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ד?N;=S ØNÖE5(361EP45 P4>‹ØA>@ÜE>=Bp5

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I&>@7¾IÝE7F>@ØN5ØNÖ45¶5&SrÙ43F?N3¾I&>27

PU3FBpØN?K3FÜ4ÝUØN3F;21-;‹Ú258?aØKÖ450Ú=5&?NØN3¾I5gB

C

S„Ý476ØN3FÙ47FL³36Ø"ï(36ØNÖ

1 − c

C >21EP>=P4P

c

ï5&3F<2ÖOØØN;kÚ258?pØK5

u

ê Q"Ö43¾B(?K58îOÝ43F?N5gB;2145P4>@ØK>@Ü¿>2BN5Å>=I&I5gBNB sPU3¾BpÛBN5&58Û ê

Q$;À?K58>2IAÖr>ÀÙ4?K58I&3FBN36;=1rÜ¿58L2;21¿PØKÖ451dÝ4S„܆5&?a;@׆ëE14<258?NÙE?N3F1OØKB|BK>zÚ258Pf3F1rØNÖ45P4>@ØK>2ÜE>2BN5

ï'5I&>21-Ý¿Bp5ÀØKÖ45?N5gIÝ4?ABN36Ú=50Ù4?K;2ن5&?NØ^L;@×

PPV

áç é‹å

PPV(u) = c

u +(1 − c) 1

|O(u)|

X

v ∈ O(u)

PPV(v)

ï(Ö45&?K5

u

PU5814;@ØK58B"ØKÖ45Sr58>=BpÝE?N5I;=1EI581OØN?A>‹ØN5gP->@ضÚ258?pØK5

u

“3ê5 ê6C ØNÖ45ÝE143{Ø0Ú258IØN;=?(;@×

u

C >@1EP

O(u)

3FB(ØNÖ45BN5Ø(;@×#;2Ý4Øpæt145&3F<2Öd܆;2?AB';2×

u

ê

Q"Ö43FB<23FÚ258BÝEB|ØKÖ45(ד;27F7F;‹ï(3614<>276<=;2?K3{ØKÖ4S @Ý4ن;21³>îOÝ458?NL

u

ï576;O>2PfØNÖE5(ëE14<=5&?KÙ4?N3F1OØ 5&1EPU3F14<OB"ד;2?

u

C ØNÖE5Bp5&Ø;@×#;2ÝUØNæƒ5gPU<25gB";@×

u

C >21EP-ØNÖ45ÀëE1E<25&?KÙ4?K361OØ(5&1¿PU361E<=B"ד;=?"ØNÖ45Ú=5&?Næ ØN3¾I58BÀ7F3F14Û25gP ÜdL

u

ê 94?K;2S ØKÖ43FBBN5Ø;2ׁëE14<=5&?KÙ4?K361OØKBÅï5rÝEBp5kØKÖ45>@܆;‹Ú25k58îOÝE>‹ØK36;=1½ØK;

>@Ù4Ù4?K; U36S>@ØN5

PPV(u)

Ý¿Bp3F14<º>-ÖE36<=Ö45&?>@Sr;=Ý41OØÅ;2×BK>@SrÙ4765gB

C

ØKÖOÝ¿B>2IAÖ43F5&Úd3F14<-ÖE36<=Ö45&?

Ù4?N5gI3¾Bp3F;21

ê

Q"Ö43¾B03FBÀ>ØK?K>=PU5&;È܆5Ø^ï5&581ÈîOÝ45&?KLØN3FSr5 sP4>‹ØA>@ÜE>=Bp5„>=I&I&58BKBp5gB (>21EP½ÙE?N5gI36æ

Bp3F;21 =ï(3{ØKÖ

k

P4>@ØK>2ÜE>2BN5¶>2I8I5gBNBN58B|ï50I8>@1³>2Ù4Ù4?K; d3FS>‹ØK5"ØNÖ450Ú258IØN;=?×“?N;=S

kN

BN>2SrÙ4765gB

ê

(8)

PPV

ï(3{ØKÖ7¾>@?K<25145836<=ÖdÜ¿;=?NÖ4;d;UP

C

>2Baד?N;=S

N

BN>2SkÙE765gB'>@Ø'Sr;=BpØ

N

Ù¿>@<25gBaï(3F7F7¿ÖE>zÚ=5ن;=BN3{ØK36Ú=5

>@Ù4Ù4?K; U36S>@ØN58PàaàaHÁڋ>276Ý45gB

ê

94;=?pØKÝ41E>@ØN5&7FL

C

ØNÖE3FBkBp5&Øk3¾Bf763FÛ25876L°ØN;I;=1OØK>@3F1ØNÖE5-ÙE>2<25gB

ï(3{ØKÖ³Ö43F<2Ö45gB^ØààaH!BKI;=?N5gB

ê-

Bp3F14<„ØNÖ45ÅBN>2SkÙE765gB;@×wØKÖ45¶145836<=ÖdÜ¿;=?N3F14<Ú=5&?NØN3¾I58Bï(367F7¿<=36Ú=5

Sk;=?N5>=PU58îOÝE>@ØN5À?N5gBpÝE7{Ø

C

>2B"36Øï(367F7:܆5Àד;2?KSr>2767FL³>@1¿>@7FLOG858P-361ØKÖ45À145 dضBN58IØK36;=1

ê

ìÈ5I&;2Ý47¾Pr>@7¾Bp;ÅÝEBN5"ØNÖ45(5 dÙ¿>@1EPU58?|Ù4?K;2ن5&?NØ^L;@׆ØNÖ45(ï5&Ür<=?K>2Ù4Ö @>‹×ØK5&?|14;2ØaBp;S>21OL

?K>21EPU;2S B^ØK5&ÙEBØNÖ45-PU3FBpØN?K3FÜ4ÝUØN3F;21ð;2×"ØNÖ45IÝ4?K?N581OØÚ25&?NØN5 Èï(3F767'Ü¿5I7F;=BN5rØN;ÄØKÖ45BpØK>@ØN3¾I

à#>2<25gH>@14ÛrÚ=58IØK;2?

ê

ñm1¿B^ØK58>2P;@×R>2767F;‹ï(3F14<„Ú=5&?KLk7F;21E<„ëE1E<25&?KÙ4?K361OØÙE>‹ØKÖEB"ï'5ÀI;=Ý47¾PI;=Skæ

Ü4361E5ØNÖE5„àaH Ú=58IØN;2?¶ï(3{ØKÖ I;d5 6³I&36581=Ø

(1 − c) L

ØN;ØNÖ45„>2Ù4Ù4?K; U36S>‹ØK36;=1Ä>@1EPÄPU?K;2Ù½>2767 ëE14<258?NÙE?N3F1OØKB7F;214<=5&?„ØNÖE>21

L

ê Q"Ö43FBfï;2Ý47¾Pð>@7¾Bp;½BN;27FÚ25³ØNÖ45-Ù4?N;=Ü47F5&S ;@×(ØNÖE5->@ÙEÙ4?N; U36æ Sr>@ØN5gPÄ361¿PU36Úd3¾PUÝE>27

PPV

Ú25gIØK;2?AB(ÖE>zÚd361E<³S>@1dL-G85&?K;d58B §361ÄØKÖ4;=BN5Ú=5&?NØN3¾I58B(ØNÖE>@Ø0ÖE>zÚ=5 14;³ëE1E<25&?KÙ4?K361OØAB5&1¿PU361E<³ØNÖE5&?K5#

ê

Q"Ö45f361¿PU5 U3F14<->276<=;2?K3{ØKÖ4SBï;2Ý47¾PÄ܆5&145&ë4Ø0ד?K;2S+ØKÖ43FB

ØN?KÝ41EI&>@ØN3F;21

C

ØK;d;

ê

Q"Ö45&?K5„3¾B>³×“ÝE?pØKÖ45&?À3F1=ØK5&?K58BpØN3F14<ºI;21¿Bp5gî=ÝE5&1EI&5„;@×'ØNÖ45k?K58IÝE?KBN36Ú=5„Ù4?K;2ن5&?NØ^L

ê

ñt×'36Ø3FB

ÛO1E;‹ï(1-3F1>=PUڋ>@1EI&5ÅØNÖE>@Ø"ï5Àï>21OØØK;rÙ¿58?KBN;21E>2763FG&5Å;‹Ú258?>fë U5gP §S>zLO܆5À7F>2?N<=5#'Bp5&Ø(;@×

ÙE>@<=58B

C

ï50I8>@13F1OØN?K;UPUÝEI5À>@1->@?NØN36ë¿I3¾>@7†14;UPU5Å361OØN;fØNÖE50<=?K>2Ù4Ö³ï(36ØNÖ-ØNÖ45Å?N5gBpن58IØN3FÚ25ÅBp5&Ø

;@×#145&3F<2Öd܆;2?ABaØK;r<25&1E5&?A>‹ØN50ëE14<=5&?KÙ4?K361OØKBד;=?"ØNÖE>@ضI;=S„Ü43F1E>‹ØK36;=1

ê

Ñ U|Î ÕNÎ`\R:жÐEÕNÎaÏ@T U#Ð@R R?R$Ò`RRÒ

ñm1³ØNÖE3FB"BN58IØK36;=1³ï5¶ï(3F767£PU3¾BNI&ÝEBKBØKÖ45I;21dÚ=5&?K<25&1¿I5(;@×:;=Ý4?58BpØN3FSr>@ØN5gB

C

>@1¿P³>@1E>276LdG85ØNÖE5

?N5gî=ÝE36?K58P>@Sr;2ÝE1=Ø(;2×$ëE14<=5&?KÙ4?K361OØKB'ד;=?(Ù4?N;=Ù¿58?(Ù4?K58I3¾BN36;=1

ê

ñtØf3¾BkI&765g>@?fÜOL°ØKÖ45-7¾>zï ;@×7¾>@?K<251OÝES„܆5&?ABØKÖE>‹Ør>2B„ØNÖ451dÝ4S„܆5&?k;@×(ëE1E<25&?KÙ4?K361OØAB

N → ∞

C ØKÖ45r58BpØN3FS>‹ØN5

PPV(u) d

I;=1dÚ25&?K<25gB(ØK;ØKÖ45k>=IØKÝE>@7#ن5&?ABp;=1E>@7F36G858PÄà|>2<258H¶>@14Û Ú258IØN;=?

PPV(u)

ê Q$;BNÖ4;‹ï ØKÖE>‹Ø"ØNÖE5?K>@ØN5Å;@×I&;21dÚ258?N<=5&1EI&53FB"5 UÙ¿;=145&1OØN3¾>@7

C

?K58I&>2767†ØKÖE>‹Ø

58>2IAÖ ëE14<=5&?KÙ4?K361OØ;@×

u

5&1EPEB³>‹Ø

v

ï(36ØNÖ ÙE?N;=ÜE>@Ü43F7636Ø^L

PPV(u, v)

C ï(Ö45&?K5

PPV(u, v)

PU5&14;2ØN5gB(ØNÖE5

v

I;d;2?APU3F1E>‹ØK5;@×

PPV(u)

ê Q"Ö45&?K5ד;=?N5

N · PPV(u, v) d

C ØNÖ451dÝ4SfÜ¿58?¶;2×

ëE14<258?NÙE?N3F1OØKB:;2×

u

ØNÖE>@Ø581EP4B>‹Ø

v

C ÖE>2BRÜ43F14;2Sr3¾>@7UPU3¾B^ØK?N3FÜ4ÝUØK36;=1ï(3{ØKքÙE>2?K>2Sr5ØN58?KB

N

>@1¿P

PPV(u, v)

ê Q"Ö4581ðM'Ö458?N1E; (ßBÅ3F1458îOÝE>27636Ø^L½Ld3F5&7¾P4B0ØNÖ45kד;27F7F;‹ï(3614<܆;2Ý41EPÈ;21ÈØNÖE5k58?N?K;2?

;@×#;‹Ú25&?Næt58BpØN3FSr>@ØN3F14<

PPV(u, v)

>21EP³ØKÖ45BK>@Sr5ÀÜ¿;=Ý41EPÖ4;27¾P4Bד;=?(Ý41EPU58?pæt58BpØN3FS>‹ØN3F;21

Pr{ PPV(u, v) d > (1 + δ) PPV(u, v)} = Pr{N PPV(u, v) d > N(1 + δ) PPV(u, v)}

≤ e N · PPV(u,v) · δ 2 /4 .

IØNÝ¿>@7F76L

C

ד;=?>@Ù4ÙE763¾I&>@ØN3F;21EB0ØNÖE5k5 4>2IØÀڋ>@7FÝ458B>2?N5k14;2ØÀ1E58I5gBNBK>@?KL

ê

ì°5k;=147FL½145&5gP

ØNÖE>@؁ØNÖ45¶;2?APU5&?K361E<ÀP45ëE145gPrÜdLØNÖE5¶>@Ù4ÙE?N; U3FSr>@ØN3F;21kS>‹ØKIAÖkק>236?K76LkI7F;=BN5&7FLØKÖ45;2?APU58?N3F14<

PU5ëE1E58P-ÜOLrØKÖ45ن5&?ABp;=1E>@7F36G858Pà|>2<258H¶>@14Ûrڋ>@7FÝ458B

ê

ñm1-ØNÖ43¾B(Bp581EBN50ï50Ö¿>zÚ25Å5 UÙ¿;=145&1OØK3F>27

I;21dÚ=5&?K<25&1¿I5ØK;d;

! 1$û„ý‹ù¿þ

õós÷§úAõø

u, v, w

úKû@ù4ø÷2õ

PPV(u)

ý‹ù-ýzøAø rõ„ó“ô4ý‹ó4

PPV(u, v) > PPV(u, w)

J

ôEõùÈóôEõÿ Nû Aý &÷i÷óƒþ-û^üÀ÷ù¿ótõNúAô4ý@ù 2÷ù

v

ý‹ùI

w

÷ù½ó“ô4õfýKÿ2ÿ Nû&ö=÷ý‹ómõ Ný‹ù ‹÷ù ómõùIzø

ótûõKöÿEû‹ùwõù¿ós÷§ýiþ³÷ùÈóôEõù DAõ ûpüù =õƒÿ÷ù¿ó§ø dø&õ

(9)

, δ > 0 N 0 N ≥ N 0

ù DAõ û^ü ù =õƒÿ÷ù¿ó§ø

3

üû ý@ùEþ NýAÿ4ô ý‹ù ý@ùEþ

õ óƒ÷§úAõø

u, v, w

øEúAô óôEý‹ó

PPV(u, v) − PPV(u, w) > δ

3 ó“ô4õº÷ùwõ Eý{÷ósþ

Pr{ PPV(u, v) d < PPV(u, w)} d <

ô4ûzø

Nûgû^ü

ìÈ5ÙE?N;‹Ú=5Å܆;@ØKÖÄØNÖE5&;2?K5&SB"ØK;2<=5ØNÖE5&?

ê

M';21EBN3FP45&?¶>rë¿14<258?NÙ4?K3F1=Ø(;2×

u

>@1EP7F5Ø

Z

Ü¿5kØNÖE5rד;27F76;‹ï(3F14<?K>21EPU;2Sڋ>2?N3¾>@Ü47F5

Z = 1

C 36×aØNÖE5rëE14<258?NÙE?N3F1OØÅ581EP4BÀ3F1

v

C

Z = −1

3{×ØKÖ45-ëE14<258?NÙE?N3F1OØf5&1¿P4B„3F1

w

C >21EP

Z = 0

;2ØNÖ458?Nï(3¾BN5

ê

Q"ÖE5&1

Z = PPV(u, v) − PPV(u, w) > 0

ê B^ØK36S>@ØN3F14< ØKÖ45

PPV

Úz>276ÝE58B„ד?N;=S

N

ëE14<258?NÙE?N3F1OØKBØKÖ4558Ú2581=Øk;@×

361OØN58?KIAÖ¿>@14<=3614<

v

>21EP

w

361ØNÖ45Å?K>214Ûd361E<=B3¾B58îOÝ43Fڋ>@7F5&1OØaØK;„ØA>@Ûd361E<

N

3F1EPU58Ù¿581EPU5&1OØ

Z i

Úz>2?N3¾>@ÜE765gB|>@1EPkÖE>zÚd3F14<

P N

i=1 Z i < 0

ê Q"Ö43¾BI&>21k܆5Ý4ÙEÙ¿58?a܆;2Ý41EP458PrÝEBN361E<

D

58?N1¿B^ØK5&3F1:ßB

36145gîOÝE>@7F3{Ø^LÄ>21EPÄØKÖ45„ק>=IØÀØNÖ¿>‹ØC>@?

(Z) = PPV(u, v) + PPV(u, w) − (PPV(u, v) − PPV(u, w)) 2 ≤ PPV(u, v) + PPV(u, w)

Pr{ N 1 P N

i=1 Z i < 0} ≤ e N (

Z)2

2

(Z)+4/3

Z

≤ e −N 10/3 PPV(u,v)+2/3 PPV(u,w) (PPV(u,v) PPV(u,w))2

≤ e −0.3N(PPV(u,v)−PPV(u,w)) 2

94?K;2S ØKÖ45>@܆;‹Ú25Å361E58îOÝE>@7F36Ø^L³Ü¿;2ØNÖØNÖE5&;2?K5&SBד;=767F;‹ï

ê

u t

Q"Ö45#ëE?AB^Ø:ØKÖ45&;=?N58S BNÖ4;‹ïB†ØNÖ¿>‹Ø$58Ú25&1>"Sr;UPU58BpØR>2Sk;=Ý41OØ:;@×dëE14<=5&?KÙ4?K361OØKB£>2?N5|5814;2ÝE<2Ö

ØN;fP43FBpØN3F14<2ÝE3FBNֳ܆5Ø^ï5&5&1³ØNÖ45ÅÖ43F<2Ö

C

Sr58PU3FÝ4S >@1EP³76;‹ï ?A>@14Û=58PkÙE>2<258B'>2I8I;2?APU3F14<ØK;ØNÖE5

Ù¿58?KBN;21E>2763FG&5gPà#>2<25gH>@14ÛBNI&;2?K58B

ê

;‹ï5&Ú=5&?

C

ØNÖ45„;2?APU5&?;2×#ØNÖ45f76;‹ïÉ?K>214Û25gPÙE>@<=58B(ï(3F767

ÝEBpÝ¿>@7F76Lº14;2Ø0ד;=767F;‹ï ØNÖE5fàà'HI&76;OBp5876L

ê

Q"Ö43¾BÅ3FBÅ14;2ØÀBNÝ4?KÙ4?K3FBN361E<

C

>21EP >=IØKÝE>@7F76LÄ>PU5&58Ù

Ù4?N;=Ü47F5&S ;2×à|>@<=58H>214ÛÄ36ØKBN5&76×

C

>2Brá

ã#‹å

BpÖE;‹ï'5gPÄØNÖ¿>‹Øà|>@<=58H>214Û½3¾BÀÝ41EBpØK>@ÜE765³>2?N;=Ý41EP

ØNÖ45k76;‹ï ?A>@1EÛ258P½ÙE>@<=58B

C

3F1½ØKÖ45kBN5&1¿Bp5„ØNÖE>@ØÀï(36ØNÖÈ7636ØpØK765kن5&?NØNÝ4?KÜE>‹ØK36;=1 ;2ׁØNÖ45k<2?A>@ÙEÖ½>

Ú25&?KL7F;‹ï ?A>@14Û=58P-ÙE>@<=5ÅI8>@1A^Ý4SrÙÄ361ØNÖE5?K>214Ûd3614<k;2?APU58?"BN;2Sr58ï(Ö45&?K5¶ØK;rØNÖ45Sr3¾P4PU7F5

ê

Q"Ö45Bp5gI;21¿P BpØK>‹ØK5&Sr5&1OØÖE>=BÅ>21È3FSrÙ¿;=?pØA>@1OØÅØNÖ458;2?K5ØN3¾I&>27|I;21¿Bp5gî=ÝE5&1EI&5

ê ì Ö45&1°ï5

361dÚ25gB^ØK36<O>‹ØK5rØNÖ45>2BNLOSrÙUØK;@ØK3FIr<=?N;‹ï"ØKÖ°;2×P4>‹ØA>@ÜE>=Bp5³BN36G85>2Bf>º×“Ý41EIØN3F;21ð;2×(ØNÖ45³<=?K>2Ù4Ö

Bp3FG&5

C

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>@1EP

δ

ê

„Ñ R:ÐÑaÓ¶Î0ÒVTAсÐ^Q Q ] UÏ-UVUXT R"Õ&YZR

ñm1 ØKÖ43¾BBN58IØK36;=1 ï'5ºï(3F7670Ù4?N;‹Ú=5BN5&Ú=5&?A>@7(7F;‹ï'58?rÜ¿;=Ý41EP4B;=1 ØNÖ45 I&;2SrÙ47F5 U36Ø^L ;@×ÀÙ¿58?pæ

Bp;=1E>@7F36G858P à#>2<25gH>@14Û

ê

ñm1 ÙE>@?NØN3¾IÝE7F>2?

C

ï5ï(3F7F7Ù4?K;‹Ú25³ØKÖE>‹Ø³5 4>2IسI&;2SrÙ4ÝUØA>‹ØN3F;21 ØNÖE5

1458I&58BKBN>2?NLk361¿PU5 P4>@ØK>2ÜE>2BN50BN36G85¶;2×R>דÝE767FLÙ¿58?KBN;21¿>@7F36G858Pà|>@<=58H>214ÛkS„ÝEBpØ܆5>‹Ø765g>2BpØ

Ω(V 2 )

Ü436ØKB C >21EPÄ36×aن5&?ABp;=1E>@7F36G8361E<;2147FLד;2?

H

ÙE>@<=58B

C

ØNÖE5fP4>@ØK>@Ü¿>2BN5„Bp3FG&5f3FBÅ>‹Ø0765g>2BpØ

Ω(H · V )

ê ñm1ºØNÖE5„>@Ù4ÙE?N; U3FSr>@ØN5ÀÙ4?K;2Ü47F5&SÁØNÖ45„76;‹ï5&?(܆;2Ý41¿Pד;2?(דÝ47F7$ن5&?ABN;21E>2763FG8>@ØN3F;21 3FB(7F3F1458>2?3F1

V

C ï(ÖE3FIAÖ3FB¶>2IAÖ43F5&Ú=58P³ÜdL³;=Ý4?>@7F<2;=?N36ØNÖES ;@ׁTd58IØN3F;21Äç

ê

Ä;2?K5Ù4?K58I3¾BN5&7FL-ï5ï(3F767#I;=1EBp3¾PU58?(Ø^ï';2æƒÙEÖE>2BN5>@7F<2;=?N36ØNÖ4SB 43F1ÄØNÖ45ëE?KBpØ0ÙEÖE>2BN5ØNÖE5

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P4>‹ØA>@ÜE>=Bp5

ê

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u

C

v

s>@1EP

w

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ê

ñm1ØKÖ43FB'Sr;UPU5&7

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ê

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, δ > 0

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C 3{×

| PPV(u, v) − PPV(u, w)| > δ

Ö4;=7FP4B ê

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H = V

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ê

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, δ

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w 1 , . . . , w n

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