HAL Id: hal-00639583https://hal.inria.fr/hal-00639583

Submitted on 22 Nov 2011

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Computing Time Complexity of Population Protocolswith Cover Times - The ZebraNet Example

Joffroy Beauquier, Peva Blanchard, Janna Burman, Sylvie Delaët

To cite this version:Joffroy Beauquier, Peva Blanchard, Janna Burman, Sylvie Delaët. Computing Time Complexity ofPopulation Protocols with Cover Times - The ZebraNet Example. Stabilization, Safety, and Secu-rity of Distributed Systems - 13th International Symposium, SSS 2011, Oct 2011, Grenoble, France.10.1007/978-3-642-24550-3_6. hal-00639583

♦♠♣t♥ ♠ ♦♠♣①t② ♦ P♦♣t♦♥

Pr♦t♦♦s t ♦r ♠s t ❩rt

①♠♣

♦r♦② qr1,3 P ♥r1 ⋆ ♥♥ r♠♥2 ⋆⋆ ♥ ② ët1

1 ❯♥ Prs rs② r♥ ④ ♥r t⑥rr2 ❯♥rst② ♦ ♦♣♥t♣♦s r♥

♥♥r♠♥♥rr3 r♥ r ♣r♦t ② r♥

strt P♦♣t♦♥ ♣r♦t♦♦s r ♦♠♠♥t♦♥ ♠♦ ♦r rs♥s♦r ♥t♦rs t rs♦r♠t ♠♦ ♥ts ♥ts ♠♦s②♥r♦♥♦s② ♥ ♦♠♠♥t ♣rs ♥trt♦♥s ♦r♥r♥ss ss♠♣t♦♥ ♦ ts ♠♦ ♥♦s ♦ s②♥r♦♥②♥ ♣r♥ts ♥ t♦♥ ♦ t ♦♥r♥ t♠ ♦ ♣r♦t♦♦ tr♠♥st ♠♥s ♥tr♦t♦♥ ♦ s♦♠ ♣rt s②♥r♦♥② ♥t ♠♦ ♥r t ♦r♠ ♦ ♦r t♠s s ♥ ①t♥s♦♥ tt ♦st♥ t t♠ ♦♠♣①ts♥ ts ♣♣r t ♥t ♦ ts ①t♥s♦♥ ♥ st② t

♦t♦♥ ♣r♦t♦♦ s ♥ t ❩rt ♣r♦t ♦r t tr♥♦ ③rs ♥ rsr ♥ ♥tr ♥② ♥ ❩rt s♥s♦rs r ttt♦ ③rs ♥ t s♥s t s ♦t rr② ② ♠♦ s

stt♦♥ r♦ss♥ t r t ♦t♦♥ ♣r♦t♦♦ ♦ ❩rt s♥ ♥②③ tr♦ s♠t♦♥s t t♦ ♦r ♥♦ ts s t rstt♠ tt ♣r② ♥②t st② s ♣rs♥t r rst rst s tt♥ t ♦r♥ ♣r♦t♦♦ s♦♠ t ♠② ♥r r t♦ t sstt♦♥ ❲ t♥ ♣r♦♣♦s t♦ st② ♠♦ ♥ ♦rrt ♣r♦t♦♦s ♥ ♦♠♣t tr ♦rst s t♠ ♦♠♣①ts t ♥ ♦t ss trst s r r♦♠ t ♦♣t♠

♥tr♦t♦♥

P♦♣t♦♥ Pr♦t♦♦s PP ♥ ♥tr♦ ❬❪ s ♠♦ ♦ s♥s♦r ♥t♦rs ♦♥sst♥ ♦ r② s♠♣ ♠♦ ♥ts ♥ ts ♠♦ ♥♦♥②♠♦s ♠♦♥ts ♠♦ s②♥r♦♥♦s② ♥ ♥② t♦ ♦ t♠ ♥ ①♥ ♥♦r♠t♦♥ ♥♥ tr stts ♥r t② r ♦s♥ ② sr ❲♥ ts ♣♣♥s s② tt ♥ ♥t ♦r ♠t♥ t♥ t♦ ♠♦♥ ♥ts ♣♣♥s♥t② ♦♥ ♦ t ♦s ♦ PP s t♦ tr♠♥ t ♥ ♦♠♣t ♥ s ♠♦ t ♠♥♠ ②♣♦tss t s ② ♥ts r ♥♦♥②♠♦s ♠♦

⋆ ♦r ♦ ts t♦r s s♣♣♦rt ② r♥ts r♦♠ ②⋆⋆ ♦r ♦ ts t♦r s s♣♣♦rt ② t tr♥ r♥t r♦♠ t r♥

♦r♥♠♥t

s②♥r♦♥♦s② ♥ s♠ ♠♠♦r② ♦ s♣ ss♠♣t♦♥ s ♠ ♦♥t sr ①♣t ♦r r♥ss ♦♥t♦♥ tt stts tt ♥ ♥♥t② ♦t♥r ♦♥rt♦♥ s r ♥♥t② ♦t♥ t s s♦♥ ♥ ❬❪ tt t♦♠♣tt♦♥ ♣♦r ♦ t ♠♦ s rtr ♠t ♥ r♦s ①t♥s♦♥sr sst ❬❪

♥ ts ♣♣r ss♠ rs♦♥ ♦ t PP ♠♦ r ♥ ♥t♦r ♦s♣ ♦r t♠ s ss♦t t♦ ♥t ❬❪ ♦r t♠ s t ♠♥♠♠ ♥♠r ♦ ♦ ♥ts ♣♣♥♥ ♥ t s②st♠ ♦r ♥ rt♥ tt ♥♥t s ♠t r② ♦tr ♥t sr ss ♦ ♥ts ♦r♥t♦ t ♦r t♠s ss♠♣t♦♥ tt ♥ ♥t ♦♠♠♥ts t ♦tr♥ts ♣r♦② t♥ ♥t ♣r♦ s ♥ ①♣r♠♥t② st ♦rs♦♠ t②♣s ♦ ♠♦t② ♥ ♥ t s ♦ ♠♥ ♦r ♥♠ ♠♦t② t♥ ♦♥ r ♦r t ♦♠ ♦♠♥ t♥♥② t t♥♥② t♦ rtr♥ t♦s♦♠ s♣ ♣s ♣r♦② t sttst ♥②ss ♦ ①♣r♠♥t tsts ♦♥r♠s ts ss♠♣t♦♥ ❬❪ s t sts ♦♥r♥ st♥ts♦♥ ♠♣s ❬❪ ♣rt♣♥ts t♦ ♥t♦r ♦♥r♥ ❬❪ ♦r st♦rs t s♥②♥ ①t t t tt t ♥tr♦♥tt t♠ t♥ t♦ ♥ts♦♥sr s r♥♦♠ r ♦♦s tr♥t Prt♦ strt♦♥ ♥ ♣rtr ts ♥♦s tt t s ♠sr ♥ tr♠s ♦ r t♠ r ♥t♥ ♣rt s t② r s♦ ♥t ♥ ♠sr ♥ ♥ts ♦ s t ♦rt♠ ♦ ♥ ♥t s t ♠①♠♠ ♦ ts s ♠sr ♥ ♥ts

♥♦t♦♥ ♦ ♦r t♠s ♠② s ♥ ♥tr♦t♦♥ ♦ ♣rt s②♥r♦♥② ss♠♣t♦♥s ❬❪ ♥ t ♦r♥ PP ♠♦ ♣rt s t ♦rt♠s r ♥♦t ss♠ t♦ ♥♦♥ ② t ♥ts s ①t♥s♦♥ ♦s t♦♦♠♣t tr♠♥st t♠ ♦♠♣①ts ①♣rss ♥ t ♥♠r ♦ ♥ts s♦ ♥t ♦♠♣①ts s s ♠♣♦ss ♥ t ♦r♥ PP ♠♦

s ♣♣r ♣rs♥ts ♦♥ ♥ ①♠♣ s♦♠ t♥qs ♦r ♦♠♣t♥ t ♥t♦♠♣①t② ♦ ♣♦♣t♦♥ ♣r♦t♦♦s ①♠♣ s st ♠♦t♦♥ ♦ ♥①st♥ t ♦t♦♥ ♣r♦t♦♦ s ② t ❩rt ♣r♦t ❬❪ ❩rt s ♣r♦t ♦♥t ② t Pr♥t♦♥ ❯♥rst② ♥ ♣♦② ♥ ♥tr ♥②t ♠s t st②♥ ♣♦♣t♦♥s ♦ ③rs s♥ s♥s♦rs tt t♦ t ♥♠ss ♣r♦t ss st♦r②s ♣r♦t♦♦ t♦ r t s♥s s t♦ sstt♦♥ ❲♥ ♥ ♥t x s t ♣♦sst② t♦ r② ts t t♦ ♦tr ♥tst ♠② st t ♦♥ y tt s r♥t② ♠t t s stt♦♥ ♠♦r rq♥t② ♣r♦t♦♦ ss♠s tt y ♦♥t♥ ♠t♥ t s stt♦♥ rq♥t②♥ t ♥r tr ♥ r t s♦♦♥r

rst rst ♥ ts ♣♣r t♦rt② s♦s tt t ♦r♥ ❩rt♣r♦t♦♦ ♦s ♥♦t ♥sr t r② ♦ t s t♦ t s stt♦♥ rr ♥♥t ①t♦♥s ♥ s♦♠ s ② t♥ s♦♠ ♠♦ ♥ts t tt ♦t 10 ♦ t s♥s s r ♦st s ①t ② ts♠t♦♥s ♥ ❬❪ s s♣♣♦rt r ② ♦r♠ ①♣♥t♦♥ ♦ ♥sr tr② t♦t ♠♦②♥ t ♠♥ strtr ♦ t ①t♦♥s ♣r♦♣♦s t♦st② ♠♦ rs♦♥s rs♣t② ♦ ❩rt Pr♦t♦♦s 1 ♥2 ❩P ♥ ❩P ❲ t♥ ♣r♦ ♥ ♥②ss ♦ tr ♥t ♦♠♣①tst♥s t♦ t ♥♦t♦♥ ♦ ♦r t♠s ♥ ♦t ss t ♦rst s ♦♠♣①t② s

♦rs t♥ ♦r t ♦rt♠ ♣rs♥t ♥ ❬❪ ts ♦rt♠ rs t ♦♣t♠♦rst s ♦♠♣①t② ♥ ♥r ss

♦ ♥ ♦tt♦♥s

♠♦ s s ♥ ❬❪ t A t st ♦ t ♥ts ♥ t s②st♠ r|A| = n ♥ n s ♥♥♦♥ t♦ t ♥ts s tt♦♥ BS s st♥s ♥t t ①t♥ rs♦rs ♥ ♠② s♦ ♥♦♥♠♦ ♥♦♥trst t BS t ♦tr ♥ts r ♥tstt ♥♦♥②♠♦s ♥ r rrr ♥ t ♣♣r s ♠♦ ❲ ♥♦t ② A∗ t st ♦ ♠♦ ♥ts ♦♥ts r ♥♠rt r♦♠ 1 t♦ n − 1

P♦♣t♦♥ ♣r♦t♦♦s ♥ ♠♦ s tr♥st♦♥ s②st♠s ❲ ♦♣t t♦♦♥ ♦♠♠♦♥ ♥t♦♥s ♦r ♦r♠ ♥t♦♥s rr t♦ ❬❪ stt♦ ♥ ♥t t♦r ♦ t s ♦ ts rs ♦♥rt♦♥ t♦r ♦stts ♦ t ♥ts tr♥st♦♥ t♦♠ st♣ ♦ t♦ ♦♠♠♥t♥ ♥ts♥ tr ss♦t stt ♥s ①t♦♥ ♣♦ss② ♥♥t sq♥ ♦♦♥rt♦♥s rt ② tr♥st♦♥s

♥ ♥t (x y) s ♣rs ♦♠♠♥t♦♥ ♠t♥ ♦ t♦ ♥ts x ♥y ♥ ♥t ♦rrs♣♦♥s t♦ tr♥st♦♥ ❲t♦t ♦ss ♦ ♥rt② ss♠tt ♥♦ t♦ ♥ts ♣♣♥ s♠t♥♦s② s s ♥ ♥♥t sq♥ ♦♥ts s t♦tr t ♥ ♥t ♦♥rt♦♥ ♥q② tr♠♥s♥ ①t♦♥ ② s♥ t ♥♦tt♦♥ ♦t♥ rt sq♥ ♦ ♥tst♦ r♣rs♥t ♦t s ♥ t ♦rrs♣♦♥♥ ①t♦♥ ♥tt② t s♦♥♥♥t t♦ s ①t♦♥s s sr rsr② ♦♦ss t♦♥ts ♣rt♣t ♥ t ♥①t ♥t ♦r♠② sr D s ♣rt ♦♥ss s ♦ D s s tt stss t ♣rt D ♦r ts ♦ s♠♣t② ss♠ tt ♥ts strt ♥ ①t♦♥ s♠t♥♦s② ♦♥ s♥rs ♦r♥ t♦ ♦ ♦r ♦♥ r♣t ♦ ♦ s♥ r♦♠ BS ♥♦♥s♠t♥♦s strt s trt ♥ ❬❪

♦r ♠ Pr♦♣rt② ♥ t ♠♦ ♥t x s ss♦t t ♣♦st♥tr cvx t ♦r t♠ ♦ x ♥ts r ♥♦t ss♠ t♦ ♥♦ t ♦rt♠s ❲ ♥♦t ② cv t t♦r ♦ ♥ts ♦r t♠s ♥ ② cvmin rs♣cvmax t ♠♥♠♠ rs♣ ♠①♠♠ ♦r t♠ ♥ cv

♥t♦♥ ♦r ♠ Pr♦♣rt② ♥ ♣♦♣t♦♥ A ♦ n ♥ts ♥ t♦r cv ♦ ♣♦st ♥trs sr D ♥ ♥② ♦ ts ss s st♦ sts② t ♦r t♠ ♣r♦♣rt② ♥ ♦♥② ♦r r② x ∈ A ♥ ♥② cvx

♦♥st ♥ts ♦ ♥② s ♦ D ♥t x ♠ts r② ♦tr ♥t t st♦♥

♥ t ♣♣r ♦♥sr ♦♥② t srs tt sts② t ♦r t♠♣r♦♣rt② ❲ s② tt t ♦r t♠ t♦r cv s ♥♦r♠ ts ♥trs rq cvmin = cvmax ♥ ts s ♥♦t ② cv t ♦♠♠♦♥ ♦t ♥ts ♦r t♠s

BS s rqr r ♦♥② ② t ♥tr ♦ t t ♦t♦♥ ♣r♦♠ ❲ ♦♥② ♦♥sr tr♠♥st s②st♠s

t ♦t♦♥ ♥ ♦♥r♥ ♥ t ♦♥t①t ♦ t ♦t♦♥ ♥♥t ♦♥rt♦♥ s ♦♥rt♦♥ ♥ ♠♦ ♥t ♦♥s ♥ ♥♣t ♥♣t s t♦ r t♦ BS ①t② ♦♥ ❲♥ ts♣♣♥s s② tt ♦♥rt♦♥ s r ♥ ①t♦♥ s s t♦♦♥r t rs ♦♥rt♦♥ ♥t ♦ ♥ ①t♦♥ tt♦♥rs s t ♠♥♠♠ ♥♠r ♦ ♥ts ♥t ♦♥r♥ ♦rst s♥t ♦♠♣①t② ♦ ♥ ♦rt♠ s t ♠①♠♠ ♥t ♦ ts ①t♦♥s ♣r♦t♦♦ ♦r ♥ ♦rt♠ s s t♦ ♦♥r ts ①t♦♥s ♦♥r

❲♥ sr♥ ♥ ①t♦♥ ♠② ♥♥♦tt ♥t s ♦♦s ♥♦tt♦♥ (x y) ♥ts tt tr s tr♥sr r♦♠ x t♦ y ♦ s♣② ♦♥ ♦ t

s ♥ tr♥srr v ♦r ①♠♣ ♥♦t (x y)(v) ♦t tt tr (x y)♥t x ♦s ♥♦t ♣ ♥② ♦♣② ♦ t tr♥srr s s♦ t ♥♦tt♦♥ (x y)♦s ♥♦t ♠♣② tt tr s ♥♦ tr♥sr

♦r s♦♠ ♥t sq♥s S1, S2, . . . , Sk tr ♦♥t♥t♦♥ ♥ t ♥♦rr s ♥♦t ② S1 · S2 · · ·Sk ♦r st S1S2 . . . Sk ♦r ♥② ♥t sq♥ S

♥ ♥② ♣♦st ♥tr l t sq♥ Sl s t sq♥ ♦t♥ ② r♣t♥l t♠s t sq♥ S ♥ t♦♥ t ♥♥t sq♥ Sω ♥♦ts t ♥♥tr♣tt♦♥ ♦ S

♦♥ ♦♥r♥ ♦ t r♥ Pr♦t♦♦

♥ t ♦r♥ ❩rt t ♦t♦♥ ♣r♦t♦♦ ❬❪ tt ♦♥sr ♥♥t ♦♦ss ♠♦♥ t ♥ts ♥ ts r♥ t ♦♥ s t ♠♦st ②t♦ ♠t BS ♥ ♥r tr ♥ tr♥srs ts s t♦ t ♥ ts ♣♣r ♦s t♦ s t ♠♦ t ♣rs ♦♠♠♥t♦♥s ♥ ♦♥trst t♦ t ♠ts ♦♠♠♥t♦♥s ♣♦ss ♥ ❩rt ♥ t ❩rt Pr♦t♦♦ ❩P♦rt♠ ♣rs♥t ♦ s rstrt rs♦♥ ♦ t ♦r♥ ❩rt♣r♦t♦♦ ♦r s ♥② ①t♦♥ ♦ ❩P s s♦ ♥ ①t♦♥ ♦ t ♦r♥♣r♦t♦♦ t ♥♦♥ ♦♥r♥ ♦ ❩P ♥♦s t ♥♦♥ ♦♥r♥ ♦ t ttr

♥ ❩P t stt ♦ ♥ ♥t x s ♥ ② ♥tr rs accumulationx

♥ distancex ♥ rr② ♦ t s valuesx ♥ ♥ ♥tr ♦♥st♥t decay

tt s t s♠ ♦r r② ♥t ♥tr rs r ♥t② st t♦ 0 rr② valuesx ♦s ♥t② t ♣r♦ ② t s♥s♦r t♠♣rtr ♦r rtrt ♦r t s ♦ s♠♣t② ss♠ rst tt t ♠♠♦r② ♦r ♥t s r ♥♦ s♦ tt t ♥ st♦r t s ♦ t♦trs s ss♠♣t♦♥ ♣r♥ts ♠♠♦r② ♦r♦s r♥ tr♥srs

♥ ♦rt♠ ♥ ♥ ♥t x ♠ts BS ts r accumulationx s♥r♠♥t ♥ distancex s rst t♦ 0 ❲♥ ♥ ♥t x ♠ts ♥♦tr ♠♦♥t ts r distancex s ♥r♠♥t distancex ♦♠s rr t♥decay accumulationx s r♠♥t ♥ distancex s rst t♦ 0 ❲♥ ♥

❲ ♦ ♥♦t ♥ t t②♣ ♦ ts rr②s ①♣t② ♥ ♦tr ♦rs ss♠ tt ♥ts ♥ ♥♦♥ O (n) ♠♠♦r② s♦ ♦♥ ♠♠♦r② s sss ♥

♦r ♦♥ ♦r♦ ♣r♦♠s ss♠ tt t accumulation rs r♣r♦② rst t♦

♥t x ♦s s♦♠ s ♥ valuesx ♥ ♠ts ♥♦tr ♠♦ ♥t y accumulationy s strt② rtr t♥ accumulationx t♥ ♥t x tr♥srs ts s t♦ ♥t y ♥ ♥t ②s tr♥srs ts s ♥ t ♠ts BS

♦rt♠ ❩rt Pr♦t♦♦

♥ x ♠ts BS ♦

x tr♥srs valuesx t♦ BSaccumulationx := accumulationx + 1distancex := 0

♥ ♥

♥ x ♠ts y 6= BS ♦

accumulationx < accumulationy ∧ valuesx s ♥♦t ♠♣t② t♥

x tr♥srs valuesx t♦ y♥

distancex := distancex + 1 distancex > decay t♥

accumulationx 6= 0 t♥

accumulationx := accumulationx − 1♥

distancex := 0♥

♥ ♥

t ♣♣rs tt ♥♦t ①t♦♥s ♦ ❩P ♦♥r ♥ ♥ rt t♥ ♠♦ ♥ts t♦t r ♥ r t♦ BS

♦r♠ ♦♥ ♦♥r♥ ♦ ❩P ♦r ♥② ♣♦♣t♦♥ A ♦ n ≥ 4♥ts ♦r ♥② decay ≥ 1 tr ①st ♥♦r♠ ♦r t♠ t♦r cv ♥ ♥①t♦♥ ♦ ❩P tt ♦s ♥♦t ♦♥r

Pr♦♦ ♦♥sr ♣♦♣t♦♥ A ♦ n ≥ 4 ♥ts ♥ ♦♥st♥t decay ≥ 1 ❲rst ♥ s♣ sq♥s ♦ ♥ts

U1 = (1 BS)(2 1)

V = [(2 3) . . . (2 n − 1)] · [(3 4) . . . (3 n − 1)] · . . . · (n − 2 n − 1) ♠♦ ♥ts ①♣t ♦r ♥t 1 ♠t ♦tr ♦♥

W1 = (1 2) . . . (1 n − 1)♥t 1 ♠ts r② ♦tr ♠♦ ♥t ♦♥

U2 = (2 BS)(1 2)

W2 = (2 1)(2 3) . . . (2 n − 1)♥t 2 ♠ts r② ♦tr ♠♦ ♥t ♦♥

Z = (3 BS) . . . (n − 1 BS) ♠♦ ♥ts ①♣t ♦r ♥ts 1 ♥ 2 ♠t BS

❲ ♦♦s ♥ ♥tr g s tt g · (n − 3) ≥ decay + 1 ♦ s S s ♦♦s

X = U1 V g Wg1 U2 W

g2 Z

S = Xω

② ♦♥strt♦♥ ♥ X t ♥ts ♠t ♦tr t st ♦♥ ♦r ♥②♠♦ ♥t x ♦♦s cvx = cv = |X| t ♠♣s tt S stss t

♦r t♠ ♣r♦♣rt② Prs② cv = g · (n−3)(n−2)2 + (2g + 1)(n − 2) + 3

❲ ♠ tt t ♥t v ♦ ♥t 2 s ♥r r t♦ BS ♦s tt ♦♥sr t ♣♣♥s ♥ t sq♥ X s ♣♣ t♦ ♥ ♥t♦♥rt♦♥ C0 r♥ U1 = (1 BS)(1 2) ♥t 1 rs t ♥t v

♦ ♥t 2 r♥ t sq♥ V g ♦♥② ♥ts 2 t♦ n − 1 r ♥♦ tst t ♥ ♥t 1 st ♦s v ♥ ♦♠s t sq♥ W

g1 ♥t 1 ♠ts

r② ♦tr ♠♦ ♥t g t♠s ♥ ♥ts 2 t♦ n − 1 ♥♦t ♠t BS ②ttr rs accumulation q 0 ♥ ♥t 1 ♥♥♦t tr♥sr v t♦ ♥② ♦t♠ ♥ t♦♥ s♥ ♥t 1 s ♥♦ ♥ g · (n−2) ≥ decay+1 t♥s t♦t ♦ ♦ g ♠t♥s t ② ♠♥s♠ ♦ ❩P ♠♣s tt t t ♥♦ W

g1 t r accumulation1 ♦ ♥t 1 qs 0

r♦r r♥ U2 = (2 BS)(2 1) ♥t 1 tr♥srs v t♦ ♥t 2 ♥ Wg2

♥t 2 s ♥♦ ♥ g · (n−2) ≥ decay+1 ♠t♥s t ♦tr ♠♦ ♥tst tr rs accumulation q 0 ♥ ♥t 2 ♣s v ♦t ttt ② ♠♥s♠ ♠♣s tt t t ♥ ♦ W

g2 t r accumulation2

♦ ♥t 2 qs 0 ♥② r♥ Z ♠♦ ♥ts x 6∈ 1, 2 ♠t BS ♥♥r♠♥t tr r accumulationx ♦r♥② r♦r t ♣♣t♦♥♦ t sq♥ X t♦ ♥ ♥t ♦♥rt♦♥ C0 s t♦ ♦♥rt♦♥ C1 ttstss t ♣r♦♣rt② P ♥ s ♦♦s

♥t 2 ♦s ts ♥t v

accumulation1 = accumulation2 = 0

∀x ∈ A∗ − 1, 2, accumulationx = 1

♦ ♣♣② X t♦ C1 t t ♥ ♦ U1 ♥t 1 s r v r♦♠ ♥t2 ♥ stss accumulation1 = 1 r♥ V g ♠♦ ♥t x 6= 1 s♥♦ ♥ g · (n − 3) ≥ decay + 1 ♠t♥s r♦r t♥s t♦ t ②♠♥s♠ t t ♥ ♦ V g t ♥ts ①♣t ♦r ♥t 1 trr accumulation q t♦ 0 ♥ r♥ W

g1 ♥t 1 ♥♥♦t tr♥sr v

t♦ ♥② ♦tr ♠♦ ♥ts ♥ t♦♥ t ② ♠♥s♠ ♠♣s tt tt ♥ ♦ W

g1 t r accumulation1 ♦ ♥t 1 qs 0 ♥ s

tt t s♠ r♠♥ts s ♥ t ♣r♦s ♣rr♣ ♥ ♣♣ t♦ tsq♥ U2 W

g2 Z tt ♦♦s s t ♣♣t♦♥ ♦ t sq♥ X t♦ C1

s t♦ ♦♥rt♦♥ C2 tt s♦ stss t ♣r♦♣rt② P♥ ♥♦ ♠ttr ♦ ♠♥② sq♥s X r ♣♣ t ♥t v ♦

♥t 2 s ♥r r t♦ BS ⊓⊔

♦ ❩rt Pr♦t♦♦

♦ ♥sr t ♦♥r♥ ♠♦② t ♦rt♠ ② ♥sr♥ tt ♠♦♥t tt tr♥srs t t♦ ♥♦tr ♠♦ ♥t ♥ ♥♦ ♦♥r ♣t t♦r ts ♣r♣♦s ♦♦♥ r activex ♥t② st t♦ true tt♥ts tr ♥t x s t ♦r ♥♦t ♥ ♠♣♦s tt ♦♥② t ♥ts♥ r s ♥ ♥ t ♥t s tr♥srr ts s t♦ ♥♦tr♠♦ ♥t t ♦♠s ♥t ♦r♠ sr♣t♦♥ ♦ ❩P s ♥ ♥♦rt♠

♦rt♠ ♦ ❩rt Pr♦t♦♦

♥ x ♠ts BS ♦

x tr♥srs valuesx t♦ BSaccumulationx := accumulationx + 1distancex := 0

♥ ♥

♥ x ♠ts y 6= BS ♦

accumulationx < accumulationy ∧ activey ∧ valuesx s ♥♦t ♠♣t② t♥

x tr♥srs valuesx t♦ yactivex := false

♥

distancex := distancex + 1 distancex > decay t♥

accumulationx 6= 0 t♥

accumulationx := accumulationx − 1♥

distancex := 0♥

♥ ♥

♦♥r♥ ♦ ❩P

❲ ♥♦ s♦ tt ♥② ①t♦♥ ♦ ❩P ♦♥rs ♣r♦♦ rs ♦♥ tt tt t st ♦ t ♥ts ♥♥♦t ♥rs s♦ tt t s♦♠ ♣♦♥t ♦ ♥②①t♦♥ t r♠♥s ♦♥st♥t r♦♠ tt ♣♦♥t tr s ♥♦ tr♥sr t♥t♦ ♠♦ ♥ts ♥ s♥ ♠♦ ♥ts ♥t② ♠t BS t♦ t♦r t♠ ♣r♦♣rt② s r ♥t② r

♦r♠ ♦♥r♥ ♦ ❩P ❩P ♦♥rs

Pr♦♦ t E ♥ ①t♦♥ ❲ ♥♦t ACT (k) t st ♦ t ♥ts ♥ tkt ♦♥rt♦♥ ♥ E sq♥ (ACT (1), ACT (2), . . . ) s ♥♦♥♥rs♥ts t s ♥t② ♦♥st♥t ∃k0 ∈ N,∀k ≥ k0, ACT (k) = ACT (k0) trt♥r♦♠ t k0t ♦♥rt♦♥ tr ♥♥♦t ♥② rtr tr♥sr t♥ t♦t ♥ts trs t st ♦ t ♥ts ♦ rs s♦ ♦r♥

t♦ ♦rt♠ tr ♥♥♦t ♥② tr♥sr r♦♠ ♥ t ♥t t♦ ♥♦tr♥t ♥t ♥♦r r♦♠ ♥ ♥t ♥t t♦ ♥ ♥t ♥t ♥ ♦tr ♦rs♦♥ t st ♦ t ♥ts r♠♥s ♦♥st♥t tr ♥♥♦t ♥② tr♥srt♥ t♦ ♠♦ ♥ts ♥ ♠♦ ♥ts ♠t BS ♥ t ♥①t cvmax

♥ts t s r ♥t② r ⊓⊔

❯♣♣r ♦♥ t♦ t ❩P ♦♠♣①t②

❲ ♦♠♣t ♥ ♣♣r ♦♥ t♦ t ♥♠r ♦ ♥ts ♥ t♦ ♦t ts t t s stt♦♥ rst ♥ t ♥♦t♦♥ ♦ ♣t

♥t♦♥ Pt ♦♦ ② t E ♥ ①t♦♥ ♥ v ♥ t s②st♠ ♣t ♦♦ ② v ♥ E s t sq♥ ♣♦ss② ♥♥t♦ ♠♦ ♥ts tt sss② rr② v

♦r ①♠♣ t x1 ♥ ♥t ♦s ♥t s v t s ♣♦ss tt x1

tr♥srs v t♦ s♦♠ ♥t x2 t♥ ♥t x2 tr♥srs v t♦ s♦♠ ♥t x3 ♥② rs v t♦ BS ♥ ts s t ♣t ♦♦ ② v s x1x2x3 ♦ttt t♦t t active r ♥ ❩P ♥t x1 ♥ ♥t x3 ♦ t s♠

♦r♠ ❯♣♣r ♦♥ ❩P ♦r ♥② ♣♦♣t♦♥ A ♦ n ≥ 3♥ts ♦r ♥② ♦r t♠ t♦r cv ♥ ♦r ♥② decay ≥ 1 ♥② ①t♦♥♦ ❩P ♦♥rs ♥ ♥♦ ♠♦r t♥

∑

x∈A∗ cvx ♥ts

Pr♦♦ t E ♥ ①t♦♥ ♦ ❩P ② ♦r♠ E ♦♥rs ts r ♥t② r t v ♥ ♥t ♦ s♦♠ ♥t x1 stt v s t st r ♥ E ♦♥sr t ♣t π ♦♦ ② v ♥ E ts ♦ t ♦r♠ x1x2 . . . xk ♦r s♦♠ k ≥ 1 xk ♥ t ♥t tt rs v t♦BS ♥ ♠♦ ♥t ♦♠s ♥t s s♦♦♥ s t tr♥srs s♦♠ s t ♥ts ♣♣r♥ ♥ π r r♥t ♥ 1 ≤ k ≤ n− 1 ♥t ①t♦♥ E ♥ rtt♥ s t ♦♦♥ sq♥ ♦ ♥ts

E =[

. . . (x1 x2)(v)

]

︸ ︷︷ ︸

e1

[

. . . (x2 x3)(v)

]

︸ ︷︷ ︸

e2

. . .[

. . . (xk−1 xk)(v)]

︸ ︷︷ ︸

ek−1

[

. . . (xk BS)(v)]

︸ ︷︷ ︸

ek

. . .

ssq♥ ei strts tr t tr♥sr ♦ v r♦♠ xi−1 t♦ xi ♥ ♥s tt tr♥sr ♦ v r♦♠ xi t♦ xi+1 t t ♥ ♦ ek v s r t♦ t sstt♦♥ ♦r 1 ≤ i ≤ k−1 t ♥t ♦ ei s ♣♣r ♦♥ ② cvxi

sxi ♦s ♥♦t ♠t BS ♥ ei t t ♥♥♥ ♦ ei xi s r v ♥ tr♥srst t♦ xi+1 t t r② ♥ ♦ ei ♥ t♦♥ t ♥t ♦ ek s ♣♣r ♦♥② cvxk

s tr t rst ♠t♥ ♦ xk t BS ♥ssr② ♦rs ♥ trst cvxk

♥ts tt ♦♦ t r♣t♦♥ ♦ v s ♦♥sq♥ t v sr t♦ BS ♥ ss t♥

∑

x∈π cvx ≤∑

x∈A∗ cvx ♥ ♦tr s rr ♦r v E ♦♥rs ♥

∑

x∈A∗ cvx ♥ts ⊓⊔

❲ r♠♥ t rr tt ts s ♥ s ♦ ♥♦tt♦♥ rr t♦ t♦♥

♦r ♦♥ t♦ ❩P ♦♠♣①t②

♦ ♣rs♥t ♦r ♦♥ tt ♠♦st ♠ts t ♣♣r ♦♥ ♦ t♣r♦s st♦♥ ♦r t s ♦ rt② ss♠ ♥♦r♠ ♦r t♠ t♦rcv ♥ t ♣♣r ♦♥ stt ♥ ♦r♠ ♦♠s (n − 1) · cv ♥ tsq ♥ ①t♦♥ tt ♦♥rs ♥ (n − 2) · cv s ♦s t♦ts ♣♣r ♦♥

♦r♠ ♦r ♦♥ ❩P ♦r ♥② ♣♦♣t♦♥ A ♦ n ≥ 4 ♥ts♦r ♥② decay ≥ 1 tr ①st ♥♦r♠ ♦r t♠ t♦r cv ♥ ♥ ①t♦♥♦ ❩P tt ♦s ♥♦t ♦♥r ♥ strt② ss t♥ (n − 2) · cv ♥ts

Pr♦♦ ❲ ♦♥sr ♣♦♣t♦♥ A ♦ n ≥ 4 ♥ts ♥ ♦♥st♥t decay ≥ 1t g ♥ ♥tr s tt g · (n − 3) ≥ decay + 1 ❲ ♦♥sr ♥♦r♠♦r t♠ t♦r cv t ♦ s ♥ tr

❲ ♥ ①t♦♥ ♥ t ♥t ♦ ♥t 1 s sss②rr ② r② ♦tr ♥t ♦r 1 ≤ k ≤ n − 2 ♦♥sr sq♥Ek ♦ ♥t cv ♥ t v s tr♥srr r♦♠ ♥t k t♦ k + 1 ♥♥♦tr sq♥ ∆ ♥ ♥t n − 1 rs v t♦ BS ♥ ss ♥ ♥♥t sq♥ s♦ ♦♥sr r♣t♥ ♣ttr♥ Ω ♥ ♥ s S = E1E2 · · ·En−2∆Ωω t② s ♥ t ♥t♦♥ ♦ tsq♥s Ek∆ ♥ Ω s♦ tt t s S stss t ♦r t♠ ♣r♦♣rt②♥ t v s r t t ♥ ♦ ∆

♦r ts ♣r♣♦s ♥ s♣ sq♥s s ♦♦s

♦r 1 ≤ k ≤ n−1 U(k) s sq♥ ♦ ♥ts ♥ t ♠♦ ♥ts①♣t ♦r ♥t k ♠t ♦tr ♦♥ ♥ ♠♦ ♥t ①♣t

♦r ♥t k s ♥♦ ♥ n − 3 ♠t♥s ❲ |U(k)| = (n−3)(n−2)2

♦r 1 ≤ k ≤ n − 1 V (k) s sq♥ ♥ ♥t k ♠ts r② ♦tr♠♦ ♥t ♦♥ ❲ |V (k)| = n − 2

♦r 1 ≤ p ≤ q ≤ n − 1 Bpq = (q BS)(q − 1 BS) . . . (p BS) s sq♥ ♥

♥t x r♦♠ q t♦ p sss② ♠ts BS ♥ ts ♦rr ❲ |Bp

q | = q − p + 1 ♦r 1 ≤ p ≤ q ≤ n − 1 Cp

q = [(q q + 1)(q BS)] . . . [(p p + 1)(p BS)] s sq♥ ♥ ♥t x r♦♠ q t♦ p ♠ts ts sss♦r x + 1 t♥BS ❲ |Cp

q | = 2 · (q − p + 1)

rst ♦♦ t t ♣♣♥s ♥ sq♥s s s U(k) ♦r V (k) rr♣t② ♣♣ ♥ U(k)g ♠♦ ♥t x 6= k s ♥♦ ♥ g ·(n−3) ≥decay+1 ♠t♥s s t♥s t♦ t ② ♠♥s♠ ♣♣②♥ U(k)g t♦ ♥②♦♥rt♦♥ ♦ t s②st♠ ♠s ♥♦♥③r♦ accumulationx t x 6= k rs t st ② ♦♥ s♠ r♠♥t s♦s tt ♣♣②♥ V (k)g t♦ ♥② ♦♥rt♦♥ ♠s accumulationk rs t st ② ♦♥ ♥ss accumulationk

r② qs 0 ♥ ♦tr ♦rs t sq♥s U(k)g ♥ V (k)g ♣ rstt♥t rs accumulation

♦ ♦♥sr ♦♥rt♦♥ ♥ ♦r x ∈ A∗, accumulationx = 0 ♥t♦♥ ss♠ tt s♦♠ ♠♦ ♥t k s tt 1 ≤ k ≤ n−2 ♦s

w ♥ tt ♥t k + 1 s t t ♥ r s ♥ t s s② t♦s tt r♥ t sq♥ Bk+1

n−1 ·C1k = Bk+2

n−1(k + 1 BS)(k k + 1)(k BS)C1k−1

♥t k tr♥srs w t♦ k + 1 ♦r♦r t t ♥ r② accumulationx t x

♠♦ ♥t qs 1 ♥ ♦tr ♦rs ♣♣②♥ Bk+1n−1 ·C

1k t♦ t ♣♣r♦♣rt

♦♥rt♦♥ rsts ♥ tr♥sr r♦♠ ♥t k t♦ ♥t k + 1❲ s♦ ♥ ♦r 1 ≤ k ≤ n − 2 ♥ sq♥ Fk ♦ ♠t♥s

t♥ ♠♦ ♥ts ❲ ♦♥② rqr tt |Fk| = n − 2 − k ♠♣stt Fn−2 = ∅ ♣r♣♦s ♦ t sq♥ Fk s t♦ ♥sr tt t ♥t ♦Ek s ♦♥st♥t ♥♣♥♥t ♦ k ♦ r r② t♦ ♥ t sq♥s Ek

1 ≤ k ≤ n − 2 ∆ ♥ Ω

Ek = U(k)g(k k + 1)Fk︸ ︷︷ ︸

♣r♦♦

·Bk+1n−1C

1k

︸ ︷︷ ︸

♥tr

·U(k)gV (k)g︸ ︷︷ ︸

♣♦

∆ = U(n − 1)g · (n − 1 BS)

Ω = Bn−1n−1C1

n−2 · U(n − 1)gV (n − 1)g · ∆

♥ st cv = |Ek| Prs② cv = g·(n−3)(n−2)+(g+2)(n−2)+2Pr♦♥ tt t s S stss t ♦r t♠ ♣r♦♣rt② s ♥♦t t tt♦s s ♣r♦♦ ♥ ♦♥ ♥ t ♣♣♥① ♦ ❬❪ ♥st ♦s ♦♥ trt♦♥ ♦ t ♥t v ♦ ♥t 1 t C1 ♥ ♥t ♦♥rt♦♥ ♣r♦♦ ♦ E1 ♦♥② ♥♦s ♠t♥s t♥ ♠♦ ♥ts ♥ s♥ ♠♦ ♥t s ts r accumulation q t♦ 0 tr s ♥♦ tr♥srt t ♥ ♦ t ♥tr ♦ E1 t ♣r♦s r♠rs s♦ tt ♥t 1 str♥srr v t♦ ♥t 2 ♥ ♠♦ ♥t x stss accumulationx = 1 ♣♦ ♦ E1 rst ♥s ② U(2)g t t ♥ ♦ ♠♦ ♥t x①♣t ♦r ♥t 2 s ts r accumulationx q t♦ 0 ♣♦ ♥st V (2)g r♥ tr s ♥♦ tr♥sr r♦♠ ♥t 2 t♦ ♥② ♦tr ♠♦♥ts tr accumulation ♥ q t♦ 0 ♦r♦r t t ♥ ♦ E1 ♠♦ ♥ts ♥♥ ♥t 2 tr r accumulation q t♦ 0♥ ♥t 2 ♦s t ♥t v ♦ ♥t 1 s♦ ♦♥② ♥t 1 s ♦♠♥t ❲ ♥♦t ② C2 t ♦♥rt♦♥ t t ♥ ♦ E1

♦s ♦♥ t rs accumulation s tt t ♦♥rt♦♥ C2

s s♠r t♦ t ♦♥rt♦♥ C1 ♥ t s♠ r♠♥ts s♦ tt r♥E2 ♥t 2 tr♥srs v t♦ ♥t 3 ♥ t rst♥ ♦♥rt♦♥ C3 t♥ts tr rs accumulation q t♦ 0 ♥ ♥ t ♣r♦ss ♥ trt t t ♥ ♦ En−2 ♥t n − 1 ♦s t v r♦r t v s r t♦ BS ①t② t t ♥ ♦ ∆ = U(n − 1)g(n − 1 BS) ♥s♠♠r② t t s S t ♦rt♠ ♦s ♥♦t ♦♥r ♦r t rst(n − 2) · cv ♥ts ⊓⊔

♦ ❩rt Pr♦t♦♦

s r② ①♣♥ t ♥♦♥ ♦♥r♥ ♦ ❩P s t♦ t t tt ♥ rt t♥ t♦ ♦r ♠♦r ♠♦ ♥ts t♦t r ♥ r t♦t s stt♦♥ ♦ ♣r♥t tt ♥❩P ♠♣♦s tt ♠♦ ♥t tt

tr♥srs s♦♠ s ♥♥♦t r t s tr ♥♦tr ② t♦ ♣r♥t②♥ ♦ s s t♦ ♠♣♦s tt ♠♦ ♥t r♥ s♦♠ s ♥♥♦ttr♥sr t♠ t♦ ♥② ♦tr ♠♦ ♥t tr ♦r ts ♣r♣♦s ♥ active t ss♦ ♥tr♦ t t r♥t ♥t♦♥t② t♥ ♥ ❩P rst♥♣r♦t♦♦ ❩P s ♥ ♥ ♦rt♠

♦rt♠ ♦ ❩rt Pr♦t♦♦

♥ x ♠ts BS ♦

x tr♥srs ts s t♦ BSaccumulationx := accumulationx + 1distancex := 0

♥ ♥

♥ x ♠ts y 6= BS ♦

accumulationx < accumulationy ∧ activex ∧ valuesx s ♥♦t ♠♣t② t♥

x tr♥srs ts s t♦ yactivey := false ♥t y ♦♠s ♥t

♥

distancex := distancex + 1 distancex > decay t♥

accumulationx 6= 0 t♥

accumulationx := accumulationx − 1♥

distancex := 0♥

♥ ♥

❯♣♣r ♦♥ t♦ ❩P ♦♠♣①t②

♦r♠ ❯♣♣r ♦♥ ❩P ♦r ♥② ♣♦♣t♦♥ A ♦ n ≥ 1♥ts ♦r ♥② ♦r t♠ t♦r cv ♥ ♦r ♥② decay ≥ 1 ♥② ①t♦♥♦ ❩P ♦♥rs ♥ ss t♥ 2 · cvmax ♥ts

Pr♦♦ ♦♥sr ♥ ①t♦♥ ♦ ❩P ♥ ♥ ♥t x t ♥t vr♥ t rst cvmax ♥ts tr r t♦ ♣♦ssts tr ♥t x ♦s♥♦t tr♥sr v t♦ ♥② ♦tr ♠♦ ♥t t♥ ♠t♥ BS t rs v rs♦♠ ♠♦ ♥t y s r v r♦♠ ♥t x ♥ s ♦♠ ♥t ♥♥t y ♥♥♦t tr♥sr v t♦ ♥② ♦tr ♠♦ ♥t ♠♣s tt ♥t y

tr♥sr v t♦ BS r♥ t ♥①t cvmax ♥ts ♥ ss v s r t♦t s stt♦♥ ♥ ss t♥ 2 · cvmax ♥ts ♥ v ♥ ♥② stt s r r t♦ t s stt♦♥ ♥ ss t♥ 2 ·cvmax ♥ts ⊓⊔

♦r ♦♥ t♦ ❩P ♦♠♣①t②

♦r♠ ♦r ♦♥ ❩P ♦r ♥② ♣♦♣t♦♥ A ♦ n ≥ 4 ♥ts♥ ♥② decay ≥ 1 tr ①st ♥♦r♠ ♦r t♠ t♦r cv ♥ ♥ ①t♦♥♦ ❩P tt ♦s ♥♦t ♦♥r ♥ strt② ss t♥ 2 · cv − 2 ♥ts

Pr♦♦ ❲ ♦♥sr ♥ ♥tr g s tt g · (n−3) ≥ decay+1 ♥ ♥s♣ sq♥s s ♦♦s

U = (3 BS) . . . (n − 1 BS)♥ts 3 t♦ n − 1 ♠t t s stt♦♥ ♦♥

V = [(2 3) . . . (2 n − 1)] · [(3 4) . . . (3 n − 1)] · . . . · (n − 2 n − 1)♥ V ♠♦ ♥ts ①♣t ♦r ♥t 1 ♠t ♦tr ♦♥

W = (1 3) . . . (1 n − 1)♥t 1 ♠ts r② ♦tr ♠♦ ♥t ①♣t ♦r ♥t 2 ①t② ♦♥

X = U · V g · W · (2 BS)(1 2)(1 BS)

❲ s S ② r♣t♥ X ♥♥t② ♠♥② t♠s S = Xω ❲♦♦s t s♠ ♦r t♠ cv = |X| ♦r t ♥ts s♠♣ t♦♥

s♦s tt cv = 2n − 3 + g · (n−3)(n−2)2 t s s② t♦ s tt S stss t

♦r t♠ ♣r♦♣rt②♦ ♣r♦ tt t ①t♦♥ ♦ ❩P ♥ ② S ♦s ♥♦t ♦♥r

♦r t rst 2 · cv − 2 ♥ts t t ♥ ♦ t rst U ♥ S ♥ts 3 t♦n − 1 sss② ♠t BS ♥ tr♥srr tr s t♦ t s trs accumulationx ♦r 3 ≤ x ≤ n−1 q 1 ♥ ♦♠s t sq♥ V g

♥ ♥t x 6= 1 s ♥♦ ♥ g · (n−3) ≥ decay+1 ♠t♥s ♥t♥s t♦ t ② ♠♥s♠ t t ♥ ♦ t rst V g r② ♥t x r♦♠2 t♦ n − 1 s ts r accumulationx rst t♦ 0 s ♦♥sq♥ trs ♥♦ tr♥sr r♦♠ ♥t 1 t♦ ♥② ♦tr ♠♦ ♥t r♥ t sq♥ W

tt ♦♦s V g ♥ r♥ t sq♥ (2 BS)(1 2)(1 BS) ♥t 2 rst ♥t v ♦ 1 r♦♠ ts ♣♦♥t ♥t 2 ♥♥♦t tr♥sr v t♦ ♥② ♦tr♥t t BS s ♦♥ ♣rs② cv ♥ts tr r♥ t ♥t (2 BS)♥ t s♦♥ X ♦ S r♦r t v s r t♦ BS ①t② trt (2 · cv − 2)t ♥ts ♦ t s ⊓⊔

♦♥ ♠♦r②

❯♣ t♦ ♥♦ ss♠ tt ♠♦ ♥ts ♥ ♥♦♥ O (n)♠♠♦r② ♥ ts st♦♥ sss t s ♦ ♦♥ ♠♠♦r② ♠♠♦r②s③ ♥♣♥♥t ♦ t ♥♠r ♦ ♥ts ❲ ss♠ ♥♦ tt t ♠♠♦r② ♦♥ ♥t ♥ ♦ t ♠♦st k s t k ≥ 1 ♦t ❩P ♥ ❩P ♥ ♣t t♦ ts s ♥ ♥② tr♥sr ♦ s s ♠t ② t ♠♠♦r② ♥ t tr♥sr ♠② ♣rt r♥ ♥ ♥t s ♠ s ♣♦sss r tr♥srr ♦t tt s r q♥t ♦r t t ♦t♦♥♣r♦♠ ts t s ♥♥ssr② t♦ ♣rs s r t② tr♥srr♥ ♥ ♣t ❩P ♦♥ ♥ ♥t s tr♥srr s♦♠ s ♥ ttr♥sr s ♦♥② ♣rt t ♦♠s ♥t ♥ ♥♥♦t r ♦tr s♦r r② ♥t x t s ② x r st♦r ♥ ②♥♠ rr② valuesx♦s s③ s ♥♦t ② size(valuesx) ② ♥t♦♥ size(valuesx) ≤ k♦rt♠ ♣rs♥ts ♥ ♣tt♦♥ ♦ ❩P t t s♠ ♥ ♣♣t♦❩P ♦r t s ♦ rt② ♦ ♥♦t ♣rs♥t ♥ t ♦ t ♠♥♠♥t

♦rt♠ ♦ ❩rt Pr♦t♦♦ ♦♥ ♠♠♦r②

♥ x ♠ts BS ♦

x tr♥srs ts s t♦ BSaccumulationx := accumulationx + 1distancex := 0

♥ ♥

♥ x ♠ts y 6= BS ♦

count := min(size(valuesx),k − size(valuesy)) accumulationx < accumulationy ∧ activey ∧ count > 0 t♥

x tr♥srs count s t♦ yactivex := false

♥

distancex := distancex + 1 distancex > decay t♥

accumulationx 6= 0 t♥

accumulationx := accumulationx − 1♥

distancex := 0♥

♥ ♥

♦ t ②♥♠ rr② valuesx ❲ ♥♦t ② ❩P rs♣❩P t♦♥♠♠♦r② rs♦♥ ♦ ❩P rs♣ ❩P

t ♣♣rs tt ♦r ♦t❩P ♥❩P t ♣r♦♦s ♥ ♥ t ♣r♦sst♦♥s t♦♥s ♥ r st ♣♣ ♥ t♠♠♦r② s③ tt♥s t ♦♥str♥ts ♦♥ tr♥srs t ♦ ♥♦t ♥♠♥t②t t strtrs ♦ ♦t ❩P ♥ ❩P t st t ♣r♦♦s ♦r❩P ♥ ❩P

♦r♠ ♦♥s t♦ ❩P ♦♠♣①t② ♦r ♥② ♣♦♣t♦♥ A

♦ n ≥ 1 ♥ts ♦r ♥② ♦r t♠ t♦r cv ♦r ♥② decay ≥ 1 ♥② ①t♦♥♦ ❩P ♦♥rs ♥ ss t♥

∑

x∈A∗ cvx ♥ts♦r ♥② ♣♦♣t♦♥ A ♦ n ≥ 4 ♥ts ♦r ♥② decay ≥ 1 tr ①st

♥♦r♠ ♦r t♠ t♦r cv ♥ ♥ ①t♦♥ ♦ ❩P tt ♦s ♥♦t♦♥r ♥ strt② ss t♥ (n − 2) · cv ♥ts

Pr♦♦ t tt ❩P ♦♥rs s t♦ t t tt t st ♦t ♥ts ♥♥♦t ♥rs s ♥❩P ♦♥ t st ♦ t ♥ts r♠♥s♦♥st♥t tr ♥♥♦t ♥② tr♥sr t♥ ♥② t♦ ♠♦ ♥ts ♥ ♠♦ ♥ts ♠t BS ♥ t ♥①t cvmax ♥ts t ♣r♦t♦♦ ♦♥rs

♣♣r ♦♥ t♦ t ♦♠♣①t② ♦ ❩P s ♦♠♣t ② ♦♦♥t t ♣t ♦♦ ② t st r v t ♠♦ ♥ts ttsss② rr② v ♠♠♦r② s③ ♦s ♥♦t t t t tt ♠♦♥t ♥ ts ♣t ♥♥♦t ♣♣r t t♥s t♦ t t active ♥♦r t ttt ♠♦ ♥t x ♥ ts ♣t ♦s v ♦r t ♠♦st cvx ♦♥st ♥tss ♥② ①t♦♥ ♦ ❩P ♦♥rs ♥ ss t♥

∑

x∈A∗ cvx ♥ts

♦r ♦♥ t♦ ❩P ♦♠♣①t② s ♦t♥ t♥s t♦ t s♠s sr ♥ t♦♥ ♥ ♣♣②♥ ts s t♦ ♥ ♥t♦♥rt♦♥ s ♥ ①t♦♥ ♥ ♥t ♦s t ♠♦st ♦♥ s ♦♠♣t t t ss♠♣t♦♥ k ≥ 1 ⊓⊔

♦r♠ ♦♥s t♦ ❩P ♦♠♣①t② ♦r ♥② ♣♦♣t♦♥ A

♦ n ≥ 1 ♥ts ♦r ♥② ♦r t♠ t♦r cv ♦r ♥② decay ≥ 1 ♥② ①t♦♥♦ ❩P ♦♥rs ♥ ss t♥ 2 · cvmax ♥ts

♦r ♥② ♣♦♣t♦♥ A ♦ n ≥ 4 ♥ts ♦r ♥② decay ≥ 1 tr ①st ♥♦r♠ ♦r t♠ t♦r cv ♥ ♥ ①t♦♥ ♦ ❩P tt ♦s ♥♦t♦♥r ♥ strt② ss t♥ rs♣ 2 · cv − 2 ♥ts

Pr♦♦ r♥ t rst cvmax ♥ts ♥ ♥t x tr tr♥srs ts ♥t v

t♦ BS ♦r t♦ ♥♦tr ♠♦ ♥t y ♥ t s♦♥ s ♥t y s t♥ ♥t♥ ♥♥♦t tr♥sr v t♦ ♥② ♦tr ♥t t BS s ♦♥ ♥ t ♥①tcvmax ♥ts s ❩P s♦ ♦♥rs ♥ ss t♥ 2 · cvmax ♥ts

♦r ♦♥ t♦ ❩P s ♦t♥ t♥s t♦ t s♠ s sr ♥ t♦♥ ♥ ♣♣②♥ ts s t♦ ♥ ♥t ♦♥rt♦♥s ♥ ①t♦♥ ♥ ♥t ♦s t ♠♦st ♦♥ s ♦♠♣t t t ss♠♣t♦♥ k ≥ 1 ⊓⊔

♦♥s♦♥

♥ ts ♣♣r st② t ❩rt t ♦t♦♥ ♣r♦t♦♦ ♥ t ♦♥t①t ♦P♦♣t♦♥ Pr♦t♦♦s ❲ s♦ tt t ♦r♥ rs♦♥ ♦s ♥♦t ♦♥r ♥ ss t ♣r♦♠ ♥ t ♣♦sst② ♦r t♦ ② ♠♦♥ t ♠♦♥ts t♦t r♥ t s stt♦♥

♦ ♥sr ♦♥r♥ ♣r♦♣♦s st② ♠♦ rs♦♥s ♦ t ♦r♥ ♣r♦t♦♦ ❩P ♥ ❩P ♦t tt ❩P s ♠t♦♣ ♣r♦t♦♦♥ ♦♥trst ❩P s t♦♦♣ ♦♥ ♥ ❩P ♣♣r♦①♠ts ttr t♦r♥ ❩rt ♣r♦t♦♦ t♥ ❩P ♦r ♦t ♠♦ rs♦♥s t ♦rsts ♦♠♣①t② s ♠ ♦rs t♥ ♦r t ♥r ♦♣t♠ t ♦t♦♥ ♣r♦t♦♦♣rs♥t ♥ ❬❪ ts ♦♠♣①t② s ss t♥ 2 ·cvmin ♦r ts ♣r♦t♦♦ ss♠s tt ♥ t♦ ♥ts ♠t ♦t ♥♦ ♦ t♠ s s♠r ♦rt♠ ❲ ♦ ♥♦t ♠ s ♥ ss♠♣t♦♥ r t ♦♥ ♦ ♦♥sr tt t❩rt Pr♦t♦♦ s ♥ ♣♣r♦①♠t♦♥ ♦ t ♥r ♦♣t♠ ♣r♦t♦♦ ♥ t ♦♦♥ s♥s ♥ ♥t tt s ♠t BS ♠♥② t♠s ♥ t ♣st s ♥tt②t♦ st ♥ ts ♠st s♠ ♦r t♠ ♦♠♣r♥ t s ♦ t♠t♦♥ rs ♥ t♦ ♥ts ♠t ♥ s ♥ ♣♣r♦①♠t♦♥ ♦ ♦♠♣r♥ tr ♦r t♠s s ♣♣rs s♦s tt ts ♣♣r♦①♠t♦♥s ♥ t ♦rst s ♦♠♣①t② s ♦♥sr ♦t tt ♦♣t♠ ♦♥st♦ t ♦rst s ♦♠♣①t② ♥ ♦♥ ♥ ❬❪ ♣rs②

∑

x∈A∗ cvx−2·(n−2)♦r ❩P❩P ♥ 2 · cvmax − 2 ♦r ❩P❩P ♣♦sst sr② t ①t♥s♦♥ t♦ ts ♦r ♦ t♦ ♦♠♣t t r♦♠♣①t② ♦ t ♣r♦t♦♦s Pr♣s t ♣ t♥ t ♣r♦t♦♦ ♥ ❬❪ ♥ t

♣r♦t♦♦s❩P ♥❩P s ♥♦t s♦ r ♥ ♦♥sr♥ r ♦♠♣①t② ♥ ♥②ss ♦ s♦ t t r♦ ♦ t ♠♠♦r② s③

♥♦tr ♣rs♣t ♦ t♦ ♣♣② ♦r ♣r② ♥②t ♠t♦♦♦②t♦ ♠♦r ♥trt t ♦t♦♥ ♣r♦t♦♦s s ♦r ♥st♥ PP ❬❪ ♦r ♦♥② s♠t♦♥ rsts r ♦r ts ♣r♦t♦♦ s s ♦r ♦trst ♥②t ♣♣r♦ s ♥♦t s♣♣♦s t♦ r♣ s♠t♦♥s t ♦s t♦♦t♥ s♦♠ ♥♦r♠t♦♥ q② ♥ t ss ♥st♠♥t

r♥s

♥♥ s♣♥s ❩ ♠ sr ♥ Prt ♦♠♣tt♦♥ ♥♥t♦rs ♦ ♣ss② ♠♦ ♥tstt s♥s♦rs ♥ P ♣s

♥♥ s♣♥s ♥ s♥stt st ♦♠♣tt♦♥ ② ♣♦♣t♦♥ ♣r♦t♦♦st r

♥♥ s♣♥s s♥stt ♥ ♣♣rt ♦♠♣tt♦♥ ♣♦r♦ ♣♦♣t♦♥ ♣r♦t♦♦s strt ♦♠♣t♥ ♦

qr P ♥r r♠♥ ♥ t ①t t♠ ♦♠♣①t② ♦③r♥t t ♦r t♠s ♥tr♥ ♣♦rt

qr r♠♥ é♠♥t ♥ tt♥ ♥ t③♥ s♣ ♥ ♥t♦rs♦ ♠♦ ♥ts ♥ P ♣s

qr r♠♥ ♥ tt♥ sst③♥ tr♥s♦r♠r ♦r ♣♦♣t♦♥ ♣r♦t♦♦s t ♦r♥ ♦r ♦♠♣t

qr é♠♥t ss ♦s③ ♥ ♦③♦② st③♥♦♥t♥ ♥ ♠♦ s♥s♦r ♥t♦rs t s stt♦♥ ♥ ♣s

♥ ❨ ♥ r♦ss♥ ♦r t ♦♥ ♦♠♥ r♦♠ ①♣♦♥♥t t♦♣♦r ♥tr♠t♥ t♠ ♥ ♥ ♣s

♥tr P r♦r♦t ♦t ss ♥ ♦tt ♠♣t ♦♠♥ ♠♦t② ♦♥ ♦♣♣♦rt♥st ♦rr♥ ♦rt♠s r♥st♦♥s ♦♥

♦ ♦♠♣t♥ ♥ ♦ rt♠♦t rss tr r ♦r ②♥ ♥ t♦♠②r ♦♥s♥ss ♥ t ♣rs♥ ♦ ♣rt

s②♥r♦♥② sr ♥ ♥ st③♥ r t♦♥ ♥ ♥t♦rs ♦ ♥tstt

♥♦♥②♠♦s ♥ts ♥ P ♣s rr♦ ♥ ♣♣rt ♥ s♠ rs r ♥q P♦♣t♦♥ ♣r♦t♦♦s

t ♥trs ♥ ♥ ♣♦rt ❨♦r ❯♥rst② ♦♥ ♠ ♥ ♦♥ ♦t♥ ♣r♦r♠♥ ♥②ss

♦ ♠♥r♥ ② t♦r♥t ♥t♦rs s♥ t tr♥t ② ♠♦ ♥♦t②♦s ♣s

P ♥ ❨ ❲♥ rt♦♥♦s P ♥ ♥st♥ ♥r②♥t ♦♠♣t♥ ♦r tr♥ s♥ tr♦s ♥ r② ①♣r♥s t③r♥t ♥ P ♣s

r♥♥s ♦ ♥ ❱♦♥♦ P♦r ♥ ①♣♦♥♥t ②♦ ♥tr ♦♥tt t♠s t♥ ♠♦ s ♥ ♣s

♥r♥ ♦r ♥ é♥ Pr♦st r♦t♥ ♥ ♥tr♠tt♥t②♦♥♥t ♥t♦rs ♦ ♦♠♣t ♦♠♠♥ ②

♥tr♦t♦♥ t♦ strt ♦rt♠s ♠r ❯♥rst② Prss