ ❢♦r s♠❛❧❧ ✈❛❧✉❡ ♦❢ k✳
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✺
✷✳✷ ❋♦✉♥t❛✐♥ ❞❡❝♦❞✐♥❣
❆ r❡❧✐❛❜❧❡ ❞❡❝♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r ❋♦✉♥t❛✐♥ ❝♦❞❡s s❤♦✉❧❞ ❛❧❧♦✇ t❤❡ r❡❝♦✈❡r✐♥❣♦❢ k ✐♥♣✉t s②♠❜♦❧s ❢r♦♠ ❛♥② s✉❜s❡t ♦❢ k + ǫ ♦✉t♣✉t s②♠❜♦❧s ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t②t❤❛t ✐s ❛t ♠♦st ✐♥✈❡rs❡❧② ♣♦❧②♥♦♠✐❛❧ ✐♥ k ❬✶✻❪✳
❚❤❡ str❛✐❣❤t❢♦r✇❛r❞ ❞❡❝♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ✐♥ s♦❧✈✐♥❣ t❤❡ ❛ s②st❡♠♦❢ k + ǫ ❡q✉❛t✐♦♥s ✉s✐♥❣ ❛ ●❛✉ss✐❛♥ ❡❧✐♠✐♥❛t✐♦♥✳ ❚❤✐s ❛♣♣r♦❛❝❤ ❤❛s ❛ ❧♦✇❞❡❝♦❞✐♥❣ ♦✈❡r❤❡❛❞✱ ✐✳❡✳ ǫ ✐s O( log(k)
k) ❜✉t ❛ ❤✐❣❤ ❞❡❝♦❞✐♥❣ ❝♦♠♣❧❡①✐t② ♣❛♣❡r
✬r❛♣t♦r ❝♦❞❡✬ ♦❢ ❙❤♦❦r♦❧❧❛❤✐ ♣❛❣❡ ✾✳ O((k+O( log(k)k
))k2) ▲❛t❡r✱ ▲✉❜② ❛♣♣❧✐❡❞ t♦✐ts ▲❚✲❝♦❞❡s ❬✶✷❪ ❇❡❧✐❡❢ Pr♦♣❛❣❛t✐♦♥ ✭❇P✮ ❞❡❝♦❞✐♥❣✳ ■t ♦✛❡rs ❛ ❜❡tt❡r ❞❡❝♦❞✐♥❣❝♦♠♣❧❡①✐t② ❛t t❤❡ ❝♦st ♦❢ ❞❡❝♦❞✐♥❣ ♦✈❡r❤❡❛❞ ✭s❡❡ ❚❛❜❧❡ ✶✮✳ ❘❛♣t♦r ❝♦❞❡ ✐s ❛♥♦♣t✐♠✐③❡❞ ✈❡rs✐♦♥ ♦❢ ▲❚ ❝♦❞❡ t❤❛t ❝♦♠❜✐♥❡s ❛♥ ♦✉t❡r ❝♦❞❡ t♦ ❛ r❡❣✉❧❛r ▲❚ ❝♦❞❡✳■ts ❞❡❝♦❞✐♥❣ ♦✈❡r❤❡❛❞ ✐s s♠❛❧❧❡r t❤❛♥ ❢♦r ▲❚ ✭ǫ = 1 ♦r ✷✮ ✇❤✐❧❡ ✐ts ❞❡❝♦❞✐♥❣❝♦♠♣❧❡①✐t② ✐s ❝❧♦s❡ t♦ t❤❡ ♦♥❡ ♦❢ ❛♥ ▲❚✲❝♦❞❡✶✳
❘❛♥❞♦♠ ▲✐♥❡❛r ▲❚ ❘❛♣t♦r
❉❡❝♦❞✐♥❣ ❖✈❡r❤❡❛❞ log(k)k
log2(k)√
ks❡❡ 1
❊♥❝♦❞✐♥❣ ❈♦♠♣❧❡①✐t② k2 k log(k) s❡❡ 1
❉❡❝♦❞✐♥❣ ❈♦♠♣❧❡①✐t② (k + O( log(k)k
))k2 k log(k) k
❚❛❜❧❡ ✶✿ ❈❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ♠❛✐♥ ❝❧❛ss❡s ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡s✳✳
❆t ■❙■❚ ✷✵✵✾✱ ▲✉ ❛♥❞ ❛❧✳ ❬✶✶❪ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ❜❧❛❝❦✲❜♦① ❧✐♥❡❛r ❛❧❣❡❜r❛ ❛♥❞♠♦r❡ s♣❡❝✐✜❝❛❧❧② t❤❡ ✉s❡ ♦❢ t❤❡ ❲✐❡❞❡♠❛♥♥ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❞❡❝♦❞✐♥❣ ♦❢ ❢♦✉♥✲t❛✐♥ ❝♦❞❡s✳ ❚❤✐s r❡s✉❧t ♦✛❡rs ❣r❡❛t ♣❡rs♣❡❝t✐✈❡s ❢♦r ❜♦t❤ ❡✣❝✐❡♥t ❞❡❝♦❞✐♥❣ ❛♥❞❛ ❧♦✇ ǫ ✇❤❡♥ k ✐s s♠❛❧❧✳ ❍♦✇❡✈❡r✱ t❤✐s r❡s❡❛r❝❤ ❛r❡❛ ✐s st✐❧❧ ♦♣❡♥ ❛♥❞ ♠❛♥②✇♦r❦s ❛r❡ st✐❧❧ t♦ ❜❡ ❞♦♥❡ ✐♥ ♦r❞❡r t♦ ✜♥❞ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s✳
❆♠♦♥❣st t❤❡ ❡st❛❜❧✐s❤❡❞ ❢♦✉♥t❛✐♥ ❝♦❞❡s✱ ✇❡ ❤❛✈❡ ❝❤♦s❡♥ ❘▲ ❝♦❞❡s t♦❣❡t❤❡r✇✐t❤ ●❛✉ss✐❛♥ ❞❡❝♦❞✐♥❣ s✐♥❝❡ ✐t ♣r♦✈✐❞❡s ❧♦✇ ❞❡❝♦❞✐♥❣ ♦✈❡r❤❡❛❞✳ ❚❤❡ ❝♦st ♦❢❞❡❝♦❞✐♥❣ ❛ r❛♥❞♦♠ ❧✐♥❡❛r ❝♦❞❡ ✐s ♠✐t✐❣❛t❡❞ ❜② t❤❡ ❢❛❝t t❤❛t ✇❡ ❛r❡ t❛r❣❡t✐♥❣ ❛♥❛♣♣❧✐❝❛t✐♦♥ ✇❤❡r❡ k ✐s s♠❛❧❧✳
✷✳✸ ❘❡❧❛②✐♥❣ str❛t❡❣✐❡s
❚❤❡ ♣r♦❜❧❡♠ ♦❢ r❡❧❛②✐♥❣ ❛ ❢♦✉♥t❛✐♥ ❝♦❞❡ ♦✈❡r ❛ ❧✐♥❡ ♥❡t✇♦r❦ ✇❛s ✜rst st✉❞✐❡❞❜② P❛❦③❛❞ ❛♥❞ ❛❧✳ ❬✶✺❪✳ ❚❤❡ ♠♦st s✐♠♣❧❡ r❡❧❛②✐♥❣ str❛t❡❣② ❝♦♥s✐sts t♦ ♣❡r❢♦r♠♥♦ ♣r♦❝❡ss✐♥❣✳ ❚❤r♦✉❣❤♦✉t t❤❡ ♣❛♣❡r✱ t❤✐s str❛t❡❣② ✐s r❡❢❡rr❡❞ t♦ ❛s ♣❛ss✐✈❡r❡❧❛②✐♥❣✳ ■t ✐s ✇❡❧❧✲❦♥♦✇♥ ❬✶✾✱ ✶✸❪ t❤❛t ✐❢ s♦♠❡ ♣r♦❝❡ss✐♥❣ ✐s ❛❧❧♦✇❡❞ ❛t r❡❧❛②✐♥❣♥♦❞❡s✱ t❤❡ ♠✐♥✲❝✉t ❝❛♣❛❝✐t② ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞✳ ❚❤❡ ❛♠♦✉♥t ♦❢ ♣r♦❝❡ss✐♥❣ ♠❛②✈❛r② t♦ r❡♥❞❡r t❤❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ s❝❤❡♠❡ ❛ttr❛❝t✐✈❡✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ str❛t❡❣✐❡s❛r❡ ❝♦♥s✐❞❡r❡❞✳
■♥ ❛ ❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞ r❡❧❛②✐♥❣✱ ❡❛❝❤ ✐♥t❡r♠❡❞✐❛t❡ ♥♦❞❡ ❢✉❧❧② ❞❡❝♦❞❡s ❛♥❞t❤❡♥ r❡✲❡♥❝♦❞❡s t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❜❡❢♦r❡ ❢♦r✇❛r❞✐♥❣ ♣❛❝❦❡ts t♦ ✐ts ♥❡✐❣❤❜♦r✳ ■♥t❤✐s ❝❛s❡✱ ❡❛❝❤ tr❛♥s♠✐tt❡r ✭s♦✉r❝❡ ♦r r❡❧❛②✮ ❝♦♠♣❡♥s❛t❡s ❢♦r t❤❡ ❧♦ss❡s ♦♥ t❤❡tr❛♥s♠✐ss✐♦♥ ❧✐♥❦ t♦ ✐ts ✐♠♠❡❞✐❛t❡ ♥❡✐❣❤❜♦r✳
❚❤❡ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ r❡❧❛②✐♥❣ ✐s ❛♥ ❤②❜r✐❞ str❛t❡❣② ❜❡t✇❡❡♥ ♣❛ss✐✈❡ ❛♥❞❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞✳ ❆ ♥♦❞❡ Ri r❡❧❛②s ♣❛ss✐✈❡❧② t❤❡ ♠❡ss❛❣❡s ❛♥❞ tr✐❡s t♦ ❞❡✲
✶❲❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬✶✻❪ t♦ ❣❡t ❢✉rt❤❡r ❞❡t❛✐❧s ♦♥ ❘❛♣t♦r ❝♦❞❡ ❝♦♠♣❧❡①✐t② ❛s ✐t ✐s r❡❧❛t❡❞t♦ t❤❡ ♣r❡❝♦❞❡✳
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✻
❝♦❞❡ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❲❤❡♥ ✐t s✉❝❝❡ss❢✉❧❧② ❞❡❝♦❞❡s t❤❡ ✐♥❢♦r♠❛t✐♦♥✱ ✐t s❡♥❞s❛♥ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t t♦ Ri−1✳ ◆♦✇✱ Ri ❜❡❝♦♠❡s ❛ ♥❡✇ s♦✉r❝❡ ❛♥❞ ❡♥❝♦❞❡s✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦❞❡s✳
❚❤❡ ❣r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣✱ ❛s ❞❡✜♥❡❞ ✐♥ ❬✶✺❪✱ ✐s ❝♦♥s✐❞❡r❡❞ ❛s ❛ ❢♦r♠ ♦❢♥❡t✇♦r❦ ❝♦❞✐♥❣ ❬✽❪ ✇❤❡r❡ r❡❧❛②s Ri ❢♦r✇❛r❞ r❛♥❞♦♠ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦✈❡rF2 ♦❢ t❤❡ r❡❝❡✐✈❡❞ ❞❛t❛✳
✸ ❆❈❑ ❛♥❞ ✢♦♦❞✐♥❣ ♦✈❡r❤❡❛❞
❆♥ ✐♠♣♦rt❛♥t ♦❜st❛❝❧❡ ❢♦r t❤❡ r♦❧❧✲♦✉t ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡s ✐♥ ❛ ❲❙◆ ✐s t❤❡ ❝♦st♦❢ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t✳ ❚❤❡ ❛✐♠ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ ♣r♦✈✐❞❡ ❛♥ ❛♥❛❧②s✐s ♦❢ t❤❡✢♦♦❞✐♥❣ tr❛✣❝ ✐♥ ❛♥ ✐❞❡❛❧✐③❡❞ ❝♦♠♠✉♥✐❝❛t✐♦♥ s❝❤❡♠❡✳
❚❤❡ ❝♦st ♦❢ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t ✐s r❡❧❛t❡❞ t♦ t✇♦ t②♣❡s ♦❢ ♣❛❝❦❡ts t❤❛t ❛r❡tr❛♥s♠✐tt❡❞ ❛❢t❡r ❛ s✉❝❝❡ss❢✉❧ ❞❡❝♦❞✐♥❣ ❜② Ri✳ ❆❝❦♥♦✇❧❡❞❣♠❡♥t ✭❆❈❑✮ ♣❛❝❦❡ts❛r❡ ❛♥ ✐♥❝♦♠♣r❡ss✐❜❧❡ ❝♦st✿ t❤❡② ❛r❡ ♥❡❝❡ss❛r② t♦ ❡♥❞ t❤❡ ♦✈❡r❛❧❧ ❞❛t❛ tr❛♥s♠✐s✲s✐♦♥ ❛♥❞ ❢♦r S t♦ st❛rt tr❛♥s♠✐tt✐♥❣ ♦t❤❡r ❞❛t❛✳ ❙✐♥❝❡ t❤❡ s♦✉r❝❡ ❛♥❞ t❤❡ r❡❧❛②sRi ❛r❡ ❝♦♥st❛♥t❧② tr❛♥s♠✐tt✐♥❣ ❡♥❝♦❞❡❞ ♣❛❝❦❡ts ✉♥t✐❧ ❛♥ ❆❈❑ ♠❡ss❛❣❡ ✐s ❜❡✐♥❣r❡❝❡✐✈❡❞✱ t❤❡r❡ ✐s ❛♥♦t❤❡r ❝♦st r❡❢❡rr❡❞ t♦ ❛s t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝✳ ❚❤✐s ♦✈❡r❤❡❛❞tr❛✣❝ ✐s ♠❡❛s✉r❡❞ ❜② t❤❡ ♥✉♠❜❡r t❤❡ ♣❛❝❦❡ts s❡♥t ❜② S ❛♥❞ t❤❡ r❡❧❛②✐♥❣ ♥♦❞❡sRi ❛❢t❡r ❛ s✉❝❝❡ss❢✉❧ ❞❡❝♦❞✐♥❣ ♦❢ t❤❡ ✐♥♣✉t ♣❛❝❦❡ts ❜② D✳ ❚❤❡② ❛r❡ ❛ s✐❞❡✲❡✛❡❝t♦❢ ❛♥ ❤♦♣✲❜②✲❤♦♣ ♠♦❞❡ ♦❢ tr❛♥s♠✐ss✐♦♥ ♦❢ t❤❡ ❆❈❑✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ❞❡♥♦t❡ ✐♥ t❤❡❢♦❧❧♦✇✐♥❣ ❜② ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t tr❛✣❝ t❤❡ ♥✉♠❜❡r ♦❢ ❆❈❑ ♣❛❝❦❡ts s❡♥t ✐♥ t❤❡♥❡t✇♦r❦✳
❚♦ ❣✐✈❡ ❛♥ ✐♥t✉✐t✐✈❡ ✐❞❡❛ ♦❢ ✇❤❛t ✐s t❤❡ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t✴✢♦♦❞✐♥❣ tr❛✣❝✱✇❡ ✜rst ❝♦♥s✐❞❡r tr❛♥s♠✐ss✐♦♥ ❝❤❛♥♥❡❧s ❢♦r ✇❤✐❝❤ ♥♦ ❡rr♦rs ♦❝❝✉r✳ ▼♦r❡♦✈❡r✱t❤❡ ♠❡ss❛❣❡ s❝❤❡❞✉❧✐♥❣ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ✐❞❡❛❧✳ ❲❤❡♥ t❤❡ ♥♦❞❡s Ri−1 ❛♥❞Ri+1 s❡♥❞ s✐♠✉❧t❛♥❡♦✉s❧② ❞❛t❛ t♦ Ri✱ Ri r❡❝❡✐✈❡s ❛t ❧❡❛st ♦♥❡ ♠❡ss❛❣❡✳ ❆❣❛✐♥✱t❤❡s❡ ❛ss✉♠♣t✐♦♥s ❛r❡ ♥♦t ♠❛❞❡ t♦ ❡①♣r❡ss ❛ r❡❛❧✐st✐❝ tr❛♥s♠✐ss✐♦♥ s❝❤❡♠❡ ❜✉tt❤❡② ❛r❡ ✈❡r② ✉s❡❢✉❧ t♦ ❣✐✈❡ ❛ r♦✉❣❤ ♠♦❞❡❧ ♦❢ ✇❤❛t ❤❛♣♣❡♥s ❛❢t❡r ❛ s✉❝❝❡ss❢✉❧❞❡❝♦❞✐♥❣✳ ❚r❛♥s♠✐ss✐♦♥ ❞❡❧❛②s ❛❧♦♥❣ ❡❛❝❤ ❝❤❛♥♥❡❧ ♦❢ t❤❡ ♠❛✐♥ ❛♥❞ ❢❡❡❞❜❛❝❦♣❛t❤s ❛r❡ ❛ss✉♠❡❞ ❝♦♥st❛♥t ❛♥❞ ✐❞❡♥t✐❝❛❧ ❢♦r ❛❧❧ ♣❛❝❦❡ts✳
❚❤❡ ♣r♦❣r❡ss✐♦♥ ♦❢ t❤❡ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t ✐s h t✐♠❡s ❢❛st❡r t❤❛♥ t❤❡ ♣r♦❣r❡s✲s✐♦♥ ♦❢ t❤❡ ❞❛t❛✿ ✇❤❡♥ ❛ ❞❛t❛ ♣❛❝❦❡t ♣r♦❣r❡ss❡s ♦❢ ♦♥❡ ❤♦♣✱ t❤❡ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t♣r♦❣r❡ss❡s ♦❢ h ❤♦♣s✳ ❚❤✐s ❛ss✉♠♣t✐♦♥ ✐s r❡❛s♦♥❛❜❧❡ ❛♥❞ ✐s ♠♦t✐✈❛t❡❞ ❜② t❤❡❢❛❝t t❤❛t ❆❈❑ ♣❛❝❦❡ts ❛r❡ ✉s✉❛❧❧② s♠❛❧❧❡r t❤❛♥ r❡❣✉❧❛r ❞❛t❛ ♣❛❝❦❡ts✱ ❛♥❞ ❤❡♥❝❡s✉✛❡r ❢r♦♠ ❛ ❧♦✇❡r ♣❛❝❦❡t ❡rr♦r r❛t❡✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ tr❛♥s♠✐ss✐♦♥ ✐s ❣✐✈❡♥ ✐♥❋✐❣✳ ✷ ❢♦r n = 4 ❛♥❞ h = 1✳
❖✈❡r ❛ ❧✐♥❡ ♥❡t✇♦r❦✱ ✐t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ s❡❡ t❤❛t ❢♦r ❛♥② r❡❧❛②✐♥❣ str❛t❡❣②✱t❤❡ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t tr❛✣❝ ✐s ❛t ❧❡❛st ❧✐♥❡❛r ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❤♦♣s n s✐♥❝❡ S❛♥❞ ❡❛❝❤ r❡❧❛② Ri ❤❛s t♦ r❡❝❡✐✈❡ t❤❡ ❆❈❑✳ ❋♦r ♣❛ss✐✈❡ ❛♥❞ ❣r❡❡❞② r❛♥❞♦♠r❡❧❛②✐♥❣✱ t❤❡ ❆❈❑ ✐s ❞✐r❡❝t❧② ❢♦r✇❛r❞❡❞ ❢r♦♠ D t♦ S✳ ❋♦r ❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞❛♥❞ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ r❡❧❛②✐♥❣✱ t❤❡ s♦✉r❝❡ S ✐s ✈✐rt✉❛❧❧② ♠♦✈✐♥❣ t♦✇❛r❞s D✿S r❡❝❡✐✈❡s t❤❡ ❆❈❑ ❢r♦♠ R1✱ R1 ❢r♦♠ R2 ❛♥❞ s♦ ♦♥✳
❚❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ✐s ✐♥ O(n) ❢♦r ❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞✳ ❚❤❡ ♥♦❞❡ Ri ❛❝✲❦♥♦✇❧❡❞❣❡s Ri−1 ❛♥❞ ❝❛♥ ♦♥❧② r❡❝❡✐✈❡ s♦♠❡ ✢♦♦❞✐♥❣ tr❛✣❝ ❢r♦♠ Ri−1✳ ❚❤❡❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ ❢♦❧❧♦✇s t❤❡ s❛♠❡ ♠♦❞❡❧✿ ✐❢ D ❞❡❝♦❞❡s s✉❝❝❡ss❢✉❧❧②✱ t❤❡♥∀i ∈ [1, n − 1],Ri ❤❛s ❞❡❝♦❞❡❞ s✉❝❝❡ss❢✉❧❧②✳
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✼
S R1 R2 R3 D
❋✐❣✉r❡ ✷✿ ▼❡ss❛❣❡ t✐♠❡❧✐♥❡✳ ❚❤❡ ❜❧❛❝❦ ♣❧❛✐♥ ❛rr♦✇s r❡♣r❡s❡♥t t❤❡ tr❛♥s♠✐ss✐♦♥t❤❛t ❝❛rr✐❡s ✉s❡❢✉❧ ✐♥❢♦r♠❛t✐♦♥ t♦ D✳ ❆❢t❡r ❛♥❞ ❞✉r✐♥❣ ❛ s✉❝❝❡ss❢✉❧ ❞❡❝♦❞✐♥❣ ❛❧❧t❤❡ tr❛♥s♠✐ss✐♦♥s ❛r❡ ✉s❡❧❡ss t♦ D ✭❞♦tt❡❞ ❛rr♦✇s✮✳ ❆❢t❡r ❛ s✉❝❝❡ss❢✉❧ ❞❡❝♦❞✐♥❣✱D ❛❝❦♥♦✇❧❡❞❣❡s t❤❡ ✐♥❢♦r♠❛t✐♦♥ r❡❝❡♣t✐♦♥✳ ❚❤❡ ❡♠✐ss✐♦♥ ♦❢ t❤❡ ❛❝❦♥♦✇❧❡❞❣❡✲♠❡♥t ❝❛♥ ❝♦❧❧✐❞❡ ✇✐t❤ s♦♠❡ ❡♠✐ss✐♦♥s ❛t t❤❡ ♥♦❞❡ ❧❡✈❡❧✳ ■t ✐s ❛ss✉♠❡❞ ❤❡r❡ ❢♦rs✐♠♣❧✐❝✐t② t❤❛t ❝♦❧❧✐s✐♦♥ ❛r❡ ❤❛♥❞❧❡❞ ✐♥ ❢❛✈♦r t♦ t❤❡ ❆❈❑✳
❋♦r ♣❛ss✐✈❡ ❛♥❞ ❣r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣✱ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ❝❛♥ ❜❡ ❛♣♣r♦①✲✐♠❛t❡❞ ❜② ✭✐♥ ♥✉♠❜❡r ♦❢ ♣❛❝❦❡ts✮✿
n⌊n
h⌋ − h · ⌊
n
h⌋(⌊n
h⌋ + 1)
2. ✭✶✮
■❢ t❤❡ ❆❈❑ ❤❛s ❜❡❡♥ r❡❝❡✐✈❡❞ ❜② r ♥♦❞❡s✱ t❤❡r❡ ❛r❡ st✐❧❧ (n − rh) ♣♦t❡♥t✐❛❧♥♦❞❡s r❡❧❛②✐♥❣ ✐♥❢♦r♠❛t✐♦♥✳ ❊q✉❛t✐♦♥ ✶ ✐s ♦❜t❛✐♥❡❞ ❜② t❤❡ s✉♠♠❛t✐♦♥ ♦❢ ❛❧❧ t❤❡♥♦❞❡s t❤❛t ❝❛♥ ♣♦t❡♥t✐❛❧❧② ❡♠✐t ✐♥❢♦r♠❛t✐♦♥ ❜❡❢♦r❡ t❤❡ ❆❈❑ r❡❛❝❤❡s S✳
❚❤❡ ♠♦❞❡❧ ❝♦♥s✐❞❡r❡❞ ❤❡r❡ ✐s ❢❛r ❢r♦♠ ❜❡✐♥❣ r❡❛❧✐st✐❝ ❜✉t ✐t ❣✐✈❡s ✉s ❛ ❧♦✇❡r❜♦✉♥❞ ♦♥ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝✳ ◆❡①t ❙❡❝t✐♦♥ ✹ ❝❤❛❧❧❡♥❣❡s t❤❡s❡ ❜♦✉♥❞s ❢♦r ❛✽✵✷✳✶✺✳✹ ❲❙◆✳
✹ ❙✐♠✉❧❛t✐♦♥
✹✳✶ ❙✐♠✉❧❛t✐♦♥ s❡t✉♣
❖✉r s✐♠✉❧❛t✐♦♥s ❛r❡ ❞♦♥❡ ♦♥ ❲❙◆❡t ❬✼❪✱ ❛♥ ❡✈❡♥t✲❞r✐✈❡♥ ♥❡t✇♦r❦ s✐♠✉❧❛t♦r✳❚❤❡ ▼❆❈ ♣r♦t♦❝♦❧ ✉s❡❞ ❜② t❤❡ s❡♥s♦rs ❢♦❧❧♦✇s t❤❡ ■❊❊❊ ✽✵✷✳✶✺✳✹ st❛♥❞❛r❞ ❬✶✼❪❛♥❞ t❤❡ ♣❤②s✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❈❈✶✶✵✵ ❈❤✐♣s❡ts ❢r♦♠ ❚❡①❛s■♥str✉♠❡♥ts ❬✶✵❪✳ ❚❤❡ ❛❝❝❡ss ♠❡t❤♦❞ ♦❢ ■❊❊❊ ✽✵✷✳✶✺✳✹ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ♣❛♣❡r✐s ✉♥s❧♦tt❡❞ ❈❙▼❆✴❈❆✳ ❆♥ ❆❈❑ ♣❛❝❦❡t ✐s ❛ r❡❣✉❧❛r ❞❛t❛ ♣❛❝❦❡t ✇✐t❤ ❛ s✐♥❣❧❡❜✐t ♣❛②❧♦❛❞✳ ❙♦ ❢❛r✱ ♠♦st ♦❢ t❤❡ st✉❞✐❡s ♦♥ ❢♦✉♥t❛✐♥ ❝♦❞❡s ❤❛✈❡ ❛ss✉♠❡❞ ♣❡r✲❢❡❝t ❢❡❡❞❜❛❝❦ ♠❡❝❤❛♥✐s♠s✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❝♦♥s✐❞❡r ❛ r❡❛❧✐st✐❝ tr❛♥s♠✐ss✐♦♥s❝❤❡♠❡ ✇❤❡r❡ ❛❝❦♥♦✇❧❡❞❣♠❡♥ts ❝❛♥ ❛❧s♦ s✉✛❡r ❢r♦♠ ❧♦ss ❜❛s❡❞ ♦♥ ✈❛r✐♦✉s ❝❤❛♥✲♥❡❧ st❛t✐st✐❝s✳ ❍♦✇❡✈❡r✱ ❡✈❡♥ ✐♥ ❛ r❡❛❧✐st✐❝ ❝♦♥✜❣✉r❛t✐♦♥✱ ❆❈❑ ♣❛❝❦❡ts ❤❛✈❡ ❛❧♦✇❡r P❊❘ ❜❡❝❛✉s❡ ♦❢ t❤❡✐r s♠❛❧❧❡r s✐③❡✳ ■♥ ♦✉r ❝♦♥✜❣✉r❛t✐♦♥✱ D r❡♣❡❛ts t❤❡❛❝❦♥♦✇❧❡❞❣♠❡♥t ✉♥t✐❧ ✐t st♦♣s r❡❝❡✐✈✐♥❣ ❞❛t❛ ♣❛❝❦❡ts✳
❚❤❡ tr❛♥s♠✐ss✐♦♥ ❝❤❛♥♥❡❧ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ✐ts P❊❘ ❞❡r✐✈❡❞ ❛s ❛ ❢✉♥❝t✐♦♥♦❢ t❤❡ ❙✐❣♥❛❧ t♦ ◆♦✐s❡ ❘❛t✐♦ ✭❙◆❘✮ γ ♦♥ ❛ ❧✐♥❦✳ ❚❤❡ ❙◆❘ ✐s ❞❡r✐✈❡❞ ✉s✐♥❣❛♥ ✐s♦tr♦♣✐❝ ♣r♦♣❛❣❛t✐♦♥ ♠♦❞❡❧ ✇✐t❤ ❛ ♣❛t❤❧♦ss ❡①♣♦♥❡♥t ♦❢ α = 2✳ ❚❤❡ P❊❘
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✽
▲❛②❡r ❈♦♥✜❣✉r❛t✐♦♥s❆♣♣❧✐❝❛t✐♦♥ ◆❡t✇♦r❦ s✐③❡ ✭✐♥ ❤♦♣s✮✿ n
❉✐st❛♥❝❡ ❜❡t✇❡❡♥ ❡❛❝❤ ♥♦❞❡ ❂ ✽✺♠◆❡t✇♦r❦✐♥❣ ▼❆❈ ♣r♦t♦❝♦❧✿ ❯♥s❧♦tt❡❞ ■❊❊❊ ✽✵✷✳✶✺✳✹ ❈❙▼❆✴❈❆
❚r❛♥s♠✐ss✐♦♥ ♣❡r✐♦❞ ♦❢ s♦✉r❝❡ ❂ ✶s❈♦❞✐♥❣✿ ❘▲ ❝♦❞❡P❉❯ s✐③❡ ❂ ✶✷✽❜②t❡s❇❧♦❝❦ ❧❡♥❣t❤ ❂ k
❘❛❞✐♦ ❘❛❞✐♦ ❞❡✈✐❝❡✿ ❈❤✐♣s❡t ❈❈✶✶✵✵▼♦❞✉❧❛t✐♦♥✿ ❇P❙❑❚r❛♥s♠✐tt❡❞ ♣♦✇❡r ❂ ✶✵❞❇❚r❛♥s♠✐ss✐♦♥ r❛t❡ ❂ ✷✵❑❜✐t✴s❋r❡q✉❡♥❝② ❂ ✽✻✽▼❍③
Pr♦♣❛❣❛t✐♦♥ Pr♦♣❛❣❛t✐♦♥ ♠♦❞❡❧✿ ❆❲●◆P❛t❤❧♦ss ❡①♣♦♥❡♥t α ❂ ✷❲❤✐t❡ ♥♦✐s❡ ❂ ✲✶✶✶❞❇♠✴❍③❋❛❞✐♥❣✿ ♥♦♥❡
❚❛❜❧❡ ✷✿ ❙✐♠✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs✳
❞❡♣❡♥❞s ♦♥ ❇✐t ❊rr♦r ❘❛t❡ ✭❇❊❘✮ ❛s ❢♦❧❧♦✇s✿ PER = 1− (1−BER(γ))ℓ✱ ✇✐t❤ℓ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♣❛❝❦❡t✳ ❚❤❡ ❇❊❘ ❞❡♣❡♥❞s ♦♥ t❤❡ t②♣❡ ♦❢ ♠♦❞✉❧❛t✐♦♥ ❛♥❞t❤❡ t②♣❡ ♦❢ ❝❤❛♥♥❡❧ ❝♦♥s✐❞❡r❡❞✳ ■t ✐s ❞❡r✐✈❡❞ ✐♥ ♦✉r ❝❛s❡ ❢♦r ❛ ❇P❙❑ ♠♦❞✉❧❛t✐♦♥❛♥❞ ❆❲●◆ ❝❤❛♥♥❡❧ ♠♦❞❡❧ ✉s✐♥❣ BER(γ) = Q
(√2γ
)
= 0.5 ∗ erfc(√
γ)✳❚❤❡ ♣❛r❛♠❡t❡rs ✉s❡❞ ✐♥ ♦✉r s✐♠✉❧❛t✐♦♥s ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ ❚❛❜❧❡ ✷✳ ❚❤❡
s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❛r❡ ❛✈❡r❛❣❡❞ ♦✈❡r ✶✵✵✵ tr✐❛❧s✳ ❆ ✜rst s✐♠✉❧❛t✐♦♥ s❡t ❞❡❛❧s✇✐t❤ t❤❡ tr❛♥s♠✐ss✐♦♥ ❞❡❧❛② ♦❢ t❤❡ ❞✐✛❡r❡♥t str❛t❡❣✐❡s ❢♦r k = 10 ✐♥ ❚❛❜❧❡ ✸✳❚❤❡ ❜❡st s♦❧✉t✐♦♥ ✐s ❣r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣✿ ✐t ✐s ❢❛st ❛♥❞ s❝❛❧❡s ✇❡❧❧✳ ❆t t❤❡♦t❤❡r ❡♥❞✱ ❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞ ✐s t❤❡ s❧♦✇❡st ❛♥❞ ✐ts ❞❡❧❛② ❣r♦✇s ❧✐♥❡❛r❧② ✇✐t❤n✳
n = 5 n = 10 n = 15❉❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞ ✻✾✳✾ ✶✸✽✳✷ ✷✵✻✳✺❋♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ ✻✻✳✽ ✶✸✹✳✶ ✷✵✷✳✼P❛ss✐✈❡ r❡❧❛②✐♥❣ ✷✹✳✹ ✺✸✳✵ ✶✶✶✳✸●r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣ ✷✵✳✷ ✷✾✳✵ ✺✼✳✾
❚❛❜❧❡ ✸✿ ❚r❛♥s♠✐ss✐♦♥ ❞❡❧❛② ✐♥ s❡❝♦♥❞s ❢♦r k = 10✳
❲❡ ♥♦✇ t✉r♥ ♦✉r ❛tt❡♥t✐♦♥ t♦ ❝❤❛r❛❝t❡r✐③✐♥❣ ✢♦♦❞✐♥❣ tr❛✣❝✳
✹✳✷ ❆ss❡ss✐♥❣ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ♠♦❞❡❧
❚❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ❤❛s ❜❡❡♥ ❝♦♠♣✉t❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ n ∈ [2, 15]✱ ❢♦r k = 10✳❚❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ ♦❜s❡r✈❡❞ ❢♦r ❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞ ❛♥❞ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡✐s ♠❛r❣✐♥❛❧✳ ❋♦r ♣❛ss✐✈❡ ❛♥❞ ❣r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣✱ t❤❡ ❛♥❛❧②t✐❝❛❧ ❡st✐♠❛t✐♦♥♦❢ ❊q✉❛t✐♦♥ ✶ ✐s ❝♦♥❢r♦♥t❡❞ ✇✐t❤ ♦✉r s✐♠✉❧❛t✐♦♥s ✐♥ ❋✐❣✳ ✸ ♦♥ ❛✈❡r❛❣❡ r❡s✉❧ts✳P❛r❛♠❡t❡r h ❢♦r ❡❛❝❤ str❛t❡❣② ❤❛s ❜❡❡♥ ♦❜t❛✐♥❡❞ t❤r♦✉❣❤ ❛ ❧✐♥❡❛r r❡❣r❡ss✐♦♥
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✾
♠❡t❤♦❞✳ ■ts ❛♣♣❡❛rs t❤❛t t❤❡ ♠♦❞❡❧ ♠❛t❝❤❡s r❡❧❛t✐✈❡❧② ✇❡❧❧ t❤❡ s✐♠✉❧❛t✐♦♥✳P❛ss✐✈❡ r❡❧❛②✐♥❣ ✐s ♠♦r❡ ✐♠♣❛❝t❡❞ ❜② t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ t❤❛♥ ❣r❡❡❞② r❛♥❞♦♠r❡❧❛②✐♥❣ ✐s✳
P❛ss✐✈❡ r❡❧❛②✐♥❣●r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣
❚❤❡♦r② ✭h ≈ 3.5✮❚❤❡♦r② ✭h ≈ 2.1✮
✵
✶✵
✸✵
✹✵
✺✵
✹ ✶✵ ✶✹n
✻ ✽ ✶✻
✷✵
❆✈❡r❛❣❡
♥✉♠❜❡r
♦❢✢♦♦❞✐♥❣❡♠
✐ss✐♦♥s
✶✷✷
❋✐❣✉r❡ ✸✿ ❱❛❧✐❞❛t✐♦♥ ♦❢ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ f ❢♦r ♣❛ss✐✈❡ ❛♥❞ ❣r❡❡❞② r❛♥❞♦♠r❡❧❛②✐♥❣ ✭❛✈❡r❛❣❡ r❡s✉❧ts✮✳
❲❡ ♦❜s❡r✈❡ ❛ ❤✐❣❤ ✈❛r✐❛❜✐❧✐t② ✐♥ ♦✉r r❡s✉❧ts✳ ❚❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ✐♥ ♦✉r♣r❡✈✐♦✉s❧② ♠❡♥t✐♦♥❡❞ s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❣r♦✇s q✉✐❝❦❧② ✇✐t❤ n ✭s❡❡ ❋✐❣✳ ✹✮✳ ❲❡♦❜s❡r✈❡ ✇✐t❤ ♥❡❛r❧② t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② tr✐❛❧s ✐♥ ✇❤✐❝❤ h ≈ n ♦r h ≈ 1✳ ❚❤✐s ✐s❡♥t✐r❡❧② ❞✉❡ t♦ r❛♥❞♦♠ ❜❛❝❦♦✛ ♦❢ ✽✵✷✳✶✺✳✹ t❤❛t ✐♥❞✉❝❡s ❤✐❣❤ ✈❛r✐❛❜✐❧✐t② ✐♥ t❤❡♠❡❞✐✉♠ ❛❝❝❡ss✳ ❲❡ ❡♥s✉r❡ t❤❛t ♥♦ ❝♦♥❣❡st✐♦♥ ♦❝❝✉rs ❜② s❡tt✐♥❣ t❤❡ tr❛♥s♠✐ss✐♦♥♣❡r✐♦❞ ♦❢ t❤❡ s♦✉r❝❡ t♦ ♦♥❡ s❡❝♦♥❞✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s t❤❛t t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ▼❆❈❧❛②❡r ♠❛② ❣r❡❛t❧② ✐♥✢✉❡♥❝❡ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝✳
Passive
✵
✺
✶✵
✶✺
✷✵
✷✺
✸✺
✹✵
♦❢t❤❡✢♦♦❞✐♥❣tr❛✣
❝❙t❛♥❞❛r❞
❞❡✈✐❛t✐♦♥ ✸✵
n✻ ✽ ✶✵ ✶✷ ✶✹ ✶✻
●r❡❡❞② r❛♥❞♦♠
✹✷
❋✐❣✉r❡ ✹✿ ❙t❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝✳
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✶✵
✹✳✸ ❈♦♠♣❛r✐♥❣ t❤❡ str❛t❡❣✐❡s
❋✐❣✉r❡ ✺ ♣r♦✈✐❞❡s ❛ ❞❡t❛✐❧❡❞ tr❛✣❝ ❛♥❛❧②s✐s ♦❢ t❤❡ ❞✐✛❡r❡♥t str❛t❡❣✐❡s ❢♦r n ={5, 10, 15, 20} ❛♥❞ k = 5✳ ■♥ t❤✐s ❞✐♠❡♥s✐♦♥✱ ❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞ r❡❧❛②✐♥❣ ♦✛❡rst❤❡ ❜❡st r❡s✉❧ts ❛♥❞ ❛ ❤✐❣❤ s❝❛❧❛❜✐❧✐t②✳ P❛ss✐✈❡ ❛♥❞ ❣r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣ ❤❛✈❡❛♥ ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ ❛♥❞ ❢♦r n = 20✱ ♣❛ss✐✈❡ r❡❧❛②✐♥❣ ✐s ❛❧♠♦st ✉♥tr❛❝t❛❜❧❡✳❋♦r n = 15✱ t❤❡ ✢♦♦❞✐♥❣ tr❛✣❝ r❡♣r❡s❡♥ts ✶✹✪ ✭r❡s♣✳ ✽✪✮ ♦❢ t❤❡ ♦✈❡r❛❧❧ tr❛✣❝❢♦r ♣❛ss✐✈❡ r❡❧❛②✐♥❣ ✭r❡s♣✳ ❣r❡❡❞② r❛♥❞♦♠ r❡❧❛②✐♥❣✮✳
✭❛✮✭❜✮✭❝✮✭❞✮ ✭❛✮✭❜✮✭❝✮✭❞✮✭❛✮✭❜✮✭❝✮✭❞✮✭❛✮✭❜✮✭❝✮✭❞✮✵
✷✵✵
✹✵✵
✻✵✵
✽✵✵
✶✵✵✵
✶✷✵✵❯s❡❢✉❧ ❞❛t❛
❆❈❑❋❧♦♦❞✐♥❣
✶✹✵✵
♦❢❡♠
✐ss✐♦♥s
❆✈❡r❛❣❡
♥✉♠❜❡r
n = 10 n = 20n = 15n = 5
❋✐❣✉r❡ ✺✿ ❚r❛✣❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❞❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞ ✭❛✮✱ ♣❛ss✐✈❡ r❡❧❛②✐♥❣ ✭❜✮✱❣r❡❡❞② r❛♥❞♦♠ ✭❝✮ ❛♥❞ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ ✭❞✮ r❡❧❛②✐♥❣ ❢♦r n = {5, 10, 15, 20}❛♥❞ k = 5✳
❚❤❡ ❋✐❣✳ ✻ ✭❛✮ t♦ ✭❝✮ ♣r♦✈✐❞❡ r❡❧❛② ❜② r❡❧❛② t❤❡ ❡♠✐tt❡❞ tr❛✣❝ r❡❧❛t✐✈❡ t♦♣❛ss✐✈❡✱ ❣r❡❡❞② r❛♥❞♦♠ ❛♥❞ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ str❛t❡❣✐❡s ❢♦r k = 5 ❛♥❞ n = 10✳■♥ t❤✐s ❝❛s❡✱ ❣r❡❡❞② r❛♥❞♦♠ ♣❡r❢♦r♠s t❤❡ ❜❡st✳ ■t ❤❛s ❛❧s♦ t❤❡ ❜❡st ❢❛✐r♥❡ss✭❏❛✐♥✬s ❢❛✐r♥❡ss ❡q✉❛❧s t♦ ✵✳✾✷✮✳ P❛ss✐✈❡ r❡❧❛②✐♥❣ ❛♥❞ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ ❤❛✈❡r❡s♣❡❝t✐✈❡❧② ✵✳✽✵ ❛♥❞ ✵✳✼✾ ❢♦r ❢❛✐r♥❡ss✳ ❋♦r ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡✱ ♠❛♥② ♣❛❝❦❡ts❛r❡ r❡❞✉♥❞❛♥t ✇❤✐❝❤ ❡①♣❧❛✐♥s t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❝✉r✈❡✳ ❘❡♠♦✈✐♥❣ t❤✐s r❡❞✉♥❞❛♥❝②✐s ♣♦ss✐❜❧❡✱ ❜✉t ❛t t❤❡ ♣r✐❝❡ ♦❢ ❛ q✉❛❞r❛t✐❝ ❡♥❝♦❞✐♥❣ ❝♦st✳
✺ ❈♦♥❝❧✉s✐♦♥
❋♦✉♥t❛✐♥ ❝♦❞❡s ❛r❡ ❛ttr❛❝t✐✈❡ ❢♦r ❲❙◆s ❜❡❝❛✉s❡ ♦❢ t❤❡✐r r❛t❡❧❡ss ♣r♦♣❡rt②✳ ❖✉r❛♥❛❧②s✐s ✐❞❡♥t✐✜❡❞ t❤❡ ♠❛✐♥ ♦❜st❛❝❧❡s t♦ t❤❡✐r ❞❡♣❧♦②♠❡♥ts ♦♥ ■❊❊❊ ✽✵✷✳✶✺✳✹✱✐✳❡✳ t❤❡ ❞❛t❛ s✐③❡ k ❛♥❞ t❤❡ ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t ♣❛t❤ ✭n✲❤♦♣✮✳ ❉❡❝♦❞❡✲❛♥❞✲❢♦r✇❛r❞r❡❧❛②✐♥❣ ✐s t♦ ❜❡ ♣r❡❢❡rr❡❞ ✐❢ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ✐s ❞❡❧❛②✲t♦❧❡r❛♥t s✐♥❝❡ ✐t ❣✉❛r❛♥✲t❡❡s ❛ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ tr❛♥s♠✐ss✐♦♥s✳ ■t ♣r❡s❡r✈❡s t❤❡ s❡♥s♦rs✬ ❡♥❡r❣②✳ ❋♦r❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✇❤✐❝❤ ❜♦t❤ ❞❡❧❛② ❛♥❞ ❡♥❡r❣② ❛r❡ ❝r✐t✐❝❛❧✱ t❤❡ s✐t✉❛t✐♦♥ ✐s ♠♦r❡❝♦♠♣❧✐❝❛t❡❞✳ ■❢ t❤❡ ♥✉♠❜❡r ♦❢ ❤♦♣s ♦♥ t❤❡ r♦✉t❡ ✐s ❜❡❧♦✇ ✜❢t❡❡♥✱ ❣r❡❡❞② r❛♥❞♦♠r❡❧❛②✐♥❣ ✐s ✇❡❧❧ ❛❞❛♣t❡❞✳ ❖t❤❡r✇✐s❡✱ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ ✐s ❜❡tt❡r✳
❆♥ ✐♠♣♦rt❛♥t q✉❡st✐♦♥ ❧❡❢t ❜② t❤❡ ♣❛♣❡r ✐s ❤♦✇ ❞♦❡s ✐t s❝❛❧❡ ✇✐t❤ ♦t❤❡r ▼❆❈❧❛②❡rs✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ♦✉r ♠♦❞❡❧ ❛♣♣❧✐❡s ✇❡❧❧ ♦♥ ❚❉▼❆ s❝❤❡♠❡s✳ ▼❛♥②▼❆❈ ❧❛②❡rs ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ ♣r❡s❡r✈❡ t❤❡ s❡♥s♦rs ❡♥❡r❣② t❤r♦✉❣❤ ❡✣❝✐❡♥t
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✶✶
✷✵
✹✵
✺✵❆✈❡r❛❣❡
♥✉♠❜❡r
♦❢❡♠
✐ss✐♦♥s
✸✵
✶✵
✵S R2 R4 R6 R8 D
✭❛✮ ♣❛ss✐✈❡
10
ACK
Useful data
✵
✷✵
✸✵
✹✵
❆✈❡r❛❣❡
♥✉♠❜❡r
♦❢❡♠
✐ss✐♦♥s
✺✵
✭❜✮ ●r❡❡❞② r❛♥❞♦♠
❋❧♦♦❞✐♥❣
DS R2 R4 R8R6
✶✵
✷✵
✸✵
✹✵
✵S R4 DR6 R8R2
❆✈❡r❛❣❡
♥✉♠❜❡r
♦❢❡♠
✐ss✐♦♥s
✺✵
✭❝✮ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡
❋✐❣✉r❡ ✻✿ ❊♠✐tt❡❞ tr❛✣❝ ❢♦r ❡❛❝❤ ♥♦❞❡ ✭n = 10✮✳ ❲❡ ❤❛✈❡ k = 5 ❛♥❞ ✇❡ ❝♦♥s✐❞❡r♣❛ss✐✈❡ ✭❧❡❢t✮✱ ❣r❡❡❞② ✭❝❡♥t❡r✮ ❛♥❞ ❢♦r✇❛r❞✲❛♥❞✲❞❡❝♦❞❡ ✭r✐❣❤t✮ r❡❧❛②✐♥❣✳
❞✉t②✲❝②❝❧✐♥❣✳ ❚❤❡ ❛✉t❤♦rs ✇✐❧❧ ❧♦♦❦ ♥❡①t ✐♥t♦ t❤❡ s✉✐t❛❜✐❧✐t② ♦❢ s✉❝❤ s♦❧✉t✐♦♥s❢♦r t❤❡ ❞❡♣❧♦②♠❡♥t ♦❢ ❢♦✉♥t❛✐♥ ❝♦❞❡s✳
❘❡❢❡r❡♥❝❡s
❬✶❪ ❆♣❛✈❛t❥r✉t ❆♥②❛✱ ❈❧❛✐r❡ ●♦✉rs❛✉❞✱ ❑❛t✐❛ ❏❛✛rès ❘✉♥s❡r✱ ❈r✐st✐♥❛ ❈♦♠❛♥✐✲❝✐✉✱ ❛♥❞ ❏❡❛♥✲▼❛r✐❡ ●♦r❝❡✳ ❚♦✇❛r❞s ✐♥❝r❡❛s✐♥❣ ❞✐✈❡rs✐t② ❢♦r t❤❡ r❡❧❛②✐♥❣♦❢ ▲❚ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s✳ ❘❡s❡❛r❝❤ ❘❡♣♦rt ❘❘✲✼✷✸✶✱ ■◆❘■❆✱ ▼❛r❝❤ ✷✵✶✵✳
❬✷❪ ❆♠♦s ❇❡✐♠❡❧✱ ❙❤❧♦♠✐ ❉♦❧❡✈✱ ❛♥❞ ◆♦❛♠ ❙✐♥❣❡r✳ ❘❚ ♦❜❧✐✈✐♦✉s ❡r❛s✉r❡ ❝♦r✲r❡❝t✐♥❣✳ ■❊❊❊✴❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ◆❡t✇♦r❦✐♥❣✱ ✶✺✭✻✮✿✶✸✷✶✕✶✸✸✷✱ ✷✵✵✼✳
❬✸❪ ❏♦❤♥❲✳ ❇②❡rs✱ ▼✐❝❤❛❡❧ ▲✉❜②✱ ❛♥❞ ▼✐❝❤❛❡❧ ▼✐t③❡♥♠❛❝❤❡r✳ ❆❝❝❡ss✐♥❣ ▼✉❧t✐✲♣❧❡ ▼✐rr♦r ❙✐t❡s ✐♥ P❛r❛❧❧❡❧✿ ❯s✐♥❣ ❚♦r♥❛❞♦ ❈♦❞❡s t♦ ❙♣❡❡❞ ❯♣ ❉♦✇♥❧♦❛❞s✳■♥ ■◆❋❖❈❖▼ ✬✾✾✱ ♣❛❣❡s ✷✼✺✕✷✽✸✱ ◆❡✇ ❨♦r❦✱ ◆❨✱ ❯❙❆✱ ✶✾✾✾✳ ■❊❊❊✳
❬✹❪ ❏♦❤♥ ❲✳ ❇②❡rs✱ ▼✐❝❤❛❡❧ ▲✉❜②✱ ▼✐❝❤❛❡❧ ▼✐t③❡♥♠❛❝❤❡r✱ ❛♥❞ ❆s❤✉t♦s❤ ❘❡❣❡✳❆ ❉✐❣✐t❛❧ ❋♦✉♥t❛✐♥ ❆♣♣r♦❛❝❤ t♦ ❘❡❧✐❛❜❧❡ ❉✐str✐❜✉t✐♦♥ ♦❢ ❇✉❧❦ ❉❛t❛✳ ■♥❈♦♥❢❡r❡♥❝❡ ♦♥ ❆♣♣❧✐❝❛t✐♦♥s✱ ❚❡❝❤♥♦❧♦❣✐❡s✱ ❆r❝❤✐t❡❝t✉r❡s✱ ❛♥❞ Pr♦t♦❝♦❧s ❢♦r❈♦♠♣✉t❡r ❈♦♠♠✉♥✐❝❛t✐♦♥ ✲ ❙■●❈❖▼▼ ✶✾✾✽✱ ♣❛❣❡s ✺✻✕✻✼✱ ❱❛♥❝♦✉✈❡r✱❇❈✱ ❈❛♥❛❞❛✱ ❆✉❣✉st ✶✾✾✽✳
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✶✷
❬✺❪ P❛♦❧♦ ❈❛s❛r✐✱ ▼✐❝❤❡❧❡ ❘♦ss✐✱ ❛♥❞ ▼✐❝❤❡❧❡ ❩♦r③✐✳ ❋♦✉♥t❛✐♥ ❝♦❞❡s ❛♥❞ t❤❡✐r❛♣♣❧✐❝❛t✐♦♥ t♦ ❜r♦❛❞❝❛st✐♥❣ ✐♥ ✉♥❞❡r✇❛t❡r ♥❡t✇♦r❦s✿ ♣❡r❢♦r♠❛♥❝❡ ♠♦❞❡❧✐♥❣❛♥❞ r❡❧❡✈❛♥t tr❛❞❡♦✛s✳ ■♥ ❆❈▼ ✐♥t❡r♥❛t✐♦♥❛❧ ✇♦r❦s❤♦♣ ♦♥ ❯♥❞❡r✇❛t❡r◆❡t✇♦r❦s ✲ ❲✉❲◆❡❚ ✬✵✽✱ ♣❛❣❡s ✶✶✕✶✽✱ ❙❛♥ ❋r❛♥❝✐s❝♦✱ ❈❛❧✐❢♦r♥✐❛✱ ❯❙❆✱✷✵✵✽✳ ❆❈▼✳
❬✻❪ ▼❛r②✲▲✉❝ ❈❤❛♠♣❡❧✱ ❑é✈✐♥ ❍✉❣✉❡♥✐♥✱ ❆♥♥❡✲▼❛r✐❡ ❑❡r♠❛rr❡❝✱ ❛♥❞ ◆✐❝♦✲❧❛s ▲❡ ❙❝♦✉❛r♥❡❝✳ ▲❚ ◆❡t✇♦r❦ ❈♦❞❡s✳ ■♥ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥❉✐str✐❜✉t❡❞ ❈♦♠♣✉t✐♥❣ ❙②st❡♠s ✲ ■❈❉❈❙ ✷✵✶✵✱ ♣❛❣❡ t♦ ❛♣♣❡❛r✱ ●❡♥♦❛✱■t❛❧②✱ ❏✉♥❡ ✷✵✶✵✳ ■❊❊❊ ❈♦♠♣✉t❡r ❙♦❝✐❡t②✳
❬✼❪ ❆♥t♦✐♥❡ ❋r❛❜♦✉❧❡t✱ ●✉✐❧❧❛✉♠❡ ❈❤❡❧✐✉s✱ ❛♥❞ ❊r✐❝ ❋❧❡✉r②✳ ❲♦r❧❞s❡♥s✿ ❞❡✲✈❡❧♦♣♠❡♥t ❛♥❞ ♣r♦t♦t②♣✐♥❣ t♦♦❧s ❢♦r ❛♣♣❧✐❝❛t✐♦♥ s♣❡❝✐✜❝ ✇✐r❡❧❡ss s❡♥s♦rs♥❡t✇♦r❦s✳ ■♥ ❆❈▼✴■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ■♥❢♦r♠❛t✐♦♥ Pr♦✲❝❡ss✐♥❣ ✐♥ ❙❡♥s♦r ◆❡t✇♦r❦s ✲■P❙◆ ✷✵✵✼✱ ♣❛❣❡s ✶✼✻✕✶✽✺✳ ❆❈▼✱ ✷✵✵✼✳
❬✽❪ ❈❤r✐st✐♥❛ ❋r❛❣♦✉❧✐✱ ❏❡❛♥✲❨✈❡s ▲❡ ❇♦✉❞❡❝✱ ❛♥❞ ❏ör❣ ❲✐❞♠❡r✳ ◆❡t✇♦r❦ ❝♦❞✲✐♥❣✿ ❛♥ ✐♥st❛♥t ♣r✐♠❡r✳ ❈♦♠♣✉t❡r ❈♦♠♠✉♥✐❝❛t✐♦♥ ❘❡✈✐❡✇✱ ✸✻✭✶✮✿✻✸✕✻✽✱✷✵✵✻✳
❬✾❪ ❊s❛ ❍②②t✐ä✱ ❚✉♦♠❛s ❚✐rr♦♥❡♥✱ ❛♥❞ ❏♦r♠❛ ❚✳ ❱✐rt❛♠♦✳ ❖♣t✐♠❛❧ ❉❡❣r❡❡❉✐str✐❜✉t✐♦♥ ❢♦r ▲❚ ❈♦❞❡s ✇✐t❤ ❙♠❛❧❧ ▼❡ss❛❣❡ ▲❡♥❣t❤✳ ■♥ ■◆❋❖❈❖▼✷✵✵✼✱ ♣❛❣❡s ✷✺✼✻✕✷✺✽✵✱ ❆♥❝❤♦r❛❣❡✱ ❆❑✱ ❯❙❆✱ ▼❛② ✷✵✵✼✳ ■❊❊❊✳
❬✶✵❪ ❚❡①❛s ■♥str✉♠❡♥ts✳ ❈❈✶✶✵✵ ❙✐♥❣❧❡ ❈❤✐♣ ▲♦✇ ❈♦st ▲♦✇ P♦✇❡r ❘❋✲❚r❛♥s❝❡✐✈❡r✱ ✷✵✵✻✳ ❤tt♣✿✴✴❢♦❝✉s✳t✐✳❝♦♠✴❧✐t✴❞s✴s②♠❧✐♥❦✴❝❝✶✶✵✵✳♣❞❢✳
❬✶✶❪ ❋❡♥❣ ▲✉✱ ❈❤✉❛♥ ❍❡♥❣ ❋♦❤✱ ❏✐❛♥❢❡✐ ❈❛✐✱ ❛♥❞ ▲✐❛♥❣✲❚✐❡♥ ❈❤✐❛✳ ▲❚ ❝♦❞❡s❞❡❝♦❞✐♥❣✿ ❉❡s✐❣♥ ❛♥❞ ❛♥❛❧②s✐s✳ ■♥ ■♥t❡r♥❛t✐♦♥❛❧ ❙②♠♣♦s✐✉♠ ♦♥ ■♥❢♦r♠❛t✐♦♥❚❤❡♦r② ✲ ■❙■❚ ✷✵✵✾✱ ♣❛❣❡s ✷✹✾✷✕✷✹✾✻✱ ❙❡♦✉❧✱ ❙♦✉t❤ ❑♦r❡❛✱ ✷✵✵✾✳ ■❊❊❊✳
❬✶✷❪ ▼✐❝❤❛❡❧ ▲✉❜②✳ ▲❚ ❈♦❞❡s✳ ■♥ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✲ ❋❖❈❙✷✵✵✷✱ ♣❛❣❡s ✷✼✶✕✱ ❱❛♥❝♦✉✈❡r✱ ❇❈✱ ❈❛♥❛❞❛✱ ◆♦✈❡♠❜❡r ✷✵✵✷✳ ■❊❊❊ ❈♦♠✲♣✉t❡r ❙♦❝✐❡t②✳
❬✶✸❪ ❯rs ◆✐❡s❡♥✱ ❈❤r✐st✐♥❛ ❋r❛❣♦✉❧✐✱ ❛♥❞ ❉❛♥✐❡❧❛ ❚✉♥✐♥❡tt✐✳ ❖♥ ❈❛♣❛❝✐t② ♦❢ ▲✐♥❡◆❡t✇♦r❦s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✱ ✺✸✭✶✶✮✿✹✵✸✾✕✹✵✺✽✱✷✵✵✼✳
❬✶✹❪ ❆✳ ❖❦❛ ❛♥❞ ▲✳ ▲❛♠♣❡✳ ❉❛t❛ ❡①tr❛❝t✐♦♥ ❢r♦♠ ✇✐r❡❧❡ss s❡♥s♦r ♥❡t✇♦r❦s✉s✐♥❣ ❞✐str✐❜✉t❡❞ ❢♦✉♥t❛✐♥ ❝♦❞❡s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♠✉♥✐❝❛t✐♦♥s✱✺✼✭✾✮✿✷✻✵✼✕✷✻✶✽✱ ✷✵✵✾✳
❬✶✺❪ P❛②❛♠ P❛❦③❛❞✱ ❈❤r✐st✐♥❛ ❋r❛❣♦✉❧✐✱ ❛♥❞ ❆♠✐♥ ❙❤♦❦r♦❧❧❛❤✐✳ ❈♦❞✐♥❣ s❝❤❡♠❡s❢♦r ❧✐♥❡ ♥❡t✇♦r❦s✳ ■♥ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❙②♠♣♦s✐✉♠ ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✲ ■❙■❚ ✷✵✵✺✱ ♣❛❣❡s ✶✽✺✸✕✶✽✺✼✱ ❆❞❡❧❛✐❞❡✱ ❆✉str❛❧✐❛✱ ✷✵✵✺✳ ■❊❊❊✳
❬✶✻❪ ❆♠✐♥ ❙❤♦❦r♦❧❧❛❤✐✳ ❘❛♣t♦r ❝♦❞❡s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡✲♦r②✱ ✺✷✭✻✮✿✷✺✺✶✕✷✺✻✼✱ ✷✵✵✻✳
❬✶✼❪ ■❊❊❊ ❈♦♠♣✉t❡r ❙♦❝✐❡t②✳ ■❊❊❊ ❙t❞ ✽✵✷✳✶✺✳✹ ✲ ❲✐r❡❧❡ss ▼❡❞✐✉♠ ❆❝❝❡ss ❈♦♥✲tr♦❧ ✭▼❆❈✮ ❛♥❞ P❤②s✐❝❛❧ ▲❛②❡r ✭P❍❨✮ ❙♣❡❝✐✜❝❛t✐♦♥s ❢♦r ▲♦✇✲❘❛t❡ ❲✐r❡❧❡ssP❡rs♦♥❛❧ ❆r❡❛ ◆❡t✇♦r❦s ✭▲❘✲❲P❆◆s✮✳ ❙t❛♥❞❛r❞✱ ✷✵✵✸✳
❘❘ ♥➦ ✼✸✾✼
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❖♥ t❤❡ ❋❧♦♦❞✐♥❣ ❖✈❡r❤❡❛❞ ♦❢ ❋♦✉♥t❛✐♥ ❈♦❞❡s ✐♥ ❲✐r❡❧❡ss ❙❡♥s♦r ◆❡t✇♦r❦s ✶✸
❬✶✽❪ ❚✉♦♠❛s ❚✐rr♦♥❡♥✳ ❋♦✉♥t❛✐♥ ❈♦❞❡s✿ P❡r❢♦r♠❛♥❝❡ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐s❛✲t✐♦♥✳ P❤❉ t❤❡s✐s✱ ✷✵✶✵✳
❬✶✾❪ ❉❛♥✐❡❧❛ ❚✉♥✐♥❡tt✐ ❛♥❞ ❈❤r✐st✐♥❛ ❋r❛❣♦✉❧✐✳ ❖♥ t❤❡ t❤r♦✉❣❤♣✉t ✐♠♣r♦✈❡♠❡♥t❞✉❡ t♦ ❧✐♠✐t❡❞ ❝♦♠♣❧❡①✐t② ♣r♦❝❡ss✐♥❣ ❛t r❡❧❛② ♥♦❞❡s✳ ■♥ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧❙②♠♣♦s✐✉♠ ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r② ✲ ■❙■❚ ✷✵✵✺✱ ♣❛❣❡s ✶✵✽✶✕✶✵✽✺✱ ❆❞❡✲❧❛✐❞❡✱ ❆✉str❛❧✐❛✱ ✷✵✵✺✳ ■❊❊❊✳
❘❘ ♥➦ ✼✸✾✼
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