Contribution to the study of weighted Petri nets

Transcription

1 Contribution to the study of weighted Petri nets Thomas Hujsa To cite this version: Thomas Hujsa. Contribution to the study of weighted Petri nets. Modeling and Simulation. Université Pierre et Marie Curie - Paris VI, 04. English. NNT : 04PA tel HAL Id: tel Submitted on 7 Mar 05 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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.

2 P é té r t q t r r t q é é t s t tr q P r s Prés té r s P r t r r P t t ès tr t à ét s rés P tr é ér sés s t t r t r sé rt r P r s sé rt r r ss s r tr P r s é tr ér r tr P P r s r t r r r r tr t ès P P r s r s r t r t ès r

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4 r ts t r r r r s t t r r q té r r t t t t s t q q r s t r t q r r t s t t é très s t t s s très r s r r trés t r tr é t tr s é s q t ssé très t t s à r r r t s s r s r r térêt r é à tr r r t ès s st ér é r rt P s é rt t s éq P r rs s r tr r rt s r s tr s éq s t é r r r t r rs té r ss r t r é r r s t r r r r s s rs s P t r rs s s ré t s rs t s ss t s q s s rt é s s r s s ts très r és ré é r t é s rs s r tré s t tt ér èr st t s s r r ré s s t t rt èr t à r s rr r s r 3 3 t r t r r r tt s r s t t r 3 t r t éq é rt t stè s s r r r r é tr ér r r t t rr rés r ê sé r s r t 3 st t r rs rs té r r s s rs s té s t q ré t t rt s sé s r ss t s t s

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6 és é t t t t s t é r s sstè s q s à é é t s r t été é é èr séq t rs è t é s è ssé r s s rés P tr t rt t st r t t ét é r s s t ès rés té r P tr tr t s rés P tr èr à t r è é ér sstè s s r ts rr ts t str és s rés s t s r s rt s q s t r ét ts st t t t t ét t sstè st é r s ét ts q ét t ét t r rés té r t s t s sstè ét t s t rs tr s t s s rés r tt t é s r s é è s rr t t s r s t r s très r ss s rés P tr st s r t t é é é té r r è s rt ts st r s s t s s è r t t t s r s r ss s st st q s q t r s t t é t r r s t s t q s r t t s t s st t t s t s rés P tr r s é t t ér t r s s t s à é r s r t s é s t sstè s q s sstè s rq és t t rs s t r ts s sstè s rq és t rés r r t t s rs t tés s r s ér s t t t s t q t té é r té t s r t s s t t s r s t s s r r étés ss é s s s rés P tr s t é s té ss t é r r é ss s s ss tè s ré s t s t s é t r s t s t é ér t ût s t t r ss té r t r r à r ét t t à rt r t t ét t ss r r été rr s t s s rés P tr st é ré rs té r rs t s s rés P tr s t s r rés t t s é s s èr str t é èr s r s rs ét t sstè s t str t s r t r t s rq s r r étés s rés é t s é t s ss s rs rq s r rés t t s rs r s s s s s r r s r s tr s t s t s t r t s s s r r s r s r s à s s rés P tr s t tés à é s t t s rq é s r tt t r t r s r r étés

7 rt t s r q s s ès s t ût s s q s t é ér t t é s st r r tt r é t s s t s t r r ss té s rés P tr t tr rt rt té s t t s r è s ss té t té s t P P r s rt à t r r r è r s tr r ss té t té st s t r é é t t ét à s s s ss s rés s r st q s s t ss réq t é s t r tt t ès st r r s r èr s ét s s q ss r t té t ré rs té s rs s s ss s rés P tr érés ss és é ér sés s t q t sstè s rq és s è s s t s s r s rès ér ts è s s t s à s r s r s t s r s r s t rs r t s t é ts s t sés s q r s rés P tr t s t s tr è rés P tr s tt t ès s r s s r s ss és Pr ss t r s t très rt r ss té t t é s r s t s t à é s t s r ss t s ét s s r s s té r st é t st s ss tr r r s r té s rs t s tés t t t s t è s q t r ss té s s t té s s rt à r s t s rq é s t s s r s s s r r s s t és t rs t t s é s tr s t rs s rs s q r ssè é r t t r q t r t q r t r s r s s tré s tt t s é s s é t s sé s s r s tré s t q s è t ê rés t t s t r ét r s r r t r A s é s à A s x = 0 q s s s t st é s s r (A,A ) s t r s à A 3 q s s s t st é s s r (A,A 3 )

8 x = 0 A A x <> 0 A 3 t r A s x = 0 rs r é à A s r é à A 3 r r t s t r A s r s r s t t t t t r s r s t tr t r t ss rs t s q s t s r s s s é t t t s s t s è s t r s t s rr t s é té s s r s r t t r s r è s s t été t sés s r s s t s s t t é èr s r r s s r s t q r tt t t r s é s tré s q r ssè é r t t r t tr r t r s s t r ss té très ré t s s s r r s s t t é r t q t té é s èr t séq sstè rt t ré r ré st ss s és s r ss té è t t r s r s ét s s t s r ss r r té t é r s t s rs é s s r s à s é t r tt r r rés t r 3 t rs q q t s rs t r é r s s s t t é r t s q t tés s é s q é s r s s s r s r s s s rs t r A t r é s s r (A,A ) t t r A st t r sé à é rr r rsq s 3 é s s t rés t s s r A A A 3 5 r r r s t 3 t rs

9 s rés P tr s rés P tr s èr t très t s r é s r s sstè s à é é ts s r ts ts s r s t s t rr s t rt r r ss t rs r s r r étés rt t s s t é s tr r t r s q t è ttr t P r rs s t é s r s r s r ç t s t rs r s tr s t s t s rs r s s q st stré r r s s ss s rés P tr érés rr s t à str t r s r s st s sstè s à s 3 B 3 A A 5 5 A 5 A 3 A 5 B 3 B 3 3 A 3 r r s r st r rés té r rés P tr s r r t s t s rt t s s rés P tr à s s t s r é s r s t rs s s q s sstè s rq és s s r t ss é s t s t t r t r ss r s r q ts s s t t s q r ss t réq t s s sstè s r s r s s rés érés s s r tt t r rés t t s t sstè s t rs s t r st s q s tér sés t tés s rèt s t q t ér r r r été r ts r è s q s s r ts t r st é ér t st t é tr s é é ts s s s s st t s tr sstè tr s rt t t q t sstè é s r ss s tr st séq ér t s t é r r t r t r ss r st rt s q t r s r ts r r r tr s rt s r ss s tr t r tr t ét t r ê s r ss r s ré t é t t s s s s r t P r t r r r été té ss r t r s q r s s s r t t rés r ss té t r ts à r q r r r été ré rs té r t t st rt t q P r s ét s s t rs s t rés tés s 3 t r s t s s

10 s sstè s rq és s sstè s t rés r r t t s rs t tés t s é r r é s s t s s t é t r s t s t ût s t t rs r ss té r t r r à r r t t t tt r r été t r s s r ts ér q s t t t r ré ér t sstè rès rr r s q ét rt t sstè t é t s s r r s s t tés t t t t êtr é sé s r s tr s t s t rs t r t s r s s t r s à s s s r rés t t s q t tés é s s é r t s rés r t t t s s t tés st rt t q st r sé r r r été té ss té r t r à ét t t à rt r t t ét t ss t r s t s ré t s t sstè é t ré rs té s r s t été t sés t r s s é s s s sstè s rq és r é s r s t s rs s t t s r r s rq és tr t t s P r r s r s ét t s r s t s t rs s s s r tt t s s r t t s t rt t q t s t été tr ts s s s ù s s t t sés r é s r r é s s s t érés t s r é s r s t s t r P rs s q r r é 3 3 t s s tr t rt è s tr t t s s r s s t ss t sés s t r t ç r ré rt r r s r r r ss rs r è s s été r P s rs ét s s é t té t été r sé s r s r s r tt t 3 3 s t rt s ét s s q t s t é t s q é ér t t r t s s rés té s s t s s sstè s r r été ré rs té st t r té t é r r é r s q st s s r s ss s s r s r tt st t t t ét r r été ré rs té s s t ès té s s ss s s r ss s r r r ss rs r t r

11 s s s ss s rés P tr rt t s r s t s s tt t ès s rés t s s rés t ts r r t rt t s rs s s ss s r ss s s rés P tr é ér sés q t r s t s rs é r s t s rs t rs ê s rs r tt t r ét r s s s s s t t s r ss s sstè s q t q t t s sstè s r r ttr t t s sstè s ss é ér s r t s sstè s t ss s r s s ét s r rs ss s sstè s r q st s s s ss s sstè s q t s sstè s q t tt ss st é t t q r str t str t r ss t èr s rés P tr érés s t q s tr s t s t tré ssè t s ê s tré s t q t s s s s rt ê s t é t r q tt tr t é té s s s rt s s st é é é té s sstè s r t s sstè s s sstè s r t s s ss s sstè s q t s q q sstè ssè s s rt s s s sstè s ssè t s tré t s rt s sstè s r t s s ss s sstè s r s sstè s r s tré s t t sstè tt ss q tr s t ssè s sstè s r ttr t s t t sstè tt ss q tr s t ssè s tré t q ssè s s rt s t t rs t s ss s r t r s ér t s ss s s t stré s r s t r sstè à st q t r t sstè st r

12 r rés s r st sstè r ttr t rés st r rés s r r t st sstè s tr t s à ét s rés P tr é ér sés s sstè s t été r t t sés r é s r s t s r q é s t s rs s t tés t s s s st q é s t r ss té s è s t rs êtr ét t st é ér s r s rés t ts s t sstè s èr à tr t r s t s s s t r s t s s t q t r s ét s s s P s ré sé t s s s s s r s t s s s sstè s t q s t str t t s r ss s str t r t t s q ss r q t r s s rs s s r r s à é r r s ê r r tt t r ét r s t s q s sstè s r t q t s sq s s t és rs s é t s s ss s s t s és t é t s é s tr r tr t tt t ès t s q tr é t r s s rés P tr à s s q s s s ss s ét é s s tt t ès t s r s rés t ts rt ts ss és s s t s s s s s r ét rés t str t r ss r st rq t t t r t t é r r é r t t rq t t str t r st t r é r s tr t s s t s t ré rt s s s tr s t s tr s rés t s s tr s r t s rés P tr ét r r été té st ét é s tr t s rés t ts s r r r été ré rs té r t s t tr tr st t tr s t s rs rs t s s ét s rès s tr t s q t t t tr s t s é s t t st tr ét é s tt

13 t ès s t s t st t q ét té s s tr s r t s s tr P t s t r s s r r étés rt t s s rés P tr s t ût s é é s r3 s tt t ès s s s s s r s r è s té t ré rs té s é s r r t r è té st P r t t s q té r è ré rs té st s èr à s r ét rt t sstè t êtr tr s r é t t t rés r t s r r étés rt s s st s t t é s r sstè tr s r é r s s tr s r t s st t rt t r t t s t é ér t é s s rè s q s q t à s t s s é q s t r q ré t s t t r t è à s t s s s t s s t s r s s s t t s s rè s ré s t s té s s rés t s s rs s tr s r t s s t s r s rés P tr érés é ér sés q rés r t rt s r r étés str t r s t rt t s s t s s r r étés s sstè s Pr èr t s é ér s s s ét s q r t ré é t é é s s t t s sstè s é ér sés r tt s tr s s s s rés P tr r t r tr s r t q s q à t s s rés P tr t q t s s t rq t é st t r s rs t q r rt s r t s str t t s t s tt ér t rés r t str t r r é t séq s r r étés té t ré rs té s ét s t t s t s r tt t ré r rq t t t rés P tr t t rés r t s tr s q s t s t s sstè à s r q t s r r à t r s t rs tr s t s t st t r s q t s r s r s s t r t tt s é ss s ér t éq r é q st s rt r s s q t à t t sstè s r t t t à sstè t s r t t r t t t s st r t rsq r t s st st t s sstè ét r r étés rt t s st r t s é tt ét s q rt r à t s s rés P tr r és r s s t s r t s s s r s s t r s t s sstè s t r s ér t

14 éq r ér t r s t t tr t s r tt s rt à q sstè r sé t ssè r r t t s s s s r ts s t s ét s té s tr s r t s s t rés té s s tr t P r s t s rés t s tr s r t t t rés P tr éré sstè r éré ss q é ér s s q t rés r t t str t r r é é st ét r s r s q ts tr s t tr s t s èr éq t t s t s s rt à q rt t s t rés r é t s ss ét r s r r étés sstè t str t r st s s q sstè t r é r s r r étés sstè t s tr s s tr s r t s tt q s q à t t ss tè r r é r t r sstè r é q rés r rt s r r étés str t r s t rt t s s s s s st t t rés r é s tr s r rs s sstè t s sstè r t tt tr s r t r ttr s t ét rt t s r s s tr s s ts t st rés té s tr t ét té tr P t s t s tr s s s s s r s sstè s r t q t érés r és r s q s r rt s rs s s ss s r t t t s sstè s r r ttr t t s sstè s s r tér s t s r ss s t s st t r t t s s ss s r r s r è é s r ss t s è s t s st rés t s q rés r é été ç é st str r rq t t t ès r ss q st t rq t s s s r tér s t s t s t s té s t s r s r ts r és à s té é s té r t éré st q st rt rt t é r è r s q st r êtr r3 st s t r 3 s séq tt té s r à t t s s ss s s r s t s r s s rq s t r s rs ss s r ss s rq ssè r t s t st é t s r st s ét s q str s t s rq s t ts r s sstè s q t t rt rs s s ss s r t

15 t s t t s t r s s s sstè s érés st s t s s s t s s r tt r t s sstè s r r és st t sq à rés t ét é é s t s s s tr s r t s tr r t r s r èr s t s s s s t s té r s r t q t r és tr st t s ré é t s ét s t s ttér t r s rés t ts r r tt t str r s sstè s ts r és t t r s r t t s t s t t r r èr t s s t té q s q r éq rés s sq s q t té t t t s r st r t s é s s s rq s ts r t s s r r és s r s s t s t é rè s s r t té s q t r és s t s s rs t é rè é s t r str r s r rs rq s tt ss q t s sstè s r s s t s s tr s r t s tt r é r tr t s s t té r s r r és s q rs t s tt t t s r s s s ét s t s tr s q s t s s t s é ss r s s s éré s rés t ts s t rt rés tés s s t s t ét ré rs té tr P t s t s tr s ét s r r été ré rs té s s t ès té s s sstè s rt t s r és q t t r s r t s s s ét s ss s sstè s q t q s t s é ss r t r és s tr s s ét r é s t s s s t s s té r s sstè s q t t r r és r tt t str r t s sstè s r ss s à s ts t r és s q s r r étés t s té t é r r é s t ss ré s ré rs té st r r r été à êtr s éré r st s t r q s r s t s ré s s s q sstè st ét t t ré rs ét s r ss té t êtr r t s é s r r étés té t ré rs té s t s t t é s st s sstè s ts q s t s ré rs s t s sstè s ré rs s q s t s ts r t ré rs té é à été ét é s s t ès té t

16 é r r é s s rs s s ss s rés P tr érés s s s sstè s ré rs té st t r té t é r r é tt t st s r r s sstè s r ttr t r r st s sstè s r ttr t t r és q s t s ré rs s s ét s s str s t s rq s ts r s sstè s r tt t ré rs té st s t ss ré r tt ss s r q s ré rs s s q r tér s t ré rs té st t r s sstè s r ts r és r t t s rq s ts ré rs s s t str ts t s t t s r t t s t st rq s ré rs s été r é s s sstè s q t ts r és r q r tér s t ré rs té t ét é é r str r s rq s ré rs s s t é s Pr èr t s rés t s s r rs rq s t à s ts t ré rs s r ss s sstè s r ttr t r és rés t t st ét é s tr t s t s ét s ss s é ér s sstè s r r q s rés t s r tér s t ré rs té r é s t t é s s s r rs rq s ts ré rs s t s t s rq s tr és r s s ss r ttr t s s ss tr t s s t r té t ré rs té s q rs t s tt t tr s r t s tt s r s t s s t s s rés t ts s t tés s tr t P r s t s ét s ré rs té s sstè s q t ts s s t ts tr s tr s t t r q tr ét é s t é é r s sstè s r t êtr ét à ss s sstè s q t t s t à é r r èr r tér s t tr ré rs té r s sstè s q t s sés ts s st ts t s sstè s r r è s tt r tér s t ét t q t é é r s rt r s sstè s r r s rés t t è r rs rq s ts ré rs s r s sstè s q t r és t s t s s s s ét s r t r s r rs rq s ts ré rs s r s r r s sq s tr t é é té st s sé s s ét s r ss té s è s t s s s s t s ré s t té é r r é t ré rs té t êtr ss ré s t

17 s rs t s tt t ès st s rt s q st s t s r st t rt s Pr èr t té r è té r t éré st s t st é r è t r s q st r êtr r3 st s t r 3 s s s q st é t é r t s t s s s t s s té t ré rs té è t s r tér s t s tr s ré rs té r s r érés è s st t s r tt ss r t r r t s r s t s rés P tr s tt r t s tr s t s q tt t q t t s rs t s tré s t s t s t s s t s r s s tr t r s tré s tr s t s r s t s st s q tt ss s t é ss r t q str t r t rt t s sstè s r s t r s s s s rés t ts str t r s t êtr é ts té t rt t s sstè s r st s èt t îtr sé s ét s s s s ss s s r s s r tér s t s s r ss t s r été tr é s r t s s rés P tr érés t r tér s t st r s rés q t t rés t t s r été t r s r à s ré t tt q st r t êtr ét é P r s t té t ré rs té t êtr ét é s s tt ss s r ss r t r é t s t s s s t s s s tr tr s r t rés P tr érés r érés rr t t r s r r t t é ér s t s r r étés tt s s ss à t s s rés P tr Pr r étés s rés P tr t r sés s t s ré s t s t t s q st é sé s rt s t s s s rés P tr t s st é r t s è é t s r r s sstè t êtr ét s r s s é t sstè st t r s sstè s à s r t t êtr ét s à s ss s s r s q s s t é s t s sstè s r s s ss s q é s t s s t q s r t t êtr ét é s èr s r tr r é t s r r s t ss s r st s rés P tr st st q s r r té sstè s ré s t êtr s ss s rs s t s ét s s t s t été ré s s s rés P tr s t s s éq ts t t rés r t s r r étés és ré s é ss t s ss s éré s q s t s r ss s q è sstè à s s té t t s ét s st t ss s ét r st s

18 é r s rt é s r s t s sés s r t r P t s t s é ér s t s s t s s r s P s rs t q s t r t é é s r rs q s st t à r r s s rt rs r rés s s rè s s é s q rés r t s r r étés s P r s r s ét t è t s t rs s rs s s s s t q s t r t ê ét é s s t t s rés P tr 3 r r t t t s s t q s s t ès t t r t êtr ss t s s ér t s tés st s t s t r t té t r s s r s r r P s t t s ss r s s ss s t P tr ts r s t s t st s s r tt t st t s ét t s tr s r t s tr tt t ès é r t s rq s ts tr t r ss t s r rs rq s ts ré rs s r s sstè s r és r ttr t tr ér s t r t s té t r 3 3 r r s s r ss t st t t r r s t t s t t r s st s r tt t r t s r èr s t s s s t s s té r s r s q t s r s t s s ét r s tés q s t s ss r t té st ss rés té r s s s r r P t t s ss r t r r st s r s t t t r t r t rr t st

19 s s r tt t rés t s tr s r t s à t s s rés P tr à s rés r t s r r étés rt s tr tt t ès t r t s r èr s t s s s t s s té r s sstè s r t r r és tr s s r s r r t rs t t r st s r s t t t r t r P tr ts s r r r tt t ét r r été ré rs té s sstè s ts r r és P r tt ss r tér s t ré rs té t s r èr s t s s s t s té t ré rs té s t rés té s tr tt t ès tr s r t r s sstè s r r és sstè s r és é r r rt t tr

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22 str t r sst s t s t r sst s sst s t P tr ts r t s sst s t r r t t s s t s st s t s t s sst s s t s s t t r s r rt s ss st t s t ss t t t t s t r t ss s r s t t t sst r r r t s t t r ss r s r rs t s st t 3 t s s s t s r rt s s r r t s st st t s t t t s s r r ss s ss s t P tr ts t rst r t s t t s r ss ss r rs t r t s ss s t s t s s t t r t t s rst r s r t tr s r t s t t r s r str t r r r rt s t P tr ts s t st t r r s tr s r t s t t s r s ss s t P tr ts t t r rt t r t s ss t r P tr ts s str t r s r s t st t r s t s tr s r t s t st t ss r q t sst s r s t rst s t t r t s r t r sst s s r r t s s st t s r s t s r s t t s ss r r r q t sst s r sst s r r r t s r t t s t tr s r t s s r s r t st t r rs t r rt r t ss ss t r s t rst r rs r s r r r ttr t sst s r r t r 3 t r rs t r r r sst s r r r rs t r s r t s ss s s t s t t ss r rs t s r t rst r t r 3 t r rs t r s st t q t sst s t rst r rs t r s r r q t sst s str t s r s r r r sst s s t s s ts s r r ss s ss s t P tr ts t t r s r t t s r t s ss s r t rst t t s t t s r t ss r rs t r rt s r r t s t s r s s t r sst s

23

24 t ts tr t t t t t t t rt st s t s t P tr ts r r t r s ts P tr ts tr t s t s s r 3 t t P tr ts t r r ts r s sst s r s q s r t r s tr t r r r rt s s ss ss r ss r s r rs t s s st s r t ss t t s t t s ts s s r 3 t s tr s t s P s ts P s sst s s s q s t t s max p gcd p t r s t ss P rt r s ss s s tr ts r ts q t ts s ts r ts ts r ts ts r ttr t ts s r t t s ss s t r ss ss t s r sst s q t sst s s ss s

25 t ts P t r t r 3 t s r ss r r ts q t ts tr t r r r ts s P t tr s r t s t P tr ts sst s t Pr rt s t s r t t Pr rt s P t r 3 t ts r t t s t s r s r t t t r sst s s tt tr s r t t Pr rt s s P s t t s ss r r r q t sst s P s t t s ss r r r sst s r ss s t t r r sst s s s r r r sst s P r s r r r sst s Pr s r s ts t ss q t sst s r t r 3 t ss r r sst s ss t t sst s P r s r r q t sst s P r s s r 3 sst s t t s ss r r sst s r s ts t ss s t t ss s tt t t s t t r s t ss t s s t t s r t ss r s

26 t ts rs t q t r sst s rs t r sst s Pr s r s ts Pr r t s r s ts P r s r r r ttr t sst s r s t t s s q s P s sst s ss r t r rs t r r sst s r t r 3 t r rs t r r sst s P r s r r r sst s s t t ss r rs t rs t s st t q t sst s rs t s t r rs P ts s q s r rs t s sst t t s t r rs s s q s t ss r t r 3 t r rs t t t r rs t sst s P r s r r sst s rs t r sst s P r s rs t s q s r t r ss s s s rs t s r

27

28 Chapter tr t t ts t t t t t t rt st s r s r s t t t t t s t P tr ts t s t P tr ts t r st s sst s r r t r s ts P tr ts tr t s t s s r 3 t

29 t r tr t t t t r rt t t r s r t t st s s t s t st t r t t P tr t r s s t rs t t s t r r st r s t s P t s s r s t r P tr tr t s tr s t s ts P tr ts r r t t r r s r t rr t str t sst s s ts r r t rt t r s t st t s s r s t t s st t rs tr s t s t t r t rr ts s r 3 t s s s t r st t s s r ss r s s t r t ts r t t t r st r s r st t s s P tr ts s t t r s r st st r ss s r r r t s st t s st t s P tr ts r s t r t t t s t s t r r t r sst s t s r ts sst s t r sst s sst s st t t r r s t t r s r r t t t r t t r t s s t t r r t s r t r t s rr s r rt s P tr ts r ss ss r r r sst s t t st t 3 t s s t t ss t t r t r t t r t st t r r st t ss t r rt P tr ts s r rs t P tr ts t s str t t r t s t s t t r s t t s t t r s s t s sst s st t r t t t P tr t s s t r s t st t r t r t r t s t r Pr rt s P tr ts r t t t ss t s t r r s r s t rs s s r r s tr s t s t r t s s r r s r s P tr ts r s t t t s s r t t s r r rt s r s t s t t s s s r s t st s t s t t s r t r r s r t rt r r ss ss P tr ts s t t st s s t r t ss r s t rs r P P r r r t t t s r tr t r ss ss t s t s r r str t t r r s ss s r st s r s s s s t t s t s s s t r t rst t t s t t s r t ss t r rs t s r r ss s ss s t P tr ts t t r t s sst s

30 t t t rt s tr t t r s r 3 s s t r s t t st t t rt r r s s t t r t P tr t t t s t P tr ts s r t r s ts P tr ts t t t r ss t t r tr t t r 3 t t t s s t t t rt st s r r t r s s r s r s t r s r t t s s r t t s t s ss t P tr ts t sst s t t r r r r t P tr t r s r s r s str r ss r t t s r t t r r s s s ss t s t ss r s t s t ss t t t t rs P r s t s t t t s s t t s t r ss ss t tr t t r r t t t s r r Pr ss t r t s r r s r t rs s t t s t t r t rs r rt t s t t s t t r s t ts t t s r s ts t s sst s t r t r s t t s r t r t t r t r s r r s t r t r r ts t s t r t t s r t t r A s s t t s t A x = 0 s t r st r t r (A,A ) t r s t s s t t A 3 s t r st r t r (A,A 3 ) x = 0 A A x <> 0 A 3 t r A x = 0 t s t t t A s s t t t A 3 r r t s t t r A

31 t r tr t r s t t t t r s r s t ts t s t t t t s s s s r s t s r rr t t s t t r r t t r s s t t t s s s P rs tr st t r s s r r str t r ss ss s r r s r r t t t t t r t t r t s r r r s t st r t r r r s q t t sst s r r r t r r r r s ts t 3 t rs t t t t r t r t rs r r t ts t t ts t r s t t s rs t r A t r t s t s s t t s t t r (A,A ) t r A s r st rt t st 3 t t s r r s t t t r 3 A 5 A A r r s t t 3 t rs r ss ss t s s t t t t s t t t s r t ss t t r s 3 s ss r r ts t r tt t s t P tr ts P tr ts r s t s r t t sst s ss t ts s r 3 t rr r t r str st t r s t r r rt s r r t s r3 r t r t r t rs t tr s t s rs t s r str t s t s r rt

32 t t t rt 3 3 A A 5 3 A A 3 5 A 5 B B 3 3 B A 3 5 r t t s r r s t t P tr t t r t P tr ts s t t s t st t t t r t sst s r r 3 t r s r sst s s s s t t t s q t st t s s sst s ts t t s r s t s t s s st t P tr t r s r s t r t st r P tr t t r r t r t r t str t r r t s ts t s t P tr ts t P tr ts r r t r st s sst s ts ss t s t r r t r s r s s t t s t t r r q t t s sst s r r r r t r r t ts t ts r t r r s t t t sst t r st s t r sst s s r t r t sst s t r t t s t r s s r t t t s t r sst rr t r r t r t s t t s r t s s r s t r rts s t s r st t s t t tr s rt sst s sst r r ss P s s q r t s r r r r t t r r t sst r s r s t t sst t t s t r t r tr s rt

33 t r tr t rr t r r ss s t r s s t r s r s ss s t t r t r rs r t ss r rt s r s t t r t r r t r ss r s r s r t t ss t t r ts t t r r rs t r rt s r s t st r r rt r r t r sst s r st 3 t r s ts sst s sst s t r s r t r t t s r t t r r r t t t st t 3 t s t t ss t r t r t t r t r t s t s r rt s r s s t t rr r r r r s s r r t t t s t t tr s t s t r t r t s s t r s ts t r s r r s t ts r r r tt t r s r t t t s s r r q r t t t s r 3 s t ss r rt ss t t r t r t t t st t r r r st t r s t t sst s r rs t r s s r rs t sst s r r s t s t Pr ss P t s P r r r s t s s s r t s r t t s t r tr s s r t r s r r r s t P t s rs r t 3 3 s s t rt t s t t t P sst s s r s s r s t t r s t r t ss P r Pr ss r rr PP r ss rs r r t t s r ss t r s r r s r tt r s 3 3 tr st t st t t s s st s t t r 3 3 r t s sst s t r rs t r rt s t ss ss r rt s r s s t t s r r r ss s r s t t t s t st t r rs t r rt r r ss ss s r t ss ss t r r r ss rs r t r

34 t t t rt r r t r s ts P tr ts r s t r t st t t rt r r r ss t t r r rst r ss r s t s t r t r s t r s st s ss s t P tr ts t t r r r ss t t sst s t s r r ss t t r s t s t str t r r r rt s r t t ss ss s s t r rs t t ss t r s ts r ss t s r t st t r P tr ts ss r s P tr ts r s sts t r t r sts r st t r t t s st t t t t r tr s t ss r s sts P tr t s t t tr s t s r t r t r st s r r r t tt t r st rt r rt r t r s t r t r r s r t t s r r r r r rt r r r s r s t r r r st t r t t r s r s s t r rs t r rt r s t r r P tr ts r s r t t tr t s tt r t t ss r t r s r P P r t r r rs t r s r r r t s s r t t r t r s r ss s ss s r r t t t t st t r st r s r r s ss s st t sst s s sst s sst t r t t t t t s s tr r tr s t sst s sst s s s t r s s r t ts t s r t r s t r sst s t t r ss ss sst s s t s r ts t r sst s t r sst s s t t str t r tr s t t ts sst s t t t s r r ss s r t t r str t r r rt s r r ttr t sst s q t sst s t t r ss r str t s t t r sst s

35 t r tr t r ss r ss s r s t t t r sts t st t 3 t t t sst s t t s r r s r t ss r rt t s r s s r t ss r rt str t r t s s st s r t ss r ss r t s r r ss s r ss s r t r 3 t r s r s ss s t r r t P tr ts r ss r r r ts r str t r r r ss t t r ts t ts t s s s s t s r q t ts r 3 t r ts r r r r ts r s r t r t ss s t r st t ss r r s t t s r ss s t s t r t r 3 t s r t ss s st t s t s s t r q t ts t t r s ss s r ss s t r r r ss t s t t t 3 t s r ss ss t r rt s ss P t r t r 3 t s ss r t r t ss s r t r ts t st s r s r t t t t r r s r3 st s t s s t t t ss r t s ss t t r t t r t s st t t s t t s t s s s s q s t s s r s r t s r t s t s r s r t r sst s r tt t r sst s s r r ss s st t s r s r s t t r t s r t r t t s r t t r sst s ss r t t s t t s r ss ss s st r t r r s ts ss r t r 3 t s ss r s r r r ss s r t s ts r t s r str t s r ss s s t rst r t r sst s r s r rs t r s r r s t t r r r r t r t t s s t t s tr t t sst t tr s t s s r rs t r rt s t r q r t s t t st r s s s r s s t st t r t r s s r t t r r rs r s r t ss r r s r t s r rs r t t P tr ts t ss t t t

36 tr t s t s s r 3 t s r rs sst s r t ss ss t s s t sst s r t r rs r s t r rt r t ss s s s t q t sst s s r s r r r st s st t t r t r 3 r r sst s r r t r t s t t r s t r t t s t st r st t ss s r ttr t sst s t s r t t st t r s rs rt r s ss s ts r t r r str t ss t r t r ts t t t ss r s t t s 3 t st t s s t r r s r t s r t t r t s tr s r t s str t r t q s r s r t s t t t t r s r r r t r str r rt s s t t t s s st s r t tr s r t tr s r t s r s r s t t t s r tt r s t r t s 3 t t t t s t t s rt t rt t r t tr t r t q s st t t t t t str t r t t r t t r t s r s t r str t r s ss s t s t t r s r rt r ss s ts r t rst t r s tr t s t s s r 3 t s r t sst s s t t s r s t t s r s st t r ss ss s t t t s t s t r 3 r s ts t s sst s r r t tr t r t s s t t r t s s t s r r s s t s s t sst s s t t r str t r r ss str t r s r t r q t ss s s r s r r s t r t t s r t t r str t s r t t r s s r t r q t sst s t s t t r r s ss t s r t t t t s r tr t s r s r t

37 t r tr t r s r t s r s t s r t tr s r t s r t P tr ts t t r s r t st r r rt s rst r 3 t s t t r t t t t sst s r tt s tr s r t s t P tr ts s t ts t t r r s r r ss s t t t s t s t r t t r r s r t s s q s t tr s r t rt r s t t s t r t s t r st t t t r sst s t st r r rt s s t t r t P tr t tr s r t t r sst r t t r t r ss r s t t s tt tr s r t t t s t r r sst t t r sst t str r rt s t t ss r rt s t tr s r t s t t t rst s t t s ss r r r q t sst s r rt r r s t r t s t r t t s t s rst t s t t t t s t r sst s r t r s r r r sst s s s t t r r r q t sst s t t rst r s r t s ss s t r sst s r r s t s tt tr s r t t t r s t t ss r r sst s s s t rs t t r t r t t s s r s ts t sst s t t r s s 3 t t str t tr st t t r r t r t s t t r t r t t r rs t r rt r s t t rst r rs t r s r t ss r ttr t s sst s t st t ss r r sst s r r s t r t r 3 t r rs t t rst r s s t r r t r ttr t s ss s r t r r s t t ss r rs t r r r sst s s s t t t s tt tr s r t r t s t s r t r st t r rs t s st t q t sst s s t t r t s t t r sst s t t t s r r ss ss s t t t s s t t rst tr r t r 3 t r rs t

38 tr t s t s s r 3 t r s st t q t sst s ss r sst s s r sst s s t r 3 s r t r 3 t r rs t t t s r r sst s s r s t s t t rst r s r r q t sst s s t s t s t t t rst r rs r s r r r sst s t t str t s r s t t r ss ss t s s r t s s ss r rs t s r t s s r 3 t rst t r r t r s t P tr ts t t s s s r rt t r s ts r r t s ss s st t s t s s tr s r t s r t s t t r st t ss r rt s t t r t r s ts t r rs t r rt r r s t t r t r r s r s s r rs t s

39

40 Chapter t P tr ts t ts t r r ts r s sst s r s q s r t r s tr t r r r rt s s ss ss r ss r s r rs t s s st s r t ss t t s t t s ts s s r 3 t s tr s t s P s ts P s sst s s s q s t t s max p gcd p t r s t ss P rt r s ss s s tr ts r ts q t ts s ts r ts ts r ts ts r ttr t ts s r t t s ss s t r ss ss t s r sst s sst s q t sst s s ss s P t r t r 3 t s r ss r r ts q t ts tr t r r r ts s

41 t r t P tr ts t s t r r t r s t P tr ts ss t s r r t r str t r r s r t t s t t s t t ss r s t t ss s P tr ts st t t s t s s q t r ts s s s t r s ss s t r r t s s r t r s r ss r t s ts t r r q t r ts ts t s s t r s t t r s t P tr ts t r r t t s r t ts t r s ss r r ts r t s r t t r s r s q s t r r t s s t r r ts t t s tr N = (P,T,W) r t s ts P T r t s t T t s tr s t s P s W : (P T) (T P) s t t P T s t s t t ts t t r s r s t r p t tr s t t r s t tr s t t t p W(p,t) > 0 r s t W(t,p) > 0 r r t s t s t t W t s s {0,} tr t N = (P,T,W) s tr s t tr C s p P t T, C[p,t] = W(t,p) W(p,t) r t t st r s 0 t t W r r s t t tr s t tr s Pre Post s s p P t T Pre[p,t] = W(p,t) Post[p,t] = W(t,p) s q t t tr s C = Post Pre r t t s r tr s C t s tr α s s t r s β s s t s t C[α,β] t r st s tr C s t s t r α t β r s t t t x P T s t s t {w W(w,x) > 0} t x t s r s s t E P r T E = x E x st s t t t x P T s t s t {y W(x,y) > 0} t x r E = x E x

42 r r s ts t t t ts tr t t t s p 3 t p t 3 t p p p p 3 t t t r t t ts tr r s t p s {t,t } ts st s t s {t 3 } r s sst s r s q s r t r s r M t N s M : P s s t M t t r s ts r t s M(p) r p P sst s (N,M 0 ) r N s t M 0 ts t r r t r s t sst r M t N s tr s t t T p t,m(p) W(p,t) r 3 t s ts s t A tr s t s s M r tr s t A s M r M s p P t p M(p) W(p,t) r 3 t s ts s t A s s M r A s M r M t r M t r tr s t t t M t M s p P,M (p) = M(p) W(p,t)+W(t,p) r s q σ t n t s t tr s t s T s {,..., n} T r tt σ = t t t n s q s t ts s t t r s q σ = t t t n s s t sst (N,M 0 ) t s ss r s t t M t 0 M M tn M n r s t t M i s t tr s t t i r σ i {,,n} t M 0 Mn sst s t s t ts t s s q s r M s s t r r t r M t r sts s r s q σ s t t M σ M s t r r s r M s t [M P r t r σ : T ss t t t s q tr s t s σ s r tr s t t T t t r rr s t σ

43 t r t P tr ts r t s t sst S = (N,M 0 ) t R(S) s t s t t r s r S r t r sst S = (N,M 0 ) t RG(S) s r t r t r (V,E,v 0 ) r V s t s t t r s R(S) t r t v 0 s t t r M 0 E = {(M,t,M ) M,M V M t M } s t s t r s t r r r t ts s ss r r s r t r s t r t r t r p 3 p t t 3 t p t 0,0,,0,,0,0 0,, t t 3 t t t,,0 r t sst S = (N,M 0 ) s t r M 0 q s (0,0,) tr s t t s t r s t t 3 r t s q σ = t t s P r t r s σ = (,0,0) s s S s t t r r (,0,0) r t r S s t r t r t tr t r r r rt s t s s t r s t t r rt s P tr ts t t s t r t t s t s s s ss ss s s t t tr s t s t sst r t t s s s t r sst s r r r s t t ss ss r t r rt s s r t t tr s t s sst s r t t t r r t s r s s r rt s r s r r t sst S = (N,M 0 ) s r r r M [M 0 r r tr s t t t r sts r M [M t

44 tr t r r r rt s S s t r sts t r k s t t t r t s r s k r k M [M 0 p P, M(p) k. S s k r p P k max{m(p) M [M 0 }. r M s r t N t sst (N, M) s s t ss t r r r rt s t sst S s t s s r rt s r r s t r ss s s r t rs s t r s t r r r r t t t s t t r r t t s 4 rt r t s st r t s t t t 3 3 t 4 t 3 r s sst s r s r r r s tr s t r t s t s r r s t tr s t s t t 4

45 t r t P tr ts r ss str t r t N st t s r t st t r M 0 s t t (N,M 0 ) s N s str t r r M 0 sts s t t (N,M 0 ) s N s str t r t sst (N,M 0 ) s r M 0 N s r t s str t r str t r t st r t r s r t s str t r s t r sts r r t s t t s r t t s str t r s ts r t s r s st t r r t r t r r s r s q r s r rs t r s r t t r r r r r M s r t sst (N,M 0 ) M [M 0,M [M sst s r rs ts t r s r s q t s t s t t sst s r rs ts r t r s str t t rt r s t r t sst s r rs s t r t r t s t s t s str t r r t t r s r rs t r t t t s r rs sst r s r rs r t 4 4 p t 3 t p 3 p 4 4 t 3,,0 t t 3 0,, t 3 t t,, 4,0,0,0,,,0,3,0 0,4,0 0,0,4 t t t t t 3 t 3 r t st r t s r r t t t r t s ts r t r t s t t r s str t r rs t s r t s r rs s ts r t r t t r t t t r t st t s str t

46 tr t r r r rt s s s st s r t ss t t P t s t s T t s t tr s t s t t s r t I(V) t s t t s t t r V s rt t r V t V s s t r st s s t I(V) ss t t 3 r ts V t t i V V[i] 0 i I(V)\ V V[i] = 0 s r rt r t r r t rs t tr C t t s s s t t P s s t r X P s t t t X C = 0 s s t r Y T s t t C Y = 0 P s s t r t st s r ts ts s q t ts s rt s t r r s rs t t s rt t r P s s t s t s s t n t t r s 3 n s ts r q t s r t ss s st r rt s r s s s t tr C t N rt r s s N s s r t P s X P sts r C s t t X P N s s st t s Y T sts r C s t t Y T t r s s r t s st t t p 3 p p 5 t 3 t 4 t p p 4 t = = t r s t t s s r t t t t r (,,,,) s P s ts ts r s st t t r t t r t (,,,) s s ts ts r

47 t r t P tr ts t t s t t s t s s t r s t s t r t s t t s r t P tr ts rst s s r s r t t t r ss ss t s s s tr s t s s s q s s s rt r s ts t s rt r s sst s s s ts s r tr s t s r t t s r r t ts r s t t t r rt r str t t t t ts s s r s rt r t r t ss ts s s r 3 t s tr s t s r ss ss P tr ts s ts r t r 3 s t t t tr s t s t t s r t s t t t r t s s t r str t r t r rt t s t t s tr s t s t r t tr s t s t t r r t t t s t t st t t t tr s t s t s P tr ts t s r 3 t s t t s s t t tr s t tr s t s tr s t t t st t t s t s s s r 3 t s t s r str t r t t t p p p r t t tr s t s t t r t p s t r t t s tr s t s r 3 t ts t s p p P s ts P s sst s s s q s s q σ s s s q t s q σ σ s t r σ r s tr s t s σ r str t σ t t s t T T tr s t s s t s s q σ s tr s t s t T t σ T r t r str t t s q σ = t t t 3 t t t s t {t, t } s t s q t t t

48 t t s t t s t N = (P,T,W ) s P s t N = (P,T,W) P s s s t P T = P P W s t r str t W t P T r t t N = (P,T,W ) s s t N = (P,T,W) T s s s t T P = T T W s t r str t W t P T sst S = (N,M 0 ) t N = (P,T,W ) s P s sst S = (N,M 0 ) N s P s t N ts t r M 0 s t r str t M 0 t P M 0 = M 0 P r s s t P s sst s sst r t sst S = (N,M 0 ) t N = (P,T,W ) s s sst S = (N,M 0 ) N s s t N ts t r M 0 s t r str t M 0 t P M 0 = M 0 P r s s t s sst s sst p 3 p 5 p 3 p 5 t p t 3 t 4 t p p 4 t p t 3 t 4 t 3 t p p 4 r P s sst s t sst t t r t r t r t t st P s sst s t s s t s {p, p, p 3 } r t st P s sst s t s s t s {p 4, p 5 } p 3 p 5 p 3 p 5 p 5 t p t 3 t 4 t p p 4 t p t 3 t p p 4 p 4 t 4 r s sst s t sst t t r s t r t t st s sst s t s s t tr s t s {t, t, t 3 } r t st s sst s t s s t tr s t s {t 4 }

49 t r t P tr ts t t s max p gcd p t max N p t t t t p t t N gcd N p t r t st s r t t t ts p t t N s r t t s max p gcd p r s s s ss r t max p = gcd p = t p s s ts t t ts r q t,t p W(p,t) = W(p,t ) t N s s r p N s s r t r s s t r t s t t r s t t s s r s s r s s t t st t t tr s t t t t tr s t r t r s s r p t r p s s r t ss t s t t s t r tr s t s t st t

50 P rt r s ss s P rt r s ss s t s s t r s t t s ss s t P tr ts st t s t s s s tr ts tn = (P,T,W) s s tr p,p P p p p p rp p r t t ts t r s tr ts r ts t N = (P,T,W) s r t,t T t t t = t q t t t s r p,p P p p p = p r r ts r s r s r r ts r st t r t s r ts s t t r ts r s t r t st r t s t s t s s s ss t r ts t s t s s s r t r ts t s str t r r tr r ts t r s r t ss t r ts s s t r r r ts st t t r r s ss s tr ts r t r s t t r ts t t r t s s q t ts s ts t N = (P,T,W) s q t t r tr s t s t t N t t s tr t t p P,W(p,t) = W(p,t ) s q t s t s t t q t ts r s t r ts r t s q t t t q r t q ss s r r t s t X s q r t t s r s tr tr s t q t a,b,c X a a r t a b t b a s tr a b b c t a c tr s t t q ss x r t [x] s s [x] = {y X x y} t r s ts q r t t tr s t s t

51 t r t P tr ts t t 3 p 3 p t t 3 p 3 p t t 3 p 4 3 p r t t t s q t t t t = {p,p } = {p } = t t t s t q t r t s s s tr t t r t t r s ts t tr s t s r q t s t t s t r r t s t q t s t s t s t t t ts p r t q t q t r t r s r t N = (P, T, W) tr s t s t t T r q t r t Pre[P,t] = Pre[P,t ] P t s q r t t s t tr s t s q ss s q t s t s t q t s ts t N s t E(N) t t q t s t s M t st tr s t t s s t s M r ts ts t N = (P,T,W) s t r t s t st t t tr s t t t p P p r ts r r r s ss q t ts t s r t N = (P,T,W) s t t s t st t tr s t t t p P p p ts r r r s ss r ts r s s r t t r ts ts t N = (P,T,W) s t r t tr s t s t st t t t t T t s r ts r r r s ss t r ts t s r ts s r 3 t s r t s

52 P rt r s ss s r t t r t t t s t t t t r t s t r ts r rt r q t ts t s r t N = (P,T,W) s t t tr s t s t st t t t t t T t t ts r r r s ss r ts r t r s t r t r t t r t t t s t t t t r t s t r ttr t ts r ttr t t r t s t r r t t s sst s s r 3 t s s r t tr s t t s r t s ts r t ts r s t s r t t tr s t s ts r t ts rs ts tr s t s s r t ts s s s ts r r s r t s t ts ts s s r ts s s ttr t s str t t t s t ss ts

53 t r t P tr ts r t r s r ttr t t r r ttr t t s r ts tr s t s r t ts s r t t s ss s r r r s ts t s r t s t t s ss s t P tr ts s r t s t s s t P tr ts q t s tr r r r ts r r ss s s ss s t sst s

54 t t t s r rs t r s tr s r s t tr s t s tr s t s t s s tr s r t ts t tr s s t tr r r ss s r ss s r s sst t r t t t t t s s tr r tr s t sst s t ts r r t r r r ss ss t s q t r sst s r t r t r t t t s s t r t s t s r sst s sst s s s t r s s r t ts t s r t r s r s r t t s P r Pr ss s PPs t t r t s t r r t sst s t t t t r r ss s r s ts r sst s t t r str t t r s r t s t t t t t t r r rs PPs s r3 sst s r s s ss t P tr ts r r s r r r t s r r s s r t r r s t s r s r r t s s r r ss rs s r s t s t t r s r 3 t ts t r ss rs t t t t s tr t t s t t r r tt r t t r s r s t r rts t t s s s ss t r t r r r r s r s t r s r r t s t r r t s t t r t t rr s t s s t t s t r r s s s r r sst s s r r s t P tr t r

55 t r t P tr ts w w w 5 w 6 P P P 3 P 4 w 3 w 4 w 7 w 8 r t s s r r s t m r s r s r r ss rs r r t t t s r s t r t ts t ts r r r tt t q t sst s s ss s t s r s t t sst s r t s t r s t ss t t r t s r s r t s s str t r r t t sst t q t s r str t s t t r P t r t r 3 t s r ss t s s t r t r t r 3 t s r ss r t s ss s t P tr ts st t s t s s rst r s t r t r 3 t r ss t t r r ts t r t r q t ts r 3 r ts s r ts r r ts t t r r ss s ss r s t t r ss r r r ts r r t s r rt t Pr s t r ss r s t t N s t str t r r r t r rt s N s s st t s r t N s r r q t r r s st s s r t ss str t r ts s r t ss s s st str t r ts r s s r r t

56 tr t r r r ts q t ts t r t r 3 t r ss r q t ts s r t Pr s t r ss r t N str t q t t r s t s r q t N s r N s s st t s r t t r ts tr s q t E(N) r t r s r q t t str t t s r s t p p 3 t t 4 p t 5 p p 4 t 3 t 3 3 t t 0 00 = = r s q t t N s s st t tt q t s r t t q t q t s ts r {t, t 4 } {t } {t 3 } {t 5 } t s E(N) = 4 r r t r ts tr q s 3 = E(N) s N s r r s t r s t t r ss s ss r t ss r q t sst s Pr s t r q t sst s r tr t r r r ts t s s t r r rt s r r t str t r r q t ts t r r s ss s s r r ts t r s t st t s t st q s r r ts s str t r

57 t r t P tr ts Pr s t q s r N = (P,T,W) s r str t r t t N s q s Y r r t s rt Y s t s t T p 3 p 5 t p t 3 t 4 t p p 4 p p p 3 p 4 p 5 t t t 3 t = r s t r t s str t s st t t s r Pr s t s s t (,,,) r s t t t s ts t t r ts s ts t s ts r s r t ts N N t t r s ts t N N = (P,T,W ) t N = (P,T,W ) s t t N = (P,T,W ) s t t P = P P T = T T t t t W r ts t ts t r s W r W r 3 t t s t C s ts t C = {C,...,C k } s t C t t r s t t C \{C } t N s r s t C s ts N s t C s t t r s r q t sst s r s 3 r r sst s r t s t t t r str t r r st rt r s sst s t r s t s s t r P s ts t t st r ss r ts Pr s t tr t r P s ts r s r str t r r t N t q s Y N = (P,T,W ) s str t P s t N t N s r t q s Y Y T s t Y r r N s r s P s ts s r s t s str t r r t P tr ts t s rt r s ts ts t ts r t P t N t N s str t s r t r P s t N t t N t N s str t s st t r s t N

58 tr t r r r ts p 3 t p t 3 t p p p p 3 t t t 3 [ ] 0 0 [ ] = t 3 t 4 p 5 p 4 [ t 3 t 4 ] p [ ] 4 p 5 [ = 0 ] 0 r r str t P s ts t r r t r s s s Y = t (,,,) r s s t r t r t t r tr r str t Y t t s t {t, t, t 3 } s Y = t (,,) s t (,,) t s t rst P s t r str t Y t t s t {t 3, t 4 } s Y = t (,) s q t t s t s P s t t s r t s t t t P t q t t s s r t t r s t st t s t t r q t ts r r P ts ts t s t r r t s t s s Pr s t r s q t ts r s r r q t t N N s r t s t ts t t s rt t ss t q s s t P ts t t s rt t ss t q P s r t r s r s t t t r r r t N s r P ts r ts s r r s t s r r r str t r r rt s r r ts r r t r rt s r r ts t r r r ts ts s st t t

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60 s

61

62 Chapter 3 P t tr s r t s t P tr ts t ts sst s t Pr rt s t s r t t Pr rt s P t r 3 t ts r t t s t s r s r t t t r sst s s tt tr s r t t Pr rt s s

63 t r P t tr s r t s t P tr ts r r rt s P tr ts r tr t s r3 t s t s s s t ss r rs t r s r t r r ss r s P P r t t t t r rs t r s t r r t s t st t r t sst t tr s r r s r t r r t r str r rt s s s r r rt s s t t tr s r sst tr s r t s st rt t rt t r t r r s r s t t t s r tt r s t r t s 3 t t t t s t t s r s t t s t s r s t r t s s t t s t r r s t s r t tr s r t s r t P tr ts t s t rt r s ss t P tr ts t r r sst s t r sst r s r str str t r r rt s t r tr s r t s t t r P tr ts r s r t s r s r t s r s q s t r ss r 3 t s t t r t t t t sst s r tt s tr s r t s r s t s q s t r t s t st r r rt s s t r s r 3 s s rst t t s P tr t t r s t t t s t tr s r t r s r s t t sst s q t t ss r rs t r rt s r r t r ss s s r s r tr t rt r s tr s r t t t t sst t s s r t t t t t s r t sst s t t r t s s r t s t s rt r t r t P tr ts s t r s r t s t r t r 3 t sst s r t t r s t t t s t s r rt s t r t t r t s P tr t r s r t t s t r s tr s r t s t tr s r t P tr t t t r sst r s r s r ss t r s t tr s r t r r sst t r sst t t r s r s t s t s s s s t s sst s t s s t r s t t s tr s r t t t t P tr t s tr s r t s t ts t

64 sst s t r t s t t t s r s r t tr s r t st t r r ss s t ss r rs t r rt s s s s s r r s s s t t r ss s r s r t t s t s ts t r s s t > 0 r t rs t t t t t t ts p t t r t ts r s t r t α p s s t p t r s t t t t ts r r t rs p sst s s s t r t α p t sst s s t s t t r α s ts r t s t rs α p r s s t s r t t 3 t 4 t 3 t 3 p 5 t 4 t 6 p 0 t 4 r r t t s s t t r t Pr rt s st s r r rt s t tr s r t r r s r t t S = ((P,T,W),M 0 ) sst α t r P s t r t ts S α r s r s t s s q s r s Pr t S = ((P,T,W ),M 0 ) t sst t s t sst S = ((P,T,W),M 0 ) α t σ = σ t t s q r s r t t s 3 σ t t σ s q t s S S r rt s tr σ s t s s t t σ s s t S S r s q σ s s S p t, M 0 (p)+ W(t i,p) σ (t i ) W(p,t i ) σ (t i ) W(p,t) t i p t i p

65 t r P t tr s r t s t P tr ts s t r s p t, α p M 0 (p)+ s q t t t i p α p W(t i,p) σ (t i ) p t, M 0(p)+ W (t i,p) σ (t i ) t i p s t s q t S t i p α p W(p,t i ) σ (t i ) α p W(p,t) t i p W (p,t i ) σ (t i ) W (p,t). s s t r s t r t r t s s t t t sst s t t t ss r rt s r s r s st t t r r r r ss r s r t sst s ts s s r t r s t s s t t t tr s r t r s r s s str r r r t s t tr t sst r s s r s r t r s r s t s t s s Pr s r sst t tr C s t r α t sst s s α t tr C s t s t s s C [p] = α p C[p] r r p s t t s r s 3 r ts α p r s r t r r s t tr r s t s s t t t sst t r s s q σ t s t t r α s t s s t sst t rst s t t sst α t r σ s r s t s r r t t t s r sst S t t r M 0 s s q σ M t r r s t t M 0 σ M t α s t r r S S t ss t s sst t t r M 0 = α M 0 r s t t s r t r r t r r M S s σ M s t s s M = α M t t M 0 Pr t r α s s S = ((P,T,W),M 0 ) t S = ((P,T,W ),M 0 ) r s r t s q s r s r t σ s q s t S S t r r p r t s t α p > 0 M (p) = M 0(p)+ W (t i,p) σ(t i ) W (p,t i ) σ(t i ) t i p t i p = α p (M 0 (p)+ t i p W(t i,p) σ(t i ) ) W(p,t i ) σ(t i ). t i p

66 sst s M 0 α σ α M M 0 σ M r r t M 0 = α M 0 M = α M M(p) = M 0 (p)+ W(t i,p) σ(t i ) t s t t M (p) = α p M(p) t i p t i p W(p,t i ) σ(t i ), r t r s t r t r s r t t r rs t r r rs t r s r t sst s r rs ts s s r r rs Pr s r sst S = (N,M 0 ) t sst S = (N,M 0 ) t s S t r v r s q σ t t s s S s s S σ t M 0 M S M 0 σ M S s s t t S s r rs t r sts s q σ t t s s (N,M) s t t M σ M 0 s q σ s t s (N,M ) t M σ M t σ r = σσ r M s q t t r s t s M 0 v s M 0 t r rs t S r r ss r s r t sst s ts s s r Pr s r sst S = (N,M 0 ) t t s S = (N,M 0 ) t s S t r v s t t S s t s t r sts p s t S t k t r p r r r s S r sts r M r S s t t M (p) > v(p) k t s r M s r S s t t M = v M r t s M (p) = v(p) M(p) t t M(p) > k tr t t t r k p t t r s r t str t r ss ss s r s sst t r t s s t r t r r tr t r ss ss r s r t sst s str t r ts s s r str t r sst s str t r ts s s r str t r s s q sst s r ts s s r r

67 t r P t tr s r t s t P tr ts Pr r r t ss r rt s r s r t s r t t s r M 0 sts r t N r s (N,M 0 ) s t sst (N,M 0 ) s t r r str t r s t r sts str t r sst S = (N,M 0 ) M 0 t t r S = (N,M 0 ) s t str t r s S t t s t r tr s r S t S v t rs t r tr s r S t S v r sts r M s t t S = (N,M ) s t r r r r r t M t sst t s t rt r r M sts s t t S = (N,M ) s t s v t M t st t t t rs v t t sst S = (N,M ) N s str t r S s tr t t r r t r r s r t t r ss s t r s t r t tr s r t t s t t r s r s s r str t r r r rt s t P tr ts t t s t r s t rt r s t r t t s r t t t t r t s sst s r s r ts tr s t s r r t st ts r r rt s s r t s s r rt s s s s r t ss s tr r t s r t ss s t s s s t s str t s r t ss t s t s s s t t s r t ss r tt r t s r t s ts s t r rst t s s s t t s t s r t sst s t t rr s t s r t sst s sst s s s r r r ts t t s r t tr s t s ts s s t tr s r t t tr s t t s t s r t W(p,t) = W(t,p). p t p t t tr s t s t r t s r t t t s s t t s r t s t s str t r tr s r s sst t t s r t sst s t s str t r

68 t s r t t 3 t 4 r t t t s r t tr s t t t r t t s r t tr s t t t sst S s sts s S t r s t r t rs s t t t r s t sst s t s r t p p 3 t t t 3 p 4 [,,] p p 3 t t t p 4 r sst t t s t t s r t s t tr s t t 3 s t t s r t t s r t sst t r t s t r r t sst s Pr rt s tr s r t s t t s r t sst s s s t t t r r sst s s r t t Pr s r s r t sst t tr C t t t r sts t r X P t r rs s t t t X C = 0 t r t C[p,t] X p s tr C s ts r t rs s t s r r tr s t t t P C [t] = 0 t r t s t r rs t sst s rs sst t r sts t r X s t r t rs t t s r ts tr t t X t st t t t rs ts ts s s r t t r t sst s s r t r t s rt r s t s r ts ts r rt s ss r rs t r s r t

69 t r P t tr s r t s t P tr ts r r s r t sst s ts s r r r s r t sst s r rs ts s r r rs P t tr s r t s t s r t t P tr ts s r t ss s ss r t r r ss s s t r t P tr ts st t t r s r r t s sst s q t s r t t r r s r t t t tr C s sts s t t t X C = 0,X P r t C X = 0,X P r t tr s C r t rs t s X r t r rs P tr t t s r t s r t s r t sst s t r t s t X t r r r t t t ts X t r t t r t rs s t s t s ts r s t t r rs t r s t r ts r t s t t s t r t s r t t r s t r s 3 s s q t t tr s r sst s t s ts r t s t t t r rt s t r sst r 3 t ts r t t r 3 t s tr t t t t sst s t t t s t t ss r tt t s tr s r t t r t s r s st t r r t t sst s t st ts r r 3 t r r t t t r r t r tt r t s tr s r t s t r t st rt r t t s st r sst t s t r t sst s r s t t r rt s ss r rs t r s r t t r r 3 t t s str t r t tr s t t s r 3 t t t t ts t r q t r sst s r 3 ts tr s t s r r 3 r 3 t tr s r s sst t r 3 s r r t sst s t t r s s t t r 3 t r r s st t sst s r r tt t sst s s st t t t r 3

70 r 3 t ts r t t t t r tr s t t t t s r 3 t r t t s t r 3 Pr s t t S = (N,M 0 ) s t s st t sst t tr C r sts t t r Y T t rs s t t C Y = 0 s s st t ts s t r t t t tr s t r r s t s t s s r r r 3 t t K t st t t s Y[t] t T s r t t s p s t t tr s t t t t tr s t t s t α p = W(t,p) Y[t ] = W(p,t) Y[t] K W(t,p) Y[t ] = K W(p,t) Y[t]. r t t t t sst S = (N,M 0 ) t s S t (α,,α P ) s r 3 s r tr s t t t r r p t r r p t W (p,t) = α p W(p,t) = K Y[t]. W (t,p) = α p W(t,p) = W(t,p) K W(t,p) Y[t] = K Y[t]. s r tr s t t s r 3 S s r 3 r str t sst s r s st t s r t Pr s t t s t t r r 3 r sst s r 3 t r t s s r t r r t t s ts t t r t s s t st t r s s t r t r 3 sst sst sst s st t t s ss r sst s s s t t r 3 t t t s r sst s s r sst s t r str t sst s t s r 3

71 t r P t tr s r t s t P tr ts t 3 t 3 p p p t p 8 p p t 7 r s st t sst s r 3 t r s s r t t s sst t t s r 3 t t r ts r s r t r r t s r t t r t p t p t q t sst t r t s t t p t s t t r 3 t t t r t s r s st t t r s q r s Pr s ts tr s t s s st t t t t sst s tr s t s q t t q s ts q t t t r t r 3 r 3 sst r t t s r 3 t r t r t ss s t s r sst s r r s st t s r t r ts r 3 sst t t t r t r ttr t sst t t s t t r r 3 r r 3 t 3 3 p t p 5 5 t p 3 p t 3 t 3 t p r t t r 3 r sst s ts str t r s r t sst s r sst s r sst t r 3 r r 3 s r 3 t s rt r s r r r r t s r t r s r t ss r rs t r r sst s ts r 3 t s r r r sst s r rs ts r 3 t s r r rs

72 s t s s t s s t t s t s t t t r sst r t t t t sst s t s s rst tr r tt r 3 t s r t t P tr ts t t P tr t s s t s t s t s t s t r p s t r t t gcd p t r s s t t t r t s t s t s t s t t t s t s s q s s t r r M 0 (p) r p sst S = (N,M 0 ) r M0 (p) gcd p gcd p t t t s r s q s S Pr r t t r M 0 t t N str t r M 0 r t p P, M 0(p) = M0 (p) gcd p. gcd p t r p t r r t s M 0 (p) gcd p M 0 (p) = M 0 (p)+r p r t t s 3 r s q σ t t σ s s r M 0 σ s s r M 0 r rt s tr σ s t t r s s s σ s s t sst s σ t M 0 M M σ 0 M s r σ = σ t t v p = W(t i,p) σ (t i ) t i p t i p W(p,t i ) σ (t i ). t t t ts p r t s gcd p v p s t gcd p t M 0 t q t ss t t r t t sst s t r t r σ p t,m 0 (p)+v p W(p,t) p t,m 0(p)+r p +v p W(p,t). M 0 (p) v p W(p,t) r t s gcd p r r p r p s str t s r t gcd p p t,m 0(p)+v p W(p,t). r r t q t r t M t s t r s t r s str t r

73 t r P t tr s r t s t P tr ts t t 3 t t 3 t 4 p 6 t 4 t 4 p 6 t 4 r gcd t s q s r t t s t t s t s t t r r s r t t r t t s t t s t s r s r t t t r sst s t s s t r s t t tr s r s t P tr t t t r sst r s r t t r ss r t t r s s t s tr s r t tr r s t s r r t t t t s ts tr s t s s t s t r str t r rst r t t t r str t r s r s ts s tr s r t t t r s r s str r rt s s sts t t ss r s t s t A tr s t s p r tr s r t s r sst S = (N,M 0 ) t N = (P,T,W) s t t M 0 t s s t s t r sts s t t s t s t T s r t tr s r t S t t s t sst S t t s s t tr s t s A p s t t A t s tr s t s t t r t t ts p gcd p = t s p r r tr s t t A r t t r p t t r t t w > 0 sts r t t p t r w w + t r s r t t r t t p t t p S S t s r r S s r t S s r Pr rst st s r t t t s s t r s r t t sst r t r ss r r r r s r s r s t s t s t s s t s t s t t r M 0 t r t t t s p r r s t p s t s s gcd p t t s s t t

74 r s r t t t r sst s t t p 3 t t r s t 3 p r rst t tr s r t t A = {t} t t t t sst t tr s r gcd p = 3 t p s t s r s r t p t t s t p t t t t t t p 3 s p 6 p 6 r s t 7 p 3 6 r s t tr s r t t A = {t} t t t t sst t tr s r gcd p = t p s s t t s sst r s r t p t t s t p r r t t p s t ts t t t t 3 t t r s p t t r r r t tr s r sst r t t tr s t r t t s s t s t t t s r r r M r t sst M(p) s t r k c + r k c r t r rs s t t c t t t r s t t r t t t r s t r s r t t r rt s t s r t t t sst s r t t s sst s s r r s r t r s t r str t t r ss t s t sst s str t r ss s r s r r s r t str t r ss s r t r s r t t s t t r ss s r s r r t t tr s r t t sst s t t r sst s r s r s t t r ss r t s s s t t t t r t s s t s t r s t st rst r

75 t r P t tr s r t s t P tr ts r r r s r t t t r sst s r sst S = (N,M 0 ) t N = (P,T,W) s t t M 0 t s s t s s r t tr s r t S t t s t sst S r r p s st s t s t t s t T t tr s t s gcd p = s t p r r tr s t t t t s t t t p t t r (p,t) t t r t t w r t t p r sts r w w + t r s r (t,p) t t t t p S S t s S s t r sst r r S s r t S s r Pr r t t r t s t tr s r t r r p s t t s t s t T t t t r ss r r s r t r t tr s r t t t s t r tr s t s t s t s P s tr s t s rt r t tr s t s s r t t s t s t s q t t tr s r sst s t r sst r t r s t tr s r t t 0 t t t 0 t t p 0 p p 0 p r sst t t s t s tr p 0 p t t tr s t t p 0 p p p 0 tr s r t s t t t r sst t r t t t s t t gcd p0 = p 0 s t t t tr s t s t s s t s t t t r s t p 0 t t t gcd p = t s t r t p s t s t t r r p t t 0 t t t t r r t 0 t p t t r r r r ss r rs t r s r t tr s r t t t r sst r s r s t ss r rs t r rt s t t sst s r s r rs t tr s r sst s r s r rs

76 s tt tr s r t Pr ss r s r t s r t r s r t r r t s r t r s r s r rs t r s tr s t s t r s t t t t t s t r t s tt tr s r t t s s t r s t tr s r t r r sst t r sst t t s s t s s s t r r rt s t r sst r t st t sst t t t ts t t s t t st r r sst s rst r s t t tr s r t t s s r rt t r rt s t s tr s r t t t r s st t r r r sst s t t t rs t r s t r t s tt tr s r t r r t t sst r r sst s t t r t s t t s s st t t s r ts s t s tt r s t sst s r t t r s r s tr s r t t t s sts s tt s s t s r s t t s t t sst t r s s t t s t st t t t r r r r t r t t r rt s sst t t s t t r r r t t t s t ts s rr t s t s t t t s t t r r ss t t s t t t str t r r t s t t r s t ts t t tr s r t str t r r sst S = ((P,T,W),M 0 ) Pr s t S s q s Y s s rt s T t U t st t t ts Y sst S = ((P,T,W ),M 0 ) s t s s t tr s t s s t r S s s tt rt r s r p t st t t tr s t s t...t k t t t s r k s p...p k P s t t r r i...k p i s t t t i t t s r t s str t r

77 t r P t tr s r t s t P tr ts t t k W(t,p) W(p,t) m p W(t k,p) t s tt t t k W (t,p ) W (t k,p k ) p m m k p k W (p,t) t W (p k,t) r p t t s k t tr s t s s tt p s t k s t r t ts r r p s t t k s p...p k r r i...k t ts s rr t p i r t r s s W (t i,p i ) = U W(t i,p) W (p i,t) = W (t i,p i ) Y(t i) Y(t). t s rt Y s T t s Y(t) s r r t U U Y(t) t s W (p i,t) t r ts r t t s t s r t t s t s t t t r t p r s r s r rt t t t ts t s r s r t s t s t s t q t W (p i,t) Y(t) = W (t i,p i ) Y(t i ) r r p i r s t r t s tt r t r r t q t W (p,t) Y(t) = W (t,p ) Y(t ) s t s t s r t r M 0 s t r t r M 0 r t M 0(p i ) = M0 (p) U W(t i,p) Y(t i ) gcd pi. gcd pi W(p,t) Y(t) s q t r tt q t s s M 0(p i ) = M0 (p) W (p i,t) gcd pi. gcd pi W(p,t) t r p i s t r t r t r t t t t t t t t t t t s t r s r

78 s tt tr s r t r t t t t t tr s t t r t t t t r t s r t t gcd pi s s r s t s s tt tr s r t s str t r t t p 0 p p 3 t 3 [ 3 ] [ ] [ 4 0 = 0 ] [ ] 0 4 = t t p 0 p 0 8 p 4 8 p t 3 r r sst t t s t s (,4,) t s U = 4 t tr s r t W (t,p 0 ) = 4 W (t,p 0 ) = 4 W (p 0,t 3) = 8 / W (p 0,t 3) = 4 4/ r p p s t gcd p 0 = 4 W(t,p 0 ) = gcd p = 4 W(t,p 0 0 ) = t M 0 (p 0 ) = 4 M 0 (p 0 ) = 8 t t sst t r t t t s s Pr rt s tr s r t r r s r s str t ss s s t t s r s r s t s t s s r ss r s r s r t s r t r str t r sst S t tr s r sst S s s S s r t sst s t s s t Pr S S t s s t tr s t s T sst S r s q s Y s s rt s T Pr s t r r tr s t t i Y(t) W (p i,t) = Y(t i ) W (t i,p i ) t s Y s s S t t S s s st t str t t s r Pr s t t s q s t s rt T Pr s t Y s s S t ts ts s t r s s r s S t s Y s t q s S s s s t Y t s t s s s r s r s r rt s str t r r t sst s r r t s s t s s t q

79 t r P t tr s r t s t P tr ts t r rt r s t r t t s t sst s r r r rt s r r str t r sst (N,M 0 ) ts tr s r t (N,M 0 ) q t s s t s r p tr s r t k s p...p k k M 0(p i ) M 0 (p) U. i= Pr t M 0 t i=...k M 0(p i ) = i=...k i=...k M0 (p) U W(t i,p) Y(t i ) gcd pi gcd pi W(p,t) Y(t) M 0 (p) U W(t i,p) Y(t i ) W(p,t) Y(t) U M 0(p) W(p,t) Y(t) i=...k W(t i,p) Y(t i ) k i= W(t i,p) Y(t i ) = W(p,t) Y(t) t s r r t r M 0 (p 0 )+M 0 (p 0 ) = M 0(p) U = 3 4 = s t S t S s s r s r s q t t s s S s s S Pr r t t t s 3 s q σ t t s s S = (N,M 0 ) σ s t t t s s s S = (N,M 0 ) s s t t σ = σ t s σ s 3 k t r rt s tr r σ t s σ s s s S t M 0 M σ M 0 M t M t t s t tr s r t t r ts t s p t ts s rr p t t rr s tr s t s r t s r t s t s M (p) = M (p) t s M S t r s t s r t s t t t r r t p t t t s tr s r t s p...p k s t t t s p i i =...k r M S t p s M S t s q t t s t t M (p i ) 0 i {p,...,p k } s M(p) 0 r M t M r r s p i i {,...,k} M 0(p i )+W (t i,p i ) σ(t i ) W (p i,t) σ(t) 0.

80 s r t t W t M 0(p i )+U W(t i,p) σ(t i ) U W(t i,p) Y(t i) Y(t) s t r q t r s p...p k t i=...k ( M 0(p i )+U i=...k W(t i,p) σ(t i ) ( i=...k σ(t) 0. W(t i,p) Y(t i )) σ(t) ) 0 Y(t) i=...k W(t i,p) Y(t i ) = W(p,t) Y(t) ( ) M 0(p i )+U W(t i,p) σ(t i ) W(p,t) σ(t) 0 i=...k i=...k r t r r rt r t r t s t t U (M 0 (p)+ ) W(t i,p) σ(t i ) W(p,t) σ(t) 0 i=...k ( ) t s U M (p) W(p,t) 0 σ s s S r t s q t 3 t t t t 3 s s t sst s t r sst r t s q t 3 t t t 3 s s t r sst r s t s t s t sst s r rt r s st r r rt s r sst s t t t rs t s r rt t t st t s t t t t sst s t r sst t s r ts r str r r t r t r r ss ss sst s t st t r rt s r sst s s r 3 t t P tr ts s r t tr s r t s t t r r t sst s r tt s s r rt t r rt s t s tr s r t t r s r t t t s s s s t ss r rs t r rt s t r ss tr rt r s tr s r s r s r t sst t t s r t sst r t t r rt s t s t r t s s r t r t s t r r s s t s r t ss r rt s t ss r r r ss t tr s r t s t r t P tr ts

81 t r P t tr s r t s t P tr ts s r s t tr s r t t P tr t t t r P tr t r s r t t r ss t s tt tr s r t s t r r sst s ts r str r rt s t r s r t t s s r ss s s t s t t tr s r sst t t t t sst

82 s

83

84 Chapter 4 P s t t s ss r r r q t sst s t ts P s t t s ss r r r sst s r ss s t t r r sst s s s r r r sst s P r s r r r sst s Pr s r s ts t ss q t sst s r t r 3 t ss r r sst s ss t t sst s P r s r r q t sst s P r s s r 3 sst s t t s ss r r sst s r s ts t ss s t t ss s tt t t s t t r s t ss t s s t t s r t ss r s

85 t r P s t t s ss r r r q t sst s t s t r s r t r q t sst s s s t r s ss s r sst s sst s sst s r ttr t sst s P t r t r 3 t s r ss st r t s ss s r r r r s q t t r t r ss s s t s s tr t r t s s q st s t t r r ss ss t s t st s r t t r t r 3 t s ss r r r t r ts t t ss t r t s q s t s str r t t t r s r s t t r3 st s t r 3 s s q t t s t r t s t r r ss s r t r st t t r t t s t r s r s r r ss ss s t t s st t t t r s r q t sst s st t r s ss s t r t t r t t s r t t r r r t t s t sst s r s t t s r r tt t s r r sst s t t s t t t s t r s t tr s r t s t r t t t rst s t t s ss r r r q t sst s s s t sst s t t r s s 3 t t str t s r s ts tr st t t r r t t s t t r t r s t r s r 3 s s rst t r t rst s t t ss r t s r t r sst s t t t t t s r s r t r t s r s t t r s t t r t r r ss r r sst s t r t r s r t ss r q t sst s t t s r s ts t r t rst s t t s r r q t sst s t t t t s tt tr s r t r s t t r t s t t ss r t s s r r sst s r t s ss t s t s t t t t s t s r t ss r t t s

86 P s t t s ss r r r sst s P s t t s ss r r r sst s s r t t r r sst s ss r r ss rst s t t ss rs t sst t s s r r r r sst s r r s ss r s t t ss t r s r t t s s str t t t t s s r r r r r r s s r ss s r t s s t s r t r sst s r r r r t r t s sst s s t s r str t r st t r sst s s t t r r sst s t r ss s s s t t r t st s t S = ((P,T,W),M 0 ) str t r sst s t s t s t s t t q t p P M 0 (p) > p P (max p gcd p ). r r r M [M 0 t r sts p P s M Pr s M 0 s t s t s t t s t t r r p M 0 (p) s t gcd p t t t t ts r p r s t s gcd p M(p) s t gcd p r r r r M r p s s tr t t t M s r r t t s t p P,M(p) max p gcd p, p P M(p) p P (max p gcd p ). S s r tr s t r t s t r t s t sst t t M 0 (p). p P M(p) = p P

87 t r P s t t s ss r r r q t sst s q t s t r r q t t M 0 (p) (max p gcd p ), p P p P tr t q t r t r t t q t r s s t t ss r str t r sst S = (N,M 0 ) s t s t s t s t s M 0 (p) > (max p gcd p ). p P p P Pr t S r sst t t t s t t r s t t S s t t s r r r r M r tr s t t t r sts t s r s q st rt t M t t s t r M t r t t r s r t t r t t t s s s q t r t s s r r tr r r t s r ts r M(p) W(p,t) t t s r t t r t s σ s t r q st r s q s r t t r t p s t L s t t t r t s p sts r r p p s p L t r st t r rs s s t r t st t t t t s st r t p t t t r s r t s t r t t s t s t r p r s s ss p s t s t r sts t st t r r s r M s r t s rt st st s d M r t t d M > lex d M t r r δ r t s r s t t d M i < d M i t t s st s t rs t t r s t s r r t r t i t q t d M > lex d M s tr t d M j r s r j > i s t r r s r δ t r t t r t s σ s t t s r s s t t t s t r t r sst sst r r s t s t s s p (max p gcd p ) = ++0 =. q t s p M 0(p) > s s t s t r t r s t t r t r sst s s t s s t r t r ts r tt s q t t t r t r r s r t t ss t t r t r t r

88 P s t t s ss r r r sst s r t r t t s s r s q st rt t M s s t tr s t t t rr t r r M t s δ t s r tr r r,...,δ q t p t δ t d M = (d M,...,dM δ ) ss t s t s rt st st dm i r t i t t p r t t r M s t t r s q σ s t t M σ M M s t σ := ǫ t t s q M(p) < W(p,t) t L t s t t s M J = {i...i k } t s t t rs t t s t s L t M t p t L t t i r i s t s st J t µ = p,t,p,...,p s rt st t r p t p r t s t t t i s s t t p t M d M σ := σ t

89 t r P s t t s ss r r r q t sst s p 3 p t t t 3 t 4 r t r t s r sst s t t s r s t s s s r r r sst s s t t t s t t ss r r r sst s s ss r s t t ss r r r r r sst s r sst s r r r str t r r t s t Pr tr t s s t t t r sts r r r str t r t N tr s t t t t > N s str t t s t r sts r t c = tp t p t...p k t k ss t r t t t k = t s N s s r t r t r sts X P s t t t X C = 0 r C s t tr N ss t t > t r s t st t r p t t p p s X[p k ] = p t X[p] > X[p ]. p r 3 r i {,...,k } X[p i ] X[p i+ ] X[p ] X[p k ] tr ts X[p k ] > X[p ] s r s t t q t r t t t r t s ss r s t t ss r r r r str t r sst s r sst s r r r r r str t r sst sst S = ((P,T,W),M 0 ) t st tr s t s p P M 0(p) P r s r r r sst s t t r str t r s s 3 t t r s s t t r r t r sst s s t r s str t ss r s M JF r r p 0

90 P s t t s ss r r r sst s r r r r t t s r r t r t s ss s s ss t s t t s t s t r r r s t r s M JF t S = (N,M 0 ) t str t s r t r r sst S s t t s r p 0 M 0 (p 0 ) = max p0 r r p P {p 0 } M 0 (p) = max p gcd p. r r r r r r s s Pr r t r sts t r α P r S s t t α t s s t t r rs S α s t sst S = (N,M 0 ) s q t t max p gcd p t t M 0(p) p P p P, max p gcd p = α p (max p gcd p ) p P {p 0 }(max p gcd p)+max p 0 > p P (max p gcd p). r r r r p max p s t gcd p s M 0 s t s t s t r t r sst S s r t r r sst r s r s t r rt ss t s S s r r r s r s s t r rs s s t s s t q t s p 3 p t t t 3 t 4 r t t r t s r r sst s t t s r t sst s p 4

91 t r P s t t s ss r r r q t sst s Pr s r s ts t ss q t sst s t s s t r r t r 3 t ss ts t t r r q t sst s s s q r s r t s ss r r r s s r t r 3 t ss r r sst s Pr s t Pr s t r r q t sst s r s t r P ts r str t s r t s r P s sst s t t r st t s t ss r q t sst s r t ss ts P ts Pr s t r s r str t r q t sst S = (N,M 0 ) S s r r r P t N = (P,T,W ) N t sst (N,M 0 P ) s s r t r 3 t ss t r s s sst s s t tr t t r t r ss s t t r s sst s ss t t sst s ss r rt s t s t s r ss s s t r r s t r r r t str t r t r q t sst s s r s t s t t s r t t t r r r t (N,M 0 ) q t sst s r r M 0 s t t M 0 M 0 (N,M 0 ) s t (N,M 0 ) s r s r r r q t t t r r r r s s P r s r r q t sst s t s s t t t r rt s t r s ts t r s s t s t t rst t r r r q t sst s r r t ss t t r r r r r s s s r t s r s ts s t t ss t t s r r r sst s r tt

92 P r s r r q t sst s p p 4 t t p p 3 p 5 t 3 r s s tr P tr t s s t t r ss t t t p 4 s r t s q t t s t P r s s ss q t ts s tr s t s t s r t s r s tr s t s r t r t s r t r s r s ts r P s ts t str t r r r q t sst s q t t s tr s t t s s r t t r r s t r st s s t t st tr s t rst r t t str t r t str t r P s ts r q t ts t N str t r q t t t t st tr s t r str t r P s t N t s t st t tr s t Pr s t r sts str t r P s t N JF t tr s t s q t N t r N JF s q t N t r t tr t r N JF s r r s t t r sts n N JF n N N JF s t t n s t n s N s str t n t t r s N JF t P s t n s tr s t t t st t t s t N JF N JF str t n s N JF t s tr s t t t s tr s t N tr t

93 t r P s t t s ss r r r q t sst s r t r t rst r r r q t sst s s s r t t r sst s st t s t r r t r s t s s s s r r r P r M EC t S = (N,M 0 ) r q t sst t t s t r sst S s r t s p i tr s t s M 0 (p i ) = max pi r t r s p M 0 (p) = max p gcd p. r r r r r s r s ss Pr t S JF = (N JF,M JF 0 ) t P ts N r N = (P,T,W ) JF JF JF JF M JF 0 s t r str t M 0 t P JF t P ts Pr s t S JF s s r t r r P s sst N r r r str t r P s t N t s t st t tr s t t s t st p s t t M 0 (p) = max p s r S JF s t t M 0 s r P t S s Pr s t r r r r r r s t sst r ts q t sst t s sst s r s r t p i tr s t s r q t max pi t t r s p t max p gcd p t s r s s t P ts t s sst t t 4 p p p t 5 p 3 3 t t 3 p 4 r q t sst s r t s t t r r s t sst r r t str t t s t 3 t t t t t t 4 r t tr s t s t s t r t s p p 4 r t r max p = max p4 = 3 t s t s t s t t r r t r p t max p gcd p t s t s t r s t r r r t s s t str ts ss

94 P r s r r q t sst s t t 4 p p p 3 5 t 5 p t t 3 t t 4 p p p t t 3 r P ts t q t sst r s r 3 sst s r sst s s t t ss s r ss r tt s t t t s r s t s s q r Pr s t t s t str t r ts str t r P s ts str t t r r ts Pr s r str t r P s t N str t t s s tr t t t tr s t t N s t st t t t s p p t p t q t t P s p p t t r t t s t p s t t s str t s t s r t t r s t ts s t t s t t r s t ts tr s t t r s t s t N s r N s t str t tr t r s t t ss r sst s t t r r t r r tt r r tt t S = ((P,T,W),M 0 ) str t r 3 sst t t s t s t t S s r r t S C S s t s s p P C M 0 (p) > r P C s t s t s t r t S C p P C (max p gcd p ), Pr t t r 3 t (P,T,W) s r 3 t t ts r ts r r r r ts r str t r ts t s r s ts r q t ts S s str t Pr s t t s r

95 t r P s t t s ss r r r q t sst s t t s ss r r sst s t s s t r r rt s r r t ss r sst s sst s r s t t ss r t rt r s ss r r sst s t t t s tt tr s r t t r t rs t s t s s t t ss t t t t sst s r tt r s ts t ss r t t r s t s r t t t ss r rt Pr s t r t r 3 t ss r r sst s r s r r sst S = (N,M 0 ) ts s t tr s t s T t tr C N S s t r st r r M [M 0 s q σ t t s s (N,M) s t t σ T C σ 0 Pr s t ss r t ss r sst s r s r sst S = (N,M 0 ) t s t tr s t s T tr C N S s t r Y T s t t C Y 0 t r sts s q σ Y t t s s S s t t σ Y = Y s t t ss s tt s t t t s tt tr s r t s s t t ss r str t r r sst s r t t ss s r str t r r sst S t S t sst t t s tt tr s r t t S S s t S s Pr t t s tt tr s r t t sst s t s s t tr s t st t r s r t s s r Pr s t S S s Y T S s t s s q σ Y s t t σ Y = Y s s Pr s t t s r σ Y s s s S s s q t t M = M 0 σ = σ Y Pr s t s s t t ss s str t r t t s t t s t t ss r r r sst s s t s O(max{ P T, P log(min p P max p )}) r tt s t t s t t ss r r r ss t s

96 t t s ss r r sst s p p 5 t p 0 p p 5 3 t 3 t t 4 p p p 6 4 p 3 3 t 5 t 6 t t p 0 p t 3 p 4 p 4 p 3 p 3 p t 5 36 t 4 3 t 6 6 p 6 p 0 p p p 3 p 4 p 5 p 6 t t t 3 t 4 t 5 t = p 0 p 0 p p p 3 p 3 p 3 p 4 p 5 p 6 t t t 3 t 4 t 5 t = r s tt tr s r t s t t r sst t t t t sst t r t t t s s t tr s t s t s sst s t r sst s r P t s t t ss s r str t r r sst S t S t sst t t s tt tr s r t t S S r 3 t r 3 sst S r r S s t s s t ss t r r sst s t S s Pr S s r S s r r t s s st t s r t r s q t S r 3 r

97 t r P s t t s ss r r r q t sst s t r s r t ss r rt r r t t t ss S s t ss S s t ss S r t t r s t S s t s t t ss r S s t r s str t r p p 5 t t p 0 p 0 p t 3 6 p 4 p 3 p 3 p 3 t 5 3 t 4 3 t 6 p 6 r s r 3 sst s t r t sst t r t r t r s t s s t s t t r t t ss t r sst t t r t r r s t ss t s r sst s r t t s t t s ss r r t r s t t t r t s s t t s s s s t t rs r s s sst s ss s t t t t r t t t t t t r t r r s s sst t t s t t t r t t r s s t s

98 s t t s r t ss r t t 3 3 p 3 p p t 3 t t 8 3 p 3 p p p 6 6 t 3 t t 3 3 p 3 p p p t 3 r sst t t s t s s t t r p P M 0(p) = 5 > p P (max p gcd p ) = 4 t s tt tr s r t r s ts t r 3 sst t r t t t r s t s s t t r p t 4 t p 4 t 3 p t p 3 p t 4 t p 4 t 3 p p 3 t p p 3 t 4 t p 4 t 3 p p 3 t p 3 r t t t r ttr t sst s t s t s t t r p P (max p gcd p ) = t t r q r s t r t s r s t t s r s t r ss sst t t r s t r r 3 sst t t s tt tr s r t t r t t r 3 rs t sst s t r t s t p t t s r r s t r r t C t r 3 sst p C (max p gcd p ) = t ss t r s t t r s t s s t t r s t t s r t ss r r s t s t t s t r s s t s s t t t sst t t s tt tr s r t s t t s r

99 t r P s t t s ss r r r q t sst s t ss t r sst s r t r s t s s t t s t ss r t r s t t s ss r s r s t ss r s s t t r r s t t t sst s r tt r t s t r t t s t t s t t r (max p gcd p ) = (4 )+( 7)+(6 3) = 9. p t 6 p 3 6 p t t p r s r t s t s t t s t t s t r s s t s r t r r s s t t r tr s t r r r r t r t r s q t s t r t r s 0,8,0 t 3 0,7, t 4,4,0 8,0,0,0,6 t t 8,4,6 t t t,4, t t 6,0, t 0,0,8 t t t t t 3 t 6,7,5,7,9 8,7,3 4,,3 4,0,4 r r t r t r t s s t s s q s t rt r r r t r sts t s s q t r tr s t t s ss

100 s s t tr s r t t t r r ts r t s ss r r q t ts t t rst s t t s ss t s t s r s t t r q r r r t t s r s r r t s s t t r t s r t r t t s s t t s tt tr s r t t r t r t s t t ss r r r sst s st t t s ss q t sst s r t s t s s t t t r t s s s t t rs t rt t r st

101

102 Chapter 5 rs t q t r sst s t ts rs t r sst s Pr s r s ts Pr r t s r s ts P r s r r r ttr t sst s r s t t s s q s P s sst s ss r t r rs t r r sst s r t r 3 t r rs t r r sst s P r s r r r sst s s t t ss r rs t t t s t t s t t s t ss r r s t t r s rs t s st t q t sst s rs t s t r rs P ts s q s r rs t s sst t t s t r rs s s q s t ss r t r 3 t r rs t t t r rs t sst s P r s r r sst s rs t r sst s P r s rs t s q s r t r ss s s

103 t r rs t q t r sst s t s t r st t r rs t r rt r t ss t s s str t r q t r sst s s s s s st q t sst s t t r t ss r r t r s t r r s t t s ss r r q t r sst s t r s s sst s t t t r rt s ss ss r s r r rs t s t t r rt t s r s t s t r t s r r sst s t r rs t st t r t r r t s ss r rs t r rt s r t t t r t t r st sst s t t r t r rs r rs sst s t t r t r t rs t s r st t r t ss t s s s r s ss s t P tr ts t s sst s r rs t s ss s t s r tr r r ttr t sst s r s t r st r ttr t sst s t t r t r rs P t s st t t r s r sst s r tt r rs t s s t s r r t s ss r s s s r t r 3 t r rs t st r r sst s r r s r s r t t r t t r t t t r t t s st r s q t sst s s r r t r t r 3 t r rs t t t t r s r r tr t s r r 3 s s rst t r s t t rst r rs t r r t ss r ttr t sst s st t r r ss r sst s r r s t r t r 3 t r rs t t rst r s s t r r t r ttr t s ss s r t r r s t t ss r rs t s s t t t s tt tr s r t t r t s t s r t r t st t r rs t s st t q t sst s t t t r rst s t t r s t t r r sst s t t t t q t ss s s t t ss s sst s t ts r rs t s t t s t r ss t s t s r t r rs t t s ss s q t t t rst tr r t r 3 t r rs t r s st t q t sst s ss r sst s s r sst s r t s r t r 3 t t t rt r s r sst s s r s t s t t rst r s r r q t sst s

104 rs t r sst s t s t s t s t t t rst r rs r s r r r sst s t t str t s r t t st r r t r 3 t r rs t r q t sst s r r ss s rs t r sst s t s s t rst r s r r rt s r sst s t t s t t r r ttr t s ss t s r s ts r r rs t r r t r ttr t ss r t ss t s s r r r sst s ss r s t t r rs t t t s r ss t r s t r rs t rt r s sst s str t P s sst s s r s t t s ss t r r r sst s r r rst t r r r t s t t r t s t t r t r 3 t r rs t s s r rt t s q s t t r s P s sst s r r t s r s t str t r t s sst s t t r t s r rs t r r s t r r t r 3 t s r t s t t t s r r r r s t t t ss r rs t t sst s ss t t r st P s ts r r ts r s t t ss r rs t t t s s t s tt tr s r t t s t t ss r rs t t t r t t r s r s s s s s t t s s r r r r s s r rs Pr s r s ts r r rt s r sst s t t r s r t r s ts t ss r rs t s ss s t s t t r s r q t sst s r s 3 r r sst s r t t s t t t r str t r r st rt r s sst s t r s t s s t r P s sst s t t st t ss r sst s Pr s t r t r 3 t ss r s r r ss t S = (N,M 0 ) t t s r s S s t str t P s sst s S = ((P,T,W ),M 0 P ) S r

105 t r rs t q t r sst s Pr s t t r rt s s s 3 t Pr s t r q t sst s r str t t r sst s s t r r P s sst s r r ttr t P s sst s rs t r t ss t s s r t r 3 t r rs t r r sst s r s t r s r s st t t r s t s t s q s P r t r s q t t s Pr s t r t r 3 t r rs t r s r str t r sst S = (N,M 0 ) t q s Y S s r rs s q σ Y s t t σ Y = Y s s t M 0 P rs st t sst s sst s rs st t r r r M t t s t tr s t s t t s t s q t t P rs st t ss t s ss t str t r rs st t r sst s t r r st s r rt s s s t r ss r r rt t t t r r t sst s str t t r r rs st t sst s sts st r t str t r Pr s t r s r r sst (N,M 0 ) σ τ M M r r r s s t t M 0 M M 0 M t r α β M 3 sts s t t M M 3 M M 3 r t s s q s α β s t s t t s r r tr s t t α(t) = max{ σ(t), τ(t)} σ(t) β(t) = max{ σ(t), τ(t)} τ(t). M 0 M σ τ α β M M 3 r t s q s σ τ r s t r sst t t s s q s α β st r t s r M 3

106 rs t r sst s Pr r t s r s ts q s s P r t r q s s r str r t t t r rs t r rt s s s q s r t r t t s r t rt r t s t st rt r s q s t t r s r t s s s r t s t t s r st r rs t t s s q s s s t s t r s t s q s sst s t S = (N,M 0 ) sst s Y s q σ s tr s t r t t Y ts P r t r s s r t t q t Y t t s σ(t) Y(t) r r tr s t t σ(t ) < Y(t ) r t st tr s t t t σ Y r σ S s t r M s tr s t sst t t q r (N,M,σ,Y) s t s str t r r r t s r st tr s t s q s sst s rt t t s tr s t s t r r 3 s s s t r r 3 s r tr s t sst S = (N,M,σ,Y) tr s t t s r S σ(t) < Y(t) t r s t tr s t s r 3 S s σ(t) = Y(t) s r S t s t tr s t t t s r S t r s t s r 3 S r t r s r r 3 s t 3 p 3 4 t 3 p 3 4 p t p t t p 4 t p 4 r s r ts t q s Y = (4,,) s s q σ = t t t s tr s t r t t Y s σ = (,,0) (4,,) = Y s t r σ t sst t t s t t sst t r t s tr s t r t t Y tr s t s t t r t r r t t Y t r t t tr s t t s r 3 s σ(t ) = = Y(t ) s q t t t s t r r 3 t p r t r s t s t s r tr s t r s r P tr t t s s s t t tr s t s t t r

107 t r rs t q t r sst s r 3 s t r s s t t r ts r 3 tr s t t s r 3 r t rt r s str t r ss t s t t s t t ts q t t tr s t t tr s t s t r s r r t tr s t s t r 3 s r r 3 s ts r t r r t t t t t t p p p p r t rt r t t t rst sst s s r t t r 3 t t t s t s r s sst s s t r 3 t ts t s t s r 3 t r t sst s r r s t t rt r s s t t t s t s t s ts t t s t s s r 3 tr s t sst t t r t ts t r t s s r sst S = (N,M 0 ) t N = (P,T,W) tr s t s q σ s t t M 0 σ M t r sts r 3 p S = (N,M,σ,Y) t tr s t s p r r 3 S t M(p) = M 0 (p) t r s M(p) < M 0 (p) Pr s q σ s tr s t t s σ(t) Y(t) r r tr s t t s s p s r 3 t s r t t tr s t t p s r t Y(t) t s t p, σ(t) = Y(t) r t tr s t t i p s r t st Y(t i ) t s t t M(p) = M 0 (p)+ W(t i,p) σ(t i ) W(p,t i ) σ(t i ) t i p t i p M 0 (p)+ W(t i,p) Y(t i ) t i p t i p W(p,t i ) Y(t i ) = M 0 (p). q t M(p) = M 0 (p) s t t tr s t s t i r r 3 t s r Y(t i ) t s q t s str t t st t s r str t s r r t s s t r t r t r p s r 3 t t r t t s r r ts t tr s t t s r t s p st t str t r t t s t s t t t t s

108 rs t r sst s t s t t ts rt r r 3 s tr s t r sst s s r str t tr s t r sst S = (N,M,σ,Y) S t s t st r 3 t t t s r 3 t t st r t S Pr S s tr s t t t s r s S s str t t r sts t r t r r t r 3 r r t q t t tr s t r s r s q t r 3 t t s t st r t tr s t tr s t sst t r t r t s t r 3 p s r t t P r s r r r ttr t sst s r ttr t sst s r s ss r sst s r t r t t r r rt s tr t r r ts r s r t t s s s rt s t s t tr s t s Pr s t t s t t s s s sts r str t r r ttr t ts r r t r s r t s r s q s P r t r s q t t s Pr s t s r s r rt s tr s t s q s s s t t s t s t t t r s r r r r sst s t r s q s P r t r s q t t s s r sst s t t t s r s t M FA r s r s r t s sst s t t t t s r t tr t t r r r sst s t t st s r 3 t s r s r s t s t t t r r t t r M 0 r r s ts t r s t s t M FA t r s t st r t r tr s t s q r s r str t r sst S FA = (N, M 0 ) t N = (P,T,W) s q σ t t s tr s t r t t s Y s t t M σ 0 M t r sts t st r (N, M,σ,Y) Pr t S σ = (N, M,σ,Y) s σ s tr s t t r sts t st r S σ t t r s t sst S FA s r s t t t tr s t t r sts t st S σ

109 t r rs t q t r sst s t t r s r tr s r r S σ r r t r s t t s r s s r t r S σ sst S FA s s r t r t r t t S = (N, M FA 0 ) t N = (P,T,W ) S FA s q σ s s S r t r sts M s t t M σ FA 0 M r S FA S t s q s Y FA σ s tr s t t sst s s q t M M t s s ts r r 3 s s t r s t st r S σ = (N, M,σ,Y) s r t tr s t s q σ t s t r 3 s r 3 ts t t t s t P f t s t t r 3 s t st r t t t ss t P fr t s t r s t t ss t P r s t r P f P fr P r r tr rt t t s S FA S FA M (p) = M (p)+ M (p)+ M (p), p P f p P fr p P r p P M (p) = M (p) M (p). p P r p P p P f P fr p f p fr p r r 3 t r 3 ts M (p f ) = M 0(p f ) r 3 t t st r t M (p fr ) < M 0(p fr ) r t r 3 r r ts r r 3 s r s t r s r t s s t r r 3 r r s t tr s t sst r tr rt t t r 3 s r 3 ts t s t P f r 3 s t st r t t s t P fr r s t s t P r r t s r 3 p s M 0 (p) s t r t s t sst r s st t t s tr rt t r s r t t t r t s r s

110 rs t r sst s s s s tr r t t r M 0 (p) = M 0(p) M 0 (p). p P r p P p P f P fr t r t s r s st t S FA p P M 0 (p) = p P M (p) t rst t r s t r t s t t q t s r q p P f M (p) = p P f M 0 (p) s P fr s t t p P fr M (p) < p P fr M 0 (p), t st t r s t r t s t q t s s t s p P f P fr M (p) < p P f P fr M 0 (p). s t r t s q s str t s r t t r t s q p P r M (p) > p P r M 0 (p) p P r (max p gcd p), r t st q t s r t t M 0 t t t t s M r s t s s M 0 t s s t s s t st P r s t M max p r r t s M t sst s r s r r r s s q s tr s t r t t t s s r t s r s S σ r s r S σ r s t st t s t t t r s t s t M FA r r str t t ts r s r s r r str t r t t M 0 t r s M FA r s t r s M FA N FA s t str t s r t t t t sst S FA = (N FA, M 0 ) s r rs r r r r r t r s s r rs

111 t r rs t q t r sst s Pr sst s r r t r tr s t s q σ t M σ 0 M t st r s M r r r tr s t s t s r tr s t s M s q σ Y s P r t r q s t s S FA r t Pr s t t s r r ss s t s r sst s r r r r r s s σ Y r s r s t r rs t t s t r r ss ss s s t tr t s t s r t r r ts s t t t r tr s t s q r tr s t s tr s t s r r r t s t r t t t r t t r t r rs str t P s ts s t st t r tr s t s q r tr s t s t s r s t r t s t t r r ts r r ts t t r r t r 3 t r s s s t Pr s t r r ts s t r t t r ts t r r s r 3 t t r s t s r t s t st t r rs t r r ss s ts t s s t s r r ts r r r P s ts Pr s t r t rst r s r sst s s t r r t r ss t r rs t r sst t r s t r rs t P s sst s r t t r s t st rst t r t s t t s q s s sst t s s q s s ts s sst s r s t t s s q s P s sst s r s t r t r s t t t r str t s q s t P s sst s s s q s s r s r sst S = (N,M 0 ) r N = (P,T,W) t t r t ts P s sst s S = (N,M 0 P ) r N = (P,T,W ) r r s s q σ S t s s q σ T s s S σ σ r r M 0 M S M T 0 P M S t M P = M Pr r t t t s 3 s s q s 3 k S σ s t t s s t sst s t r s s s t t σ = σ t s s S r σ s s 3 k t t t σ s tr t s 3 t s q s str t s r t k t M t 0 M M S t t t s s σ T s s S σ T t M 0 P M S r r p P M (p) = M (p)

112 rs t r sst s s S s P s sst S tr s t s t T t s t r t t t st P t s t T t σ T q s σ T s t s s S r r t s t t r s P t s M = M P t r s t s t T s (N,M ) r r M P = M t t t t s t t t t P r (N,M ) t s t s t M p P s t r t t t N (N,M ) t M t s r t t r N t s M P = M s s str t r p 3 p 5 t p t 3 t 4 t p p 4 t p t 3 t r P s sst t r t s t r t sst t t s r t p s q σ = t 4 t t t 3 t 4 s s t sst t t s q t t s s q σ t r str t σ t t s t tr s t s {t t t 3 } t t s t t t 3 s s t P s sst t r t ss r t r rs t r r sst s s t r r s str t r sst s r s t r t ss t rt r r t r 3 t r rs t r t t r s t r r q r r r r s r str t r r sst S = (N,M 0 ) t s Y t r sts s t t r k k s s q σ S s t t σ = k Y t t r sts s s q σ Y S s t t σ Y = Y s q t t s q (σ Y ) k s s S r r r t s r t st t r s s q σ Y t t k t s t P r t r s q t t P r t r σ t r r t t s r s σ

113 t r rs t q t r sst s r t r t ss r t r rs t r ss r t r rs t s r str t r sst S = ((P,T,W),M 0 ) rs t ts str t P s sst s r rs t S s t r Pr Pr s t s S s r s q s Y s s rt s t s t T S s r rs t s Pr s t t r sts s q σ Y t t s s S s P r t r s q t Y s r S = ((P,T,W ),M 0 P ) str t P s sst S t s Y s q σ Y T s s S r r σ Y T = Y T Y T s t Y Pr s t t t σ Y T s t Y s t r sts s q σ Y s S t P r t r q t Y Pr s t S s r rs r t r 3 t r rs t r r sst s r t s rt t r t r 3 t st t t t t r rs t t sst s t st r rs P s sst r t t r s r t r t r t r 3 t r rs t s r s q s s s t r ts r t r t r 3 t r rs t sst s tr s t r t t s Y t r r tr s t r s t r st s t tt Y r r tr s t s t s t ss t t r t t r r s r 3 t t t sst s r tr s t s t r t sst tr s t sst s t t s r tr s t r t r s sst ss t sst r r s str t r sst s r r Pr s t q s s s rt t s tr s t s Pr s t s t P r t r s s q s s r t s t t s Y s sst S = (N,M,σ,Y ) s t ss s t st r 3 tr s t S Y s t s st t Y r t r t σ r t s sst s t r t r 3 t r rs t s t r t t Pr s t s t t sst r t r t r t r 3 t r rs t s r str t r sst S S s r rs s s q S s t sst

114 rs t r sst s p 3 p 3 t p t 3 t p t p t 3 t p r r sst s t s (,, ) t t t s s t tr s t sst t r t t r t s q t t sst t t s t t r tr s t t tr s t s s t s t s r 3 s tr s t s q t t s tr s t sst s Pr r t t t sst s r rs t s s t s r t t s q s tr s t r t t t q s Y Pr s t s rr s tr s t sst s t t s r tr s t s r r tr s t s r r r s r tr s t t s q t P r t r q t Y s r Pr s t t sst s r rs S = (N,M 0 ) s r rs t r sts s s q σ Y t σ P r t r q t t s Y s t t M Y 0 M0 Pr s t s r r k t s q σ ky = (σ Y ) k s s s r s s q σ t s st t rk s t tσ s tr s t r t σ t t s Y k = k Y t M 0 M r rt Pr s t α st t s t st t s s q s α β s t t M 0 M 0 M β M 0 s t s r r tr s t t α(t) = max{ σ ky (t), σ(t)} σ ky (t) = σ ky (t) σ ky (t) = 0 β(t) = max{ σ ky (t), σ(t)} σ(t) = σ ky (t) σ(t). s s q β t s t tr s t s q σ t t s Y k σ + β = k Y t M 0 σ.β M 0 s rt r s s str t r t t t tr s t sst (N,M,σ,Y k ) t s r tr s t s t s s t st P s sst r sst t t r t r t r ts t t s r t ss r rt

115 t r rs t q t r sst s σ ky M 0 M 0 σ M r t s q s σ ky σ r s t t r sts s q β s t t σ.β s s σ ky = σ + β β st s sst s r str t tr s t r sst S = (N,M,σ,Y) S s t t t s t str t P s sst (N,M P ) t N = (P,T,W ) s t t t tr s t sst (N,M P,σ T,Y T ) s Pr str t t r t r r sst t ss t sst t r rt r s s q s sst s t r st t r t t t ts t t 3 t 4 t t 5 t 6 t t 3 t 4 t t 5 r r sst t t s t r rs r t s q t t t 3 t 5 t s sst s t t sst t r t r tr s t s r t r 3 tr s t s r tr s r tr s t s s t r s r t s s ts r q t s t t t r t t t r n tr s t s n = 0 S s sst r t r s t t tr s t t r t s r r t s r 3 t s r t p ts t s s sts s t sst s t r s t s r 3 S s str t t s tr s t t r sts S t r t t t s t t t rr s t s r r tr s t t t t t p t r t r s s t r t r t = t 3 s t t r s r s p = p 4 sst S s t t t s t t p t R t r r S t r t r s t tr t t s str t t S t r (u,v) u v t s R rr s s t r (a,b) S s t t a s t u b s t v t G R t t t t 6

116 rs t r sst s t p p 0 t 3 t 4 t t 5 p p 4 r t r t t 4 p 4 t 3 st rts r t r tr s t t 4 s t t r 3 tr s t t 3 p 4 s r s r t t r ts t 3 p 0 r t t 6 t N G = (P G,T G,W G ) t s t S rr s t G s t t G = (N G,M PG,σ TG,Y TG ) s str t s t P s sst S s t r s G s t p t 3 t 4 t t 5 p p 4 t 6 t p t p G r t t t tr s t sst t t r 3 tr s t t 3 s t t r t t r r r r s ts t sst t t G s s t G s t P s sst S S s s t t s t st tr s t r r S s str t t ts s t t t r ts t tr s t t s t t r t t s r S s t st t s t t S s t t t G t s t st tr s t r r p G S ts p S t G s G s t R t q t t p S s t G s G s str t s G s P s sst S s P s sst S s s r r G s t P s sst S

117 t r rs t q t r sst s G s ss t t S s t r S 0 = (N,M 0 ) r σ G s t P s sst S Y TG s s G σ TG s s G 0 = ((P G,T G,W G ),M 0 PG ) r r r r tr s t t T G σ TG (t) = σ(t) t s t s r S t s r G s rst t t G t s r tr s t s t t G t s r 3 tr s t s s r r 3 tr s t t f G S s str t t r t c r r tr s t t r t t f sts S s t s t st S s G s t t s c t s t t t t c = c tc t f t r t tc t f sts S s t G s G s t R t s r tr t t r s t s r 3 t ts t t s t r sts t r t c r r tr s t t r t t S t t rc tc s t G t s t s r tr s t tr t r ts t s s t f t r S c t c c p p G t r r sst S s t t t r ss p t f t r r 3 t t r t t rc t s t G t s G t s t r tr s t t r t t G t s t st r tr s t σ TG s tr s t r t t Y TG G 0 r r t tr s t s G t s ts G s S s t s t G s r t ts t s s t s t s G t r r tr s t s S r t t r t s r tr s t G s G s r s s P s sst G G s str t r P s sst S t str t r tr s t s t S t t t s s G t P s sst F = ((P,T,W ),(M PG ) P,(σ TG ) T,(Y TG ) T ) s str t sts G t s sts S s t t (M PG ) P = M P (σ TG ) T = σ T (Y TG ) T = Y T r s s P s sst F

118 rs t r sst s t p G F t 3 t 4 t t 5 p p 4 t 6 r str t P s sst F s r t r t r t r 3 t r rs t r r t r 3 t r rs t s r str t r sst S = ((P,T,W),M 0 ) S s r rs ts str t P s sst s S = ((P,T,W ),M 0 P ) s r rs Pr ss t s r r r t s t S s t r t s q r tr s t t t s tr s s t t S s tr s t s t t S s t r rs t σ r s q σ s s S s t t M 0 M t t sst S b = ((P,T,W),M,σ,Y) r Y s s S s s s S b t s t str t P s sst S b = ((P,T,W ),M P,σ T,Y T ) t r σ T S = ((P,T,W ),M 0 P ) Pr s t s S s r S s t r rs r s s s s sst s r s t t s s t sst s sst s r r r s ss r sst s str t r t r ss s q t t t ss t r r t P s sst s r r t r 3 t r rs t r t s sst s s r t r rs t sst s r t str t sst s t t s r rs t t sst s r rs ts r t P s sst s r t s r rs st t t rt r s r t r s r sst s t ss t sst s r s t ss ts P s sst s r t r rs s tr st s str t r

119 t r rs t q t r sst s S b S t t 6 t 3 t 4 t 3 t 4 t t 5 t t 5 t t 6 r sst S b t t s t r t s q t t t 3 t 5 t t sst S t r t t s t P s sst S b t sst S t r t t s t ss t P s sst S r str t t t t 3 t 5 s t T = {t 4,t 5,t 6 } s t s q t 5 s s S s t S b p 3 p 5 p p 4 t p t 3 3 t 3 t 4 t t 3 t p p 4 p p 3 p 3 p 5 p p 4 t t p p t 3 p t 4 t p t p 3 t 3 r t t t r sst s t r rs t s P s sst t t s t r rs tt t t r t t r sst s r rs s s ts r t P s sst s P r s r r r sst s r t rst r rs t r r str t r r sst s t s str t s t s t r s M FA r r r sst s t M CF t r M EC r t t r r sst s r t s t t M CF s s r rs

120 rs t r sst s r r M CF s r str t r r t N t t s t sst S = (N,M CF ) s r rs r r r r t r s t sst r rs Pr r t str t P s sst S = (N,M ) S t s t st t tr s t M s t r str t M CF t t s N t t p S t s max p t s t r s p S r ss t st max p gcd p t s s M s r t r t r q t r M FA r S s r rs s str t P s sst s S r r rs t r t r 3 t r t r sst S s r rs s s s r r sst s st t t rt r s q t sst s r s ss s t s t s sst s S s r t s q s Y Pr s t t r sts s s q σ Y s t t σ Y = Y Pr s t r r t r s t s q σ Y t s r s r s t r rs t r rt Pr s t s t r s str t r r r rs r sst s t r t ts t r rs str t P s sst s p 3 p 5 p 3 p 5 t p t 3 t 4 t p p 4 t p t 3 t 3 t 4 t p p 4 r t t r rs r sst s t s t t r t r t ts r rs str t P s sst s s t t ss r rs t Pr s r s t t ss r s t t s r rs t r str t r sst s r t s ss t s t t P s sst s s r s r r rs r t s r s s str t t t t s r t t r sst rt r t t s t st s t s s s sst s

121 t r rs t q t r sst s s t t t t s tt tr s r t s s t t ss r rs t r str t r r sst s r t t t str t t t s r s r str t r r sst S t S t sst t t s tt tr s r t t S S s t S s r rs Pr t sst s t s q s Y S s r Pr s t t s r s r t r S s t t s r rs r s q σ Y t P r t r q t t s Y t sst s s s S Pr s t t s r σ Y s s S s s q t t M = M 0 σ = σ Y Pr s t r rs Pr s t t t s t t t s t t ss r r r 3 sst s s t t t s t t ss r rs t r r r sst s r P t s t t ss r rs t s r str t r r sst S t S t sst t t s tt tr s r t t S S r 3 t r 3 sst S r r S s t s s t ss t r r sst s t S s r rs Pr S s r S s r r t s s st t s r t r s q t S r 3 r t r s r t ss r rt r r t t t ss S s t ss S s t ss r rs t S r t t r s t S t t s t t ss r rs t r S s t r s str t t r t r r t r 3 sst t s t t t sst r s r sst s r t p 4 s r rs s t t s t ss r ss t sst s t ss r r t ss r rs t t r sst s t t s r t s r t s r rt ss t r s s t t r t r s t t t r

122 rs t s st t q t sst s p 4 p 4 p 4 p 4 t t 3 3 p 3 p p 3 t t t 3 3 p p 3 p p 3 t t t 3 3 p 5 3 p p 3 t t t 3 3 p p 3 p p 3 t r t t t r sst r s r t t tt rt r t r M FA r s t r M CF r rs s tr s r t sst t r t t t r t r s r sst s t r t r t r q t r M FA r s t r M CF s max p3 gcd p3 = 3 = r t tr s r t s t sst s t s t s t t t r s r sst s r rs t t t t sst s r 3 r s t t r s t t r s s t t t r rs r s M FA r sst s r t s t t t s t t s s r r sst s t t r r t M CF s t r t t s r t tt p 4 r r t r st r s t t r r rs t t r t r t r q t s r M FA t t t s t t s t r t r t r s s s t s r t r M CF t tt p 4 s s t s t r s t t t t r s t t r s t r s t s r t r t rs t s t r t rs t s st t q t sst s r ss ss r rs t r rt s q t sst s st r r r t s r t t t ss ss t s t st r r s t tr r t r 3 t s r s s r s r r s ts r r r ss s t P tr ts

123 t r rs t q t r sst s t s r rs t r t r 3 t s t r t t st t r rs t sst s r t r s t ss t st t s t s str t t s s t r r rst t r rs t r rt r s st t q t sst s r r t t t t t str t t s r r s s s s s r q t sst s r s st t Pr s t r s s t r t r 3 t r rs t r r sst s s s t t P s sst s rst s t r t r t t t s s t t t t r q t sst s t s q s r s sst s t r rs s t s q s t t t r t s t s q s r rs t t t t r sst s r s q s r t s s t t ss r rs t s t t r t t t s t t s r s sst s s 3 t rt t ss ss t st t r rs t r rt q t sst s r t r r s s s q s t rst tr r t r 3 t r rs t r s st t q t sst s s t t s r t r 3 t r rs t r r sst s s r r t r 3 t r rs t r q t sst s s s rt r r s sst s t t rst r r r q t sst s ss s r sst s rs t s t r rs P ts s r Pr s t r q t sst s r r r P ts s r t s r P s sst s r s s t r r t r 3 t r rs t r r sst s st t t r r rs t t r s t r rs t t r str t P s sst s r t t r P ts r Pr s t t t r q t sst s ts P ts r t t r 3 t st t t t str t q t sst s r rs ts P ts r r rs r t r r rs q t sst s P ts r r rs

124 rs t s st t q t sst s r s q t sst s t r rs r t P ts t q t sst r r r rs s q s r rs t tr t t t s q s r s r st r rs t t s q s rt s q s s r P tr t t s t tr s t s T s q s s q s P r t r s q t s s s rt s T rt s q s s q s P r t r s q t s s s rt s r t r T s rt r t s q s r t s s r 3 s s r ts st rt r t s q t t s s t t t r s r rs t r tr t s q t s 3 t t t t t t s r s q ss t t s sst S s t t r sts t s s q t t r s r tr s t t st s q t t s sst S s r rs t s s q s r t ss t s s t st s s q s ss r t r r rs t s t t t r rs P tr ts r ss r s st t t s r t t t r t st s s q r t ss t s s s r rs t r t P tr ts t s t tr r r r r r r r sst s s r s r r s ts t r r r ss s t r rt r ss s

125 t r rs t q t r sst s st 3 r st t s t r t s t t r rs t r rt t st r r s t t rt s q ss t t s t r st t s s t s t t t st s s q s t r rs t s st t q t sst s s r r t t t s t s r s t s r t P tr ts t r s sst t t s t r rs r rs s sst rt r q t sst s t r r t t t t r sts s s q t t 6 t P r t r q t s s s q sts t t t 3 t 6 t 5 t 4 r s s sst s r rs s s q s t ss s st t r sst s t s s q s s P r t r s q t s r s q s r t s s r t s q t sst s s r r r sst t r t s t st s s q s t ss r rs t r r t s s t t s s sst s s s r r s q s s t r sst t t r t r 3 t r rs t s s s q s r s t t rst tr r t r 3 t r rs t r s st t q t sst s rst t r t r r r t r ss r rt t t tr t t s r

126 rs t s st t q t sst s t t t 4 t 3 r s s sst s t r t t t s q s s r t t t 4 t t t 3 s r t r r r ss r rt t t t q t ss t r ss q t sst s t r s q s r t r s r tr s t t sst t r t s t r s q s r r t q t s t s ts tr s t s s r t r t r r s t tr s t s t q t s ts t tr s t t t r s t r t s r r t r t s t r s s r t r s t r t t s t s t r t t r ss s s t ss str t ss ss t s st r ss ss t r r ss q t sst s t S q t sst s q σ t t s s S s r t s r Pr σ s r t s s t s t t σ s r t s r t rs t Q t s t t q t s ts t tr s t t t rs t t σ Q t s t t t r q t s ts s t Q s t s t r s t r q t s ts σ s t s s t t σ s r Q s t t t tr s t s t s ts Q r r t r t s σ t tr s t s t s ts Q r r t r t s r r t r t r s t r t r s q σ 0 σ t M t r r r σ 0 S σ t t s s q σ s t s σ = σ 0 σ t ss ss t t r sts tr s t t Q t s q σ s t M s t t σ t s tr s t s Q s t r r tr s t s Q r str t r t r t s r t ts t t t s t r t r s q σ σ sts s t t σ > σ s q t t s t r t r t t s q σ 0 σ tr t t t t t

127 t r rs t q t r sst s r tr s t Q st s r r t r t r σ 0 s Q s t σ s r t t s r tr s t t t E t t q t s t t t tr σ t t t t t t t s q σ t t K n t x (σ) n r r s K n x(σ) t s t r st r s q σ r t n t rr t x σ t s t n rr s t x r s r t s q σ = t t t t 3 t t t 3 K 3 t (σ) = t t t t 3 K t 3 (σ) = t t t r r t r t r 3 t r rs t s t s str t r s r q t sst S = (N,M 0 ) t N = (P,T,W) s t r sts s s q σ r S r r tr s t t M 0 s t t M 0 t M t r sts s q σ c t t s s t M t σ = tσ c t r k s t t σ s s q s t s σ = k σ r σ r M 0 t M σ c r t s q σ r s s s t s t t s q σ c s t M Pr t r st t r t K = K t(σ r ) t r st r s q σ r r t rst rr t t tσ r s t r K tσ t s q K s t t rr t s s q s s t s rt σ r s T t s t rst tr s t E t t r t r r σ r t t K s t t rr tr s t s E t t t s q K s t s t r t t s t t s t K t r t r t rst rr t t s q tk σ s s t M 0 σ c = K σ t r s t s t t rst tr s t E t t r t r r σ r t t K t s t st rr t r tr s t E t s t t t r t s t s q σ c t t r t r s t t t r str t t r r s t t rst s s t r t tr s t s r t r t E t t t r t r rr s τ q s t t K t t t tr s t t s t r r s t r tr s t s E t s r rst

128 rs t s st t q t sst s r t t t t s s q σ c t sst (N,M) t r t S t s s q σ r s t s q σ c t t s s (N,M) s t t M σc M 0 τ := t t E t \{t}, K (t ) > τ(t ) t q t s t E t s t t tr s t s t t t q t s ts r t tr s t t x s ( τ(t x )+) t rr rs t t s st s t (σ r ) τ := τ t x t tr s t s E t r t r t r t tr s t t y s t s K (t y ) > τ(t y ) s ( τ(t y )+) t rr rs t t s st s t K τ := τ t y t α t s st t r r s t t τ α σ r t κ t s q (σ r ) α τ κ r t tr s t t x s ( τ(t x )+) t rr rs t t s st s t κ τ := τ t x τ s t r tσ c r t r σ c

129 t r rs t q t r sst s t t 3 t 5 t t 3 t 5 p 0 p p p 3 p 0 p p p 3 t t 4 t 6 t t 4 t 6 r s sst t t s s t s q σ r = t t t 3 t 5 t 5 t 4 t 3 t 6 s rst t t = t 3 s r t t sst t r t t rst t t tr s t p t r σ r s t t 3 t r t str ts t s q σ c r t r t M 0 st rt r t sst t r t s r t rst τ = t 3 K = K t 3 (σ r ) = t t t t t r r σ r t s r t r t E t s r rs t s s t t t s t r r K t t r s t r tr s t s t t t t E t t ss t r r (σ r ) s s t E t t s s t t t r s t r t s s E t rst t ss ss t r r r s t st q t s t s s t t t r s t t r t s q t t s s q ν s r t t r s E t t r s t t r r (σ r ) t s rt σ r s T t s q ν s r t s r r tr t t t t t E t r s t t E t s t r t r t s r t t r t t t rst t r s t r t s tr s t t y s r t t r t r t t t r s t t K (t y ) > τ(t y ) t y s t t t t rr t τ r s t r r st s t tt K (t y ) t t r t r t s t t s s t t t r t t r t t rst t rr t s q τ t t κ t r r t r r rr s κ rst tr t t MinPos(E t,τ,κ) t r st s s t A E t s t t r tr s t t A s t s s τ(t ) < κ(t ) s t t τ κ MinPos(E t,τ,κ) r t r s t tr s t t A s ( τ(t ) + ) t rr rs t t s st s t κ r t r t I(k) t tr r k 0 I(k) t t t r t k r r tr s t t u s t t τ(t u ) < κ(t u ) t u = MinPos(E tu,τ,κ) t r r tr s t t y E tu τ(t y ) = K(t y ) r K t s t s q Ku m (κ) m s t τ(t u )+ r st rt t k = 0 s r t s s r r t u s t s τ(t u ) < κ(t u ) t u = MinPos(E tu,τ,κ) t u s t t E t t r s E tu

130 rs t s st t q t sst s t t 3 t 5 p 0 p p p 3 κ = σ r = t t t 3 t 5 t 5 t 4 t 3 t 6 τ = t 3 t 5 t 5 t 4 t t t 4 t 6 τ = t t 3 t 6 r t t t sst t t t t rst t rr s τ t r t r ss s t t rr s tr s t s κ t t t t r τ r st rt t rst K = t t t t t tr s t s p r r s t K r t s q t t rst r s t t t tr s t s p t t s r r t rst rr t 3 rr t t r t s r t st rts t r t sst r r t q t s t s {t 5 } s t rst t r t s τ q s t 3 t 5 t 5 t 4 t t t sst t t t r t rst α = κ = σ r s r s t s s q κ t rr s t t t t r τ t r r κ s t s s t r t s q τ = t t 3 t 6 t r t t s q s τ κ r r r s t s t t r t rst r st rt t s s q τ s r t s σ c = t 5 t 5 t 4 t t t 3 t 6 r t s r r r s τ s K t r t t r tr s t t y E tu s t s s τ(t y ) = K(t y ) t r s t u s t E t t rst r r s t rr s E t t t t K t t t rst q r t s r tr s t t y E t s t s s τ(t y ) = K(t y ) ss t t k t r t s t r I(k) s tr r t r t k + tr s t t x s r t r r κ t t t r t k + r r tr s t t u s t t τ(t u ) < κ(t u ) t u = MinPos(E tu,τ,κ) t K t s q Ku m (κ) r m = τ(t u ) + s r t s s rst t u s t t E tx t K s t s s q s t r s t r t r r tr s t t y E tu τ(t y ) s t t r t s τ(t y ) = K (t y ) t r s t u s t E tx E tx = E tu t K t s t s r rr s r tr s t t y E tx s t K ss t t t x t r s t r t t r t x s r s r t s s t tr s t s r rr s τ s r t s t x s q t r r tr s t t y E tu τ(t y ) = K (t y ) t t t q t s t t r s s t tr t t t r t k r tr t t t r t k + t r t s tr t r t r t t r r t t t x s t r st r t rr t m = τ(t x )+ t s q K = Kx m (κ) t rr s K r r r s t t s q τ t rr t t r t s t t r t r t r r tr s t t y T

131 t r rs t q t r sst s τ(t y ) K(t y ) r r t s q K s s t M 0 s t r t t s t x t t s s q κ s τ r t t tr s t s t t s t x t st s t s s K t r t s t t t tr s t s E tx r t s t s K s τ s t t s t x r t st t r t s t r r K r M 0 t t t x s t t t t s τ t κ t r t s s r t s r s t r ss tr s t s t r r κ t t x = t m = τ(t )+ = K = Kt (κ) = t τ s r r rr t rr t s r τ t s t s t r t r t t t t r t t x = t 3 τ = t 3 t 5 t 5 t 4 t t τ = t 3 t 6 m = τ(t 3 ) + = K = Kt 3 (κ) = t t t 3 t 5 t 5 t 4 τ s r r t rr s K r r τ(t ) = = K(t ) τ(t 3 ) = = K(t 3 ) t s t 3 s s κ s t r (σ r ) α r t r α > 0 t s s q tσ c s s q s t s t t t r t r 3 t r rs t r q t sst s r s r q t sst S = (N,M 0 ) s t t N = (P,T,W) sst S s r rs t s s q Pr s s s t ss t s t t S s r rs t ss ss t t r sts s s q σ 0 s s rt s t s t tr s t s t r rs t ss t t r sts s s q σ r t r t M 0 s t s s q σ 0 σ s s s q r t s t s t r sts s s q σ r t σ sst S s t t t r t r s s q σ t M 0 M t r sts s s q σ t t s t t t r r t t r s s t t t n σ t r rt P(n) s q σ t n s s q t sst S = (N,M 0 ) s s q t σ r sts S t t r sts s s q σ s t t M 0 σσ M 0 n = 0 σ σ r t s q s t t r s r t r s s s n > 0 t σ = tσ t t M 0 M σ M ss t r rt P(n ) t tr t r sts s q σ c t t s s t M s t t M σc M 0 t s q tσ c s s q s t s q σ c t s s t M t t t s s t s q σ s 3 n s s t sst (N,M) t s q σ d t t s s t M r t r s t M s t s q σ = σ d σ c s s t M s t M 0

132 rs t s st t q t sst s σ r t σ M 0 M M σ c σ d t t t r t r s s q S t r sts s s q t t r t r s t t t r t t S s r rs t t r rs t sst s r t r 3 t r rs t r t r s t t t t r rs t r rt t s q t sst s s st t t t r r rs t t t s r q t sst S = (N,M 0 ) t t s r rs r r M 0 M 0 t sst (N,M 0 ) s s r rs Pr Pr s t (N,M 0 ) s s r r r t r s t t s q σ s s S sst (N,M 0 ) r s σ s s q t r rs r P r s r r sst s s t r t r 3 t r rs t r q t sst s t s t t t r M EC r s s r rs r r q t sst s t t s t str t r s r s sst s s r s rr ts t t t sst s r P tr t t p t st t t t S s sst s r t p t st ts ts r t ts t max p t t t t p S max p p S gcd p t p S gcd p t p S gcd p gcd p max p gcd p max p gcd p t t t t max p max p Pr t p r ts S t S max p max p t r s r t t t max p = 0 t p r ts r ts S t S t t s s t t ts s rr t r r r q t s gcd p gcd p s t st q t

133 t r rs t q t r sst s p p r t t gcd p = max p = 6 max p gcd p = 5 t s sst t r t gcd p = max p = 4 max p gcd p = r t r s t t t s t t r s r s M EC M JF s s ts r s r str t sst S = (N,M 0 ) t t s r ts S s t r s s t t M 0 s r t r t r q t t r M EC t r s S s r s s t t M 0 s r t r t r q t r M JF S s s q Pr str t t s s t r t t r t S s s sst S t ts r t r M CF r r r r t M EC t t s t r r rs r t C...C k t k ts r t sst r s q s s s sst s s s t sst r t C i i =...k s S rt s q σ i s P r t r s q t t s C i r t s q σ = σ...σ k s s q t t s s S s t r r t t 3 t 4 t t 5 t 6 r s q t sst s r M EC ts t ts C C r s t {t, t, t 3, t 4 } {t 5, t 6 } t r t r t t t s r r t r q t M CF t s r r rs s r t rt s q σ = t 3 t t t 4 t t s s C t rt s q σ = t 5 t 6 t t s s C t s q σ = σ σ s s s q r t sst s r t t s ts r s sst s t sst s t r t t t s r t t t s t M 0 s q t r

134 rs t s st t q t sst s r t r t M EC r r t t t s t s r rs t r s t sst s r M 0 s q t r r t r t r M JF t st p s t s s M 0 (p) max p t sst t s t s t r t r t M 0 ts s s r t r t r q t r M JF s t s r rs r s q t s s s s t t S t s t st r rs t s s r r rt s q t t s r rt s q s r t ts t t r t r rs r t r st ts t t r t t r rs t t t t t r t r t t t r r s tc t r sts s q t t s s tm 0 r s t tr s t s C r t r s t M 0 r str t s t s t t t s s q t 3 t 5 t 7 t 8 t 6 t 4 t t r C 3 t t {t 7, t 8 } s t r C p t p 0 t 3 t 4 t t 5 t 6 t7 pf t 8 r r t C t t s t {t, t, t 3, t 4 } s t r rs r t t ts C C 3 r s t {t 5, t 6 } {t 7, t 8 } r r r t tr s t t 3 s t t t p C r rs t r t s q t 3 t 5 s C 3 r rs t s s t r sts S t st t t s t tr s t t t r t µ t r p 0 t 0... p f r p 0 s t 0 s t tr s t µ M 0 p f s t C t sts str t ss t r s ss t s r s r t t p 0 t 3 pt 5 p f r µ s t r t ts s tr s t s r s t t σ t s q t s ss tr s t s t r r µ t s st rt t 0 t ss s ts ts s t s t t µ s t s t D r t u t C t t r tr s t µ s t t st t ts D t ts D r r t t µ s t s q D i i =..u t D u = C r D = C D = C D u = C 3 t t µ r r tt s d...d u p f r r r i =...u d i st rts t s t tr s t d st rt t p 0 p f = d u s t t s t µ t t s t C d i s t st s t µ t t s

135 t r rs t q t r sst s t D i t t s t t s t D i+ r d = p 0 t 3 d = pt 5 d u = d 3 = p f rst s t t σ s s S t σ ts rst tr s t t 0 s r r r p i µ t s t st max pi gcd pi t s S s t t σ s t s t t t s t r τ tτ r τ r S t s t 0 t r M t t s t t t t s t r t s t r r s t s s t s tr s t t t s M 0 s ss t µ t s t tr s t t tr s t t 0 s t s q t p t s t q t t p t tr s t τ r t s r p t s p t t st max p gcd p t s t M 0 s t s M s t r ts t tr s t s τ t s tr t s s t t t r t t t p sts S t s tr s t t p t r t tr t t t µ s r t t p s p s ts q t t t t p s r τ t ts q t p s r s t tr t t t t t t µ s t r t s t t p s r τ t t t s s s s s t s M t t σ s r t u t t t r M c r r σ s C r rs t t t s s q σ s tr s t s t t ts D...D u r t r s t M 0 u = t σ r s s q D s r rs t r M t t s t p f r t M t t C s r t r t r q t r M FA ts ss r rs t s q σ s M t t s q σ t t s D r C = C t r σ = t 3 s p C r rs t s q σ = t 4 t t s s t t s t M 0 u > s s t t t r rt s tr r u r s q ν σ rr s t µ = d...d u s s t M 0 µ t r t r t D u t tr s t M 0 s t 0 t s t r M t t s D u r rs r r t s q ν ss t t d u s s t M s p f C r rs t r M s D u s r rs s q ν s s t M s t t ν ν s rt s q s P r t r s q t t s D u t s s t M t t t s s s q γ s tr s t s t t ts D...D u s M t t s t M 0 s q t t s q ν γ s s t M s t M 0 t s tr s t s t ts D...D u t t t r rt s tr r r u r t s r rt t t r r t C i t t s t t r rs t r sts s q σ s t M 0 t r M c t t s C i r rs t r sts s q σ i s P r t r s q t t s C i

136 rs t r sst s s s t M c t t r t r s t M c s q σ s s t M c r t r t M 0 t t t t s s q s γ i = σσ i σ t sst s str t r ts t t st t t r rs t t t s rt s q s γ i r r t C i t t s t t r rs t t r t t s t t t r rt s q s t r rs ts t s s q r t t sst t t t t r s M EC M JF r r rs r r q t s r sst s r s t r P r s t S = (N,M 0 ) str t r q t sst S s t r s s t t M 0 s q t t r M EC t r s S s r s s t t M 0 s t M JF t s s S s r rs r r r r r s s t ss r rs t Pr r r q t t s r ts Pr s t t r M 0 s t s s t t s r t s s q t r t r 3 t r rs t r q t sst s r t sst S s r rs r r ss r q t sst s s t s Pr s t t s s t r rs t r rs t r sst s r sts tr r t r 3 t r rs t r r r ss t s s ss t s t r ss ss s r sst s r t t str t s r sst s t st t t r r s ss t q t ss r r t str t t t rst r s r t s ss t t r t r 3 t P r s s t t t r s M JF r r r r sst s r s r rs s s t st t ss r rt t r r sst s t t t t t s t r rt t t r r r s t sst s r s r t r t t r s M JF s s t s t s r rt t t t tr s t s r sst t s t t t r r s q t t t r rs t t s t r q t s ts r t s t t t r s ts r q t sst s r r t r r t rts t t r ts

137 t r rs t q t r sst s rst t str t r s s t r rt r t r r sst s s r r r s s r str t r r sst S = (N,M 0 ) t N = (P,T,W) s t t r r r s s t r sts s s q σ r S r r tr s t t M 0 s t t M 0 t M t r sts s q σ c t t s s t M t σ = tσ c t r k s t t σ s s q s t s σ = k σ r σ r M 0 t M σ c r t s q σ r s s s t s t t s q σ c s t M Pr t t s t t s s t q t s t K = K t(σ r ) t r st r s q σ r r t rst rr t t t σ r s t r K tσ r K s t t rr t s s q s s t s rt σ r s T t p t q t t t s t rst tr s t p t r t r r σ r t t K s t t rr tr s t s p t t s q K s t s t t t t t s t K t r t r t rst rr t t s q tk σ s s t M 0 σ c = K σ t r s t s t t rst tr s t p t r t r r σ r t t K t s t st rr t r tr s t p s t t t r t s t s q σ c t t r t r s t t t r s s r t q t s ts r r t s s t t t r s t r t s s p ss t r r r s s s t t t r s t t r t s q t t s s q ν s r t t r s p s s t t t t tr s t s p r r t r t s ν t r s t t r r (σ r ) t s rt σ r s T t sst s t r r p t t tr s t r t r t s t t ts ts p r r t r t s str t ss t t tr s t s r t r t s tr t s p s t t r t s r st t r s r t r s κ s t r (σ r ) α r t r α > 0 t s s q tσ c s s q

138 rs t r sst s r t t t t s s q σ c t sst (N,M) t r t S t s s q σ r s t s q σ c t t s s (N,M) s t t M σc M 0 τ := t t p \{t}, K (t ) > τ(t ) t t p t s t t tr s t s t t r t ts s r t tr s t t x s ( τ(t x )+) t rr rs t t s st s t (σ r ) τ := τ t x t tr s t s p r t r t r t tr s t t y s t s K (t y ) > τ(t y ) s ( τ(t y )+) t rr rs t t s st s t K τ := τ t y t α t s st t r r s t t τ α σ r t κ t s q (σ r ) α τ κ r t tr s t t x s ( τ(t x )+) t rr rs t t s st s t κ τ := τ t x τ s t r tσ c r t r σ c

139 t r rs t q t r sst s s t s t t t r t r 3 t r rs t r t r r sst s s r r r s r s r r str t r sst S = (N,M 0 ) s t t N = (P,T,W) r r r s sst S s r rs t s s q Pr r s s r t t r r t t s t M t r r r t t t t r r r r M s s t t t r r sst s s t r s t M JF r r rs r P r s M JF t S = (N,M 0 ) str t r r sst s t r M 0 s t M JF S s r rs r r r r r t r s s r rs Pr st s s q s r r s t ss r t r rs t r t P tr ts r t t s s t t S s s q sst S s r r ts Pr s t r r t s str t r s s S s s q s s r r r s r s S s r rs r r r r s r s r r r s s q s r s s r r r r s r rs rs t s q s r t r ss s str t s r s P tr ts s r ss ss s t r r t q t ss r r rs s q s s t t P tr t s q s t ss r r rs r t r s r r str t sst s t t t r s q t t s t r s s M 0 (p 4 ) = 0 t t s r r s t p 4 t t sst s s s t t s t p 4 t sst r s r rs t t sst t r t t t t r s t t 3 p 4 M 0 (p 4 ) = 0 s sst s t t r rs s s q s s s t sst t t r t r p 4

140 rs t s q s r t r ss s p 0 t p p t 0 p 0 t p p t 0 p 3 t 3 t? p 4 p 3 t 3 t p 4 r t sst t t t r t t s t p 4 s ts r M 0 (p 4 ) = 0 t t s r rs M 0 (p 4 ) t t s t r rs t r t M 0 (p 4 ) = 0 t t r s t t 3 p 4 t sst t s t t r rs s t s q t 0 t 3 t t t s sst s t r t t p s t r t 0 s ss r t t t r t 3 t t r t r s t 3 t r t s t s p s M 0 (p 4 ) = r t s q (t 3 t ) k r s t t r k s t ss p t t M 0 (p 4 ) = t t r s t r s t r t s p r t t s t s t t t t r r s r t t t t r t r t p 5 t t t s r s t t r s t t s t t p s t p 0 r r t t t s s q t str t r t r r rs sst t t st s t r s q r t sst r s t t t s tr ss s p 0 p r t ts t 0 p 0 = {t 0,t } p = {t 0,t 3 } t s p 0 p p p 0 t r r r r s tr t t t s r rs s s q s s t t t r s t p 0 t t sst t r t r

141 t r rs t q t r sst s p 0 t p t 0,,,0,0, t 0 t 3,,0,,0, t p p 5 0,,,0,,,,,0,0,0 t 3 t t 0 t 0,,0,,, 0,,,0,, p 3 t 3 t p 4 t,0,0,,0, t 3 0,,0,,, r sst t t s r r rs s t s q t 0 t 3 t t ts r t r s t r t s t t r r r s s q p 0 t t 0 p p p 3 t 3 t p 4 P p t t s t p p 0 t 0 p t p t 0, t 3 p 3 t, t p 4 t r s r r s tr sst s t t r rs s t s q t 0 t 3 t t r t s t r P tr t t t s t r rs s s q t t s s t r t r t s t p 0 t r s s t t t s ss t t t t s r t s q t 3 t s t t r (,,0,,0) r t s ss t r t t r r t s q t 3 t t t 0 r t r s t t t r s t r t t s t r P tr t rs Pr r s rs r 3

142 rs t s q s r t r ss s p 0 p t 0 t p t t 3 p 0 p p p 3 p 4 t 0 t t t p 3 p4 r s P tr t s t r rs s t s q t 3 t t t 0 ts tr t r t s s t t t s t s r t t 0 t t t s t r r t s s st t t t r (,,,) t P r t r t s q t t r sst s t t t r s q r ss r r rs r s r t r sst t t t s tr s t t t ts s t str t r t s t s r s t r r t str t r r t r t t t ss t r ss t t t s t s r 3 t s ts t t s r 3 t t 0 3 t t p p p 0 t 3 p 3 r s r P tr t s t t r rs s t s q t t 3 t t 0 r r t t s tr s t t 0 s t ts p s t t p 0 s s t s t r s t r t ss t 0 s t r r t r t 3 r t r s t 3

143 t r rs t q t r sst s r ts s r s ss s r r s t t r s r ss s r r s t sst s r r s r s t s t t t sst s r t s st t s ts ss ts r rs t t tt r t s t t t r sts s q t t s s t t t r t t s r rt s s r ss r t r r s tt rs t P tr ts s tr r q t r r 3 r ts r r ss r r s ts t t r r s r r s tr sst s t s B R r ss r r s ts t tr s r t t t r r t r sst s s q t r t sst s t s R r r r r r ss 3 r r s ts t t r r s t r sst tr s t s t s B R

144 s s t s t r st t r rs t r rt s r r ss s ss s t P tr ts r t ss t s s r rs t r rt s t s t s r q r r t s sst s st r s t s s t r t r s st tr s t s t s s t r t s t t t r rs r t rst r rs r s r t ss r r ttr t s sst s t t r t r 3 t r rs t r r sst s t t r sst s s s t t s r s sst s s t s r t r 3 t t t rst r rs r s r t s ss r s r r t s r t t r t r t t s s t s tt tr s r t s t r s t t r ss r rs t r t r t s t t r t s t s t t t r rt t r st r t t r s t r s t r r ss s st t q t sst s r r s t t rst tr r t r 3 t r rs t s t s r s t s t t r r s r r q t sst s r s r rs t t t r r s t r s t r r r r sst s t s r s r rs r t s ss st t t r t s t s q s r rs t r r ss s t P tr ts r t r s s t t t st s s q P tr t s t s s t t s r r rs t sst s t t r s r r t r 3 t s t t r t t t q t str t r r s t s r s t t s r t r rs t s r r ss ss s r st rt t s r sst s t t t r rs

145

146 Chapter 6 s rs t s

147 t r s rs t s t s t s s st t t ss r rs t r rt s s r r ss ss s t P tr ts str t r t r P tr t r str r t s r r tr s r t s t t t s t st t r r rt s t r r s t t s r t t t s t s r 3 r s r s ts t t t t sst s tr t tr s r t rt r s t t s t s r t sst s s t t s r t sst s t t r t s s st t r t s t r r s r t s tr s r t t t r 3 t sst s s st t t r t s r r t s r s t tr s r t t P tr ts t t r P tr ts t s tr s r t s r s r str r rt s t ss t t r ss rt r r tr s tt tr s r t t t s t r r sst t t r str r sst s s t t t sst t r s t ss r rt s t r t t st t ss r sst s t t rst s t t ss r r r sst s r s r t r r sst s t t r t ss r s t s r s s s s t r s t r r q t sst s t rst r s r t s ss s t t r s t t ss r t s ss r sst s s t s tt tr s r t t t r t t t r s t t s t t t t s r rt t r st t r st t r rs t r rt r t ss ss t r s t rst r rs r s r t r r ttr t sst s r t r 3 t r rs t r t r sst s s s t t rt r r ttr t s sst s t rst r rs r s r r r sst s s s t s tt tr s r t t s t t s ss r rs t r t s ss t t r t t st t r str r s t sst r t r t s t t s r t t t r t s s s t t rs t r r s s t t r r t r 3 t r rs t r r ss

148 t s t tr t t t q t ss r t s r tr t t s s q s s t t s s q s t r r rs t r t ss ss t r t s ss t t rst tr r t r 3 t r rs t r s st t q t ss t s st t t t ss t s t st s s q t s r r rs t s t s r s ts t r t rst r rs r s r r q t sst s s s r r r sst s t t r t ss r s s t t t st s s q s t s t r r r ss s t P tr ts t r sst s s r tr s r t t r sst s s r r t r 3 t s t t r t t t q t str t r s r s ts t s r t t ss r rs t s r r s s s ss s t P tr ts r s r r t s r t r t r t st r tr s r t s r r t ss s t t r r r ss t q t r sst s t s r t st t r r rt s r t r ss s t s s r 3 t r s ts t t s s r s ts t r rt s r s r r t tr s r t s st s s t r rt st q t t t sst r t tr s r sst Pr rt s t tr s r t s r s s r ss r rs s t s r P s tt

149 t r s rs t s r s t r s s t t s s r r t ss r rt r t s r s t r r ts t s r s r s ts P s t t s r ss t s r s P s t t s sst s r r q t s r 3 s t r s s t t s r r rs t t t r t t r t r 3 t s r t s r s t r r ts t s r s r s ts r t r 3 t s s t t s r t ss r rs t r rt s t s P s t r t r 3 t s r s t s sst s r r q t

150 P rs t s q st s t q st s r rst t t t ss r t t r t r s s r t t t t r r s s t t r3 st s t r 3 s q st r s s r t r s t t s ss r rs t tr r t r 3 t s r rs t r s t r sst s t t r s ss r s t s r 3 t r t P tr ts t s t s r t tr s t s t t t r t r t t s t s t s r t r tt t st t st t r tr s t s s t t s st s t t s ss s s t t r ss t str t r r r sst s r rst str t r r s ts t t r r sst s s t t t st r P t s r r ss s P t r t r 3 t s r ss t r t P tr ts r t r 3 t sts r q t ts r s r r s t s t r t r ts t s q st s st t t r t ss r rs t r rt s st t s r r ss ss r s t t s r r r tr s r t t P tr ts t t r ts r r t r 3 t r rt s t s s ss t t P tr ts Pr rt s t P tr ts t s t t t s t s s P tr ts t s t t t r r t sst r r P t s t t r t sst st r t sst s r t t r r ss s t t t r sst s t ss s t t s r t s t st t s r t t r r P r r t P tr ts s st s st st P tr ts s r r t r sst s s s t ss s r s s t t s t s r t P tr ts s q t s r s r s r r rt s t ss s t t r r r ss t t t sst s t t t s t s s st t r st t s r s r s t r s P t r s t r 3 t s t r s

151 t r s rs t s r r t t q s s st s st r rt r s t r t rt r r s r s r r rt s r r s t t r t rs t s r s s s t t q s r s st t t t P tr ts s 3 r r t t s s t s s r t t q s ss r t t r t t s

152

153 t r s rs t s

154 r t P t st rst r rt r t s s r rt s t r r P tr ts r t t r s r ér r r r tr t r t r r P tr ts s s r t r st t t s r ér r r r r s ss s P tr ts t s s t r str t r r rt s r r s t s st s r t s P rt s r r s t t t t r t t t r s Pr s t t r t r P r str t Pr ss q s t s P P s r s r r s r s t str t t P tr t r t s ts r t t r s r r t r r t t ss s s ss s P tr ts s t r t r P tr ts t r t s t r s r r r r r r r ç s Pr t P r ss tr s s P tr ts t s t rs t r P tr ts t r t s t r s r r r r r r r r r r tt r r P r s s r t t t r s r s t s t t tr s r r tt r r rr t r 3 r s 3 s r t t t r s t r s st s r t s t

155 r r r s s r ss t st t t r Pr s t t s t t r s r rt t ér r rt t r rt s ts s tr s r t s 3 r t r s P tr ts s r r r rt t ér r rt t r s r t s s t s ts r r s 3 r t rs P tr ts tr s r Pr rt s s r r r st st P r s t r s r P r s st t P tr ts r r r t r t s r P tr ts s s r t s r s r s P rstr t st t t t r t r st s Pr ss P s r r r r ss tr r ts r ts r r P r t rs t s r P tr ts t r t s t r s r r r r r r r r ss r t st s t r ts r r s s t r s tr r tr r s r r r r r rt t r r s r r q r s r r ts P tr ts r3 r3 3 r t r s P tr ts t r t s t r s r r r r s ss r r 3 P tr ts t r r s r s r3 s P s r t r s ts r s ts r t s r t r t s t r r t t r t r t s t r s r r r r r3 st s t P tr r3 st s t r 3 s ss t r ts t t r r s tá s t r

156 r t s t t r t r t s t r s r r r r r tr t r t s P tr ts r st s r s r r r r t s s r r t s s r tt t Pr rt r rt P r s tr rs s ör s r s r3 r P tr ts r r ts r t t r r rs t Pr ss r s ör s s P r s t P tr ts s r3 r3 3 r t rs t r s P tr ts s s t r t s t r s r r r r s r s r t s r s P rstr t st t t t t Pr s t t s r r s st s t rs s s r3 r s r3 s s t ss s r P tr ts P tr ts s tt r s s r3 r s r3 P tr ts t t t t r r rs s P r Pr ss s t r t s s r3 r s r3 t t P tr t r s tr t s 3 r t rs t r s P tr ts s s t r t s t r s r r r r 3 t 3 t rt 3 P tr t s r t r t r sst s r s t s t s t t s r r r r s r r 3 s r s t r t s r t t t t s s r rs q t r t r t ss r r P tr ts t r t sst s r r t s t t t r s t

157 r t q st s r P tr ts r rt r t r r t r r r r st t rt t r Pr rt r t ss t r t r s Pr ss s P tr ts PP s t t t P tr ts r P rt t ss ss s tr ts r t t r s s s r r t s r s P tr ts r t t r s s s t s s r r r r Pr s P r s r rt r t t r s r s r t t s t r P r Pr ss s r r r ss rs t t t s s r s t r r s r t s r ss r s t t rs r ss rs tt r s t Pr s t s r t Pr rt s r 3 P tr ts r t s t t r t r r q r s t s r rt rt t t r rs t 3 r P 3 r sé tr t r t s r t tr s r t t st s s t s r t Pr r t3 t rs tr s r t t st s t r t s tr r t s s r r r tt r r tt r r s t t r t ss t t r s r r r t s r s

158 r r r st r r t r t r P tr t r t r r t s r r r Pr ss r r t sst s r t t t s r t t r r r Pr s t t r r ss s r s ér r ér r r r r t t r r r t r t r t s s r r r r t r t P tr ts r rt s s s t s Pr s t s P r s s P r s Pr ss t r s P t s s rt t rs t r r P t P t tt r t t r s r r r s r s t r s r s Pr ss st s P s t P s s P r tt r r r r t r ss r s r r r s r s t r s r rt rs t r r r 3 s r 3 s t t r s r t r s r r q r r ss s s t s t r st st s q t Pr ss s s t r tr t r s rr r r s s t s r r r r q r ss r r r r t sst s t s t rs t r P tr ts t r t s t r s r r r r r r q r s s s q t r ss s t t r t t r rs r s t s t s t t s s s s tr t r r rt s P tr ts s t r t t t s t r s r r r

159 r rt tt P tr ts t r s P tr ts s r r r r q r sé r r r r r t q s r t s s tr s t t sst s s 3 r t rs t r s P tr ts s s t r t s t r s r r r r r r r r r r r tt r t r s s rs s s r 3 t Pr ss t s Pr ss 3 r 3 r t t r st s r t str t P tr ts r t r st s s r r q r P tr r3 st s t sé t sst s s t r t r P tr ts s r r r r r q r ss st t s q t sst s rs t r t r P tr ts s r r r r r q r r ss q t sst s rt tt t r t r P tr ts t r t s t r s r r r r r r q r tr t r t r q t sst s r t t r s r r q r sé r P tr ts r t r st rr t sst s t s r s rr s r s t s st s r t s P rt s r s t r s r s r rt rs r s t t r s t r r t r r r r s r s t s t st s s r rs rt rs r rr t P rr s r r t t t t r t s r t t t

160 r r s Pr s t t s t t r s rs rt rs r rr t P rr s r r t t t t r t s r st t r t sst s t r ss r t t t s s s r

161

162 st t s t r t r t r r s s r s r r P s t t s ss r s s ss s t P tr ts r s t s t st s s r s t s t s t t tr s r t s t r t s t s s r t r s r t rst r rs r s r r r ttr t sst s t r t r s t r r 3 3 r r s s r ss t st t t r r s t t s t t r s st s r s t r s s t t s ss r r s r r s t s s r t t t s t t r t s s r ss s s r s t r s s s r r P t t s ss r t r r st s r s t t t r t r t rr t st s s r s t r s ts t tr s r t s t P tr ts r s r str r rt s t r t s t s s r s t rst s t t s ss r r r r sst s s s r s r r t rs t t r st s r s t t t r t r P tr ts s r r r s t st s t r rs t r rt r r sst s r t s ss r t r 3 t r rs t t rst s t t s ss r rs t r r s t t r t s t s s r tr s r t r r sst s t r sst s s s t t t r r

163