HAL Id: hal-00639583 https://hal.inria.fr/hal-00639583 Submitted on 22 Nov 2011 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Computing Time Complexity of Population Protocols with 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 of Population 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
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Computing Time Complexity of Population Protocols with ... · Computing Time Complexity of Population Protocols with Cover Times - the ZebraNet Example Jo roy Beauquier 1,3, Peva
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HAL Id: hal-00639583https://hal.inria.fr/hal-00639583
Submitted on 22 Nov 2011
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Computing Time Complexity of Population Protocolswith Cover Times - The ZebraNet Example
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
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
♥ ♥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 ♥
❲ ♠ 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♠
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♥
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
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 ⊓⊔
❲ 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
♦ 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
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 ⊓⊔