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100000
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100
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10
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10
100
1000
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1e+06
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100000
10000
1000
100
y=x
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10
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100
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1e+06
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D
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3 3 ) g >t? 5;3[8g A^ :M: ?8) 145;-k143 ) -0= 3 - G />:*3 CkA();?>+ ' -#/ Ct!) 1 MC = g ! = 14' *) 3 - G /8:M3 :M- + ) 1 C =>3 )qG -05 3L7 / 14'l145 -0= > :M3 ) -05;3s-k: ' 3 -k:*14' Ct? > '‹1;'>3 lG 365n36: '.- 3s1 / C
A -0: ) 33 3 ) g ! =J14'>3E- C 3E-0: >0C 5 14' G 1;' M) / Ct;) 1 *C = M) 7839-07 3 + - ?>) 3 1 M)f+6C = ;) 7>365n367
1 C '.- 3[= C 3 3 ) /.- + 30g %(' *)u+ - ) 3 M)q+6? 5;5;3L=h14: > = C 5n367 3 + - ?>) 3 1 C2+L+6? 5 ) 365 54-k5;36: g ?>+ ' - + - ) 3 +LCt? :*7 3W783L143 + 1;367 uMG />:M3 G 3L=h1 = > ,3L= );C = ) -0: >tC 5 14' G )  C 5 ? :*:M3L5 )BG 3o14' C 7 ADC 5 ) 14-k1 *+ 7>3L1;3 + 1 MC = C0A`) - A 3 ) 1 C =83 ) ) • = - g
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7>3o143 + 1 *C = C0A :ˆ-05 > 3y3 3 ) g ! A 1 / C -07 - + 36=h1B/ C 1;36=h1 -0:.3 3W/ C =h1 ) -05;3 );? 5;5 C0? =>7>3L7 7>3 A 36=>7>3L5 ) 1 C =>3 ) 1;'>3 -05;3 +6C = ;) 7>365n367‹1 C 3Q-[:ˆ-k5 > 3x3 30g f3L1WŽJ 3Q-</ C 1436=h1 -0:
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f3L1BŽ 3JŽW C 5,Ž g “ '83 + 1;'>3 ADC :*: C / = >u+LC =>7 1 MC = )EADC 5 C 14'OŽJ‘-0=87 Ž 8
! A Ž *) -k1e14'>3(3L7 > 3 C0A 14'83 C -05;7 -0:M:@7 - >tC =.-0:@=>3 > ' C 5 )C0A Ž y3 + 3L/81
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1 / C :ˆ-05 > 3 ) - A 3K3 3 ) C =>3K:M-05 > 3 ) - A 3K3 3s-0=>7 C =>3 ) - A 3K3 3 C 5^1 / C ) - A 3Q3 3 ) 14'83
7>3 A 36=>7>3L5 / = ) g %('>3 ) 3W1 / C 3 3 ) -05;3‰-0: )nCK) - A 3 3 + - ?>) 3‰14'>3 -k5;3W- ) /r3 +2 -0: + - ) 3 C0A
,3L= );C = )B? = +6C =87 1 MC =.-k: ) - A 3L1 C 5Š14'83‰3 *) 143L= + 3 CkAfG[C 3 ) 1;'.-k1‘:M39-k7[1 C ,36= );C = )
) - A 3L1 -0: 14365n=.-k1 = > />:M- =M)A>0? -054-k=c1;363L7 g %(' M):M) /85 C 3L=‹- )(ADC :*: C / () 6
= + 3F1 / Cz) - A 3 !) = > :*3^/ C =c1‘3 3 ) -05n3 ) - A 3 1;'>3 + - ) 3 )BC0A 1 / C :ˆ-05 > 3 ) - A 3W3 3 ) -0=87 C =83J:ˆ-05 > 3 ) - A 3q3 3u-0=>7 C =>3 ) - A 3x3 3u-05;3 +LC = ;) 7>365n367 g
%('>3 ) 14-k143 = / ' *+ '#1;'>365n3Q-05n3x1 / C<) 1 C =83 )FC0A 1;'>3Q-¥1;1¡- + 365 =m-w:ˆ-05 > 3J3 3 *)
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! A 1;'>365n3 M) - ) 1 C =>3 C0A 1;'>3l-k1n1¡- + 365 = 14'>3 :ˆ-05 > 3#3 3 1;'>3 A - + 1<14'>-k1O14'83
7>3 A 36=>78365”'.- ) -0= C 14'>3L5 ) - A 3,3 3 C 5 C =>3 ) - A 3E3 3 >t? -k54-0=h143L3 ) 1;'.-k1 14'>3,-k1;14- + 3L5
+ -0=>= C 1e/>:ˆ- \ = ;) 783‘14'83,:ˆ-k5 > 3E3 30g 3L= + 3 14'>3‘7>3 A 3L=>7>3L5 : C@+ '.- 3,1 / CJ);? 5n3
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-k1n1¡- + 365 3 :*: = > 1;'>3J:ˆ-k5 > 3W3 3 G - k = > 14'83W:ˆ-05 > 3W3 3x- ) - A 3 ;) = > :M3W/ C =c1
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7>3 A 36=>7>3L5 G - 3m1 / C 3 3 ) -k5;3 = C 1 *G />:*3 G 36=h143L7 3L1 3 + - ?>) 3 C0A 14'83 ADC :M: C / = >
+6C0G />:*3 1 3 () 6
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:*3 G );?8+ '‹- ) 6
=>3l5n3 >*C = +LC =h1¡- = = > 1 / C +6C0G />:*3L1;3‹3 3 ) M) 5;3 +LC > = 3L7!-k=>7 +6C =@H
=83 + 1;367m1 C - + 5 ?>V+ -0: : C@+ g
% / C};) = > :M3x3 3 ) -05n3Q5;3 +6C > = 367j-0=>7 +6C =>=83 + 1;367 F
g %('>3L= 14'83K3 3 ) -05n3
+LC =>=>3 + 1;367 1 C -k= + 5 ?>V+ -0: : C@+ g
! =}14'>3q7>3 3 = 1 MC = CkA 14'83 C =>3oHŒ3 3x/>5 C :M3 G -0:M: + 5 ?>V+ -0: ) 1 C =>3 )FGK?>) 1 3 +6C =@H
=>3 + 1;367<1 C -0=w3 30g %('>3^-0: >0C 5 14' G 1 Cx+6CtG / ? 1;3 +LC =>=>3 + 1 MC = `) M)S = 0C 3L7 C = + 3
1 C^+ '>3 + 4 A -0:*: C0A 1;'>3 G -05n3 +LC =>=>3 + 1;367 g =x14'>3 C 14'8365 '.-0=>7 ADC :M: C / = > C : A )
7>3 3 = 1 MC = ADC 5U1 )n?>G 3IH JC 1;'>3,783 A 36=87>365 / = Z) A -0= uC =>3 CkA 14'>3 + 5 ?82+ -k: ) 1 C =>3 )
*) -0: ! 3kg C0G 3 + 5 ?>V+ -0: ) 1 C =>3 )+ -0=z/ Ct);!) : 3 + -k/81 ? 5n367 g ! A 1;'>365n3y-05;3 ! + 5 ? H
V+ -0: : C2+ c) +LC =>=>3 + 1 MC = )FGK?>) 1 3 +6CtG / ? 1;367m1 Cw+ '>3 + ^*A`C =>3 C0A 1;'>3 : C@+ c)
-05n3[/85 C 3L= 1 C '.- 3<1 / C ) - A 3<3 3 ) 7g %('8365;3 ADC 5;3 14'83<-0: >tC 5 14' G+6C0G / ? 1 = >
+LC =>=>3 + 1 *C = )”GQ?>) 1 3 = 0C 3L"7 !<1 *G 3 ) / ' M+ ' G - \ = +6? 5-^' > '8365 C 365;'839-07
1 Cw+LCtG / ? 143kg
# > $ %&' .C 5 + 367 GwC 3 )(!) MG :ˆ-k5Š1 C 14'>3 C =>3oHŒ3 3 )nC : 365(-05n3 = +LC 5;/ C 5;-k143L7
=h1 C 1;'>3‘1 );?>G 3oH qCF);C : 3L56g ˜9¤¥ 4£¡)™ (x—0  —@+£ *h™o- ,</¤ .0™ *)@> 36=>3L54-k1;367 C =Q-^/ C =h1 0y'>365n3
14'83Q7>3 A 36=>7>3L5 +6C0? :M7 G - 1k3x21 0 C 43 3t3 )J)n?>+ 'j- ) / C =h1 = >t? 5;3u8g - g ˜9¤¥ 4£¡)™ (
(@™D˜9™o¦(@™o 5,<¤/.0™ M):> 36=83654-¥14367 C = -u/ C =c160y'>3L5;3‰1;'>3‰783 A 36=87>365 GQ?>) 1,/>:ˆ-73 + '>3 + 1 = >
14'83 ADC :M: C 0 = >K+6C =>7 1 MC = 9) 8
5 =8: C@+ 1@?A0y' M+ 'l'>- ) - ;) = > :*3oH GwC 3 +6C =@H
;: '>3L5;3 *)‰C =><: 3 C =>3 ? = ) - A 3s-k1n1¡- + 1k36>
=>3 + 1 MC = 1 Cw) - A 3 ) 1 C =>3 ) g
;B A 14'83l7>3 A 3L=>7>365}/>:M7- 3 ) -0= 3 C 1;'>365 GwC 3j-0=>7 1;'>3j-k1n1¡- + 1k365 +6C =8=>3 + 1 ) ?m1 C
) - A 3L21 3 14'83Q7>3 A 36=>7>3L5W'.- ) 3 14'>3L5^= C / C 1;36=h1 -0:U3 303K/ C =h1 C =>3u/ C 1436=h1 -0:U3 303
/ C =h1 C 5B21 0 C -07 - + 3L=c1F/ C 143L=h1 -k: 3 303u/ C =h1 ) g
8
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- = 0>t? 5n3q8g 8 .C 5 + 367 G[C 3 ) g
<= 1;' M)Š+ - ) 3 14'>3^7>3 A 3L=>7>3L5 M)ADC 5 + 367[1 C /8:ˆ-73 );C 14'.-k1E14'83W-k1n1¡- + 1k365 + -0=>= C 1 +6C =>=>3 + 1
?x1 C ) - A 3 ) 1 C =>3 ) g 14'>3L5 0 *) 3 1;'>3z-k1n1¡- + 1k365 +6C0? :M7 ) 1439-k:`14'83z/ C 1;36=h1 -0:e343t3z/ C =h1 )
14'>-k1q-05n3z=>3 + 3 );) -05 3 ADC 5‰14'>3z783 A 36=87>365 =3‹/>:M-73 = > C =‹1;'.-k1q/ C =h19g = 3 - G /8:M3 *)
/ C =h1 = >0? 5;3J2g = g
MGK? :ˆ-k1 MC = M)g = 0C 103L7 =O14'>3 ) - G 3>0B-73}- )R =O14'83 C =>3oHŒ33t3 )nC : 365Lg
> ^););?8G 3J14'>-k1B14'83q-k1;14- + 103L5 *) 14'83J/>:ˆ-73t3L5B1 C />5 C 3x/ Ct;) 1 MC = -0=>7<1;'>3‰7>3 A 36=>78365 *),C =>3F1 C 7 M) /85 C 3kg B = C 5;7>3L5‘1 4C = 1 -0: 3B/85 C@C0A = ?>G H
) M)S = +LC 5;/ C 5;-k143L7
=r365 ) 1;'>3 ) - G 3F3 -0: ? -k1 *C = AD? = + 1 MC =w1;'.-k1Š14'>3 C =83oHi343t3 );C : 365 ?>) 3 *
=h1 C 14'83F1 );?>G 3oH qCu);C : 365Lg : '>3 !) = > :M3oHŒ/>:ˆ7- 30365Š7 M) 1¡-0= + 3^1 CQG - 103y21 0 C 3 303 g) *)‘+LCtG H
/ ? 1;367‹1 uC = 1 -0: 3W7 M) /85 C@C0A = ?>G =r365 ) g : ' M)Q*)^!) MG :M-05,1 C 14'>3u'>3 ? 5 M) 1 M+qAD? = + 1 *C =
A 5 C0G 14'83 C :M7m\^]`_bacZedfVbdl1 );?>G 3oH qCs)nC : 365 g : '83x'83 ? 5 M) 1 M+WAD? = + 1 *C = 145 3 ) 1 C
-0/>/85 C MG -k143w14'>3 +L? 5;5n36=h1 n7 M) 14-0= + 3 ‹1 C + 5;36-k143[21 0 C 43 3t3 )sC =21 0 C / C 143L=c1 -0:E3 303
/ C =h1 ) Ž z-0=87Ž ¡g : '>3L= =.- ) 3L7 C =j1;'>3 +6? 5n5;3L=c1x7 *) 14-0= + 3 14'>3s7 *) 1¡-k= + 3K1 C 21 0 C
3303 ) - A 1;365 G[C 3 M) />:ˆ-73t3L7 M)F+6CtG / ? 1;367 0y' M+ ' *)^?8) 367 - ) -0= = 1 -0: 7 M) /85 C@C0A
= ?>G =r365 ADC 5 lg : ' *) /85 C@+ 3 )n)y+ -0= = 3 )n?>G[G -05 3L7}- )(ADC :M: C 0 9) 8
Ž ) 1;'>3W= ?>G =r365 CkAU+6C = ) 3 +L? 1 ! 3
C g .C 5‘39- + ' / C 143L=h1 -k: 43 3t3J/ C =h1,Ž !"$# !&%(' *
7>3 A 36=>78365 GwC 3 ) 1 CJG - 103(-0=Q3 3t3y-¥1ŠŽ *)e+LCtG / ? 143L7s- ) 3 />:M- =8367 = 3 + 1 *C =
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2g f3L1 Ž l-k=>7 Ž "=r3‹/ C 143L=c1 -0:^3 303 / C =c1 ) g : '>3l7 *) 14-0= + 3‹1 C 21 0 C 43 3t3 B
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=>3 > ' = C 5 C0ABGwC 3 lg f3L1QŽy =r3[-}/ C 143L=c1 -0:43 3t3[/ C =h1x1;'.-k1 +6C =c14- = ) -¥1
:*39- ) 1 C =>3 +6CtGwG[C =m783 A 36=87>365 ) 1 C =>3q1 CwG -1k3x- ) - A 3u343t3u-k1^ŽJ-0 1;'#5n3 ) /r3 + 1
1 C ŽFg bC 5z-0:M: +6CtG = =.-¥1 MC = )QC0AF)n?>+ ' ŽWQ-0=>7 ŽF 1;'>3 );G -0:M:*3 ) 1 -0: ? 3 C0A
Ž $' ŽW ¡Žy ) M)z+LCtG / ? 143L7 1 C) 3L1[- ) -0= = 1 -0: 3677 M) />5 C@C0A = ?>G =r365Lg
Ž $' ŽW ¡Žy ) M) 783 =>3L7‹- )BADC :M: C 0 ) 8
B A '>3L:M/ ) 1 CwG - 1k3 C =>3q3 3t3 = = C 14'#ŽW(-0=87 Žy ADC 5y3 - G />:*3 *A *)
-s7 5;3 + 1F=>3 > ' = C 5 C0A = C 14'#ŽJ(-0=>7#ŽF Ž $' ŽJ ¡Žy ) *)F+LCtG / ? 143L7
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-k=>7 !"$# ! %(' Žy )w-05;3 ?>) 3L7 1 C G 39- )n? 5;3 !" ' ŽW ¡Žy ) -k=>7 '836:M/ ) 1 C 783 + 5n39- ) 3x1;'>3q7 M) 1¡-0= + 3J1 CsG -103x-0=}3303 C = = C 14'mŽW(-k=>7 Žyg
1;'>365 0 M) 3 Ž ' ŽJ 4ŽF ) ' C !" ' ŽW ¡Žy ) C )og
,>g B A ŽW‘-0=>7 Žyy-05n3W= C 1 ADCt? =>7 G[C 3 7 C 3 ) = C 1B'>3L:M/O7>3 + 5n39- ) 3W14'>3W7 M) 14-0= + 3
1 C 21 0 C 43 3t3 ) g B = 14'>3 MG />:M3 G 3L=c14-k1 *C = C / C =h1 ) />: ?>) 14'83 :ˆ-k5 > 3 ) 1 )n+6C 5n3
C0A 14'83 GwC 3 0y' *+ '7 5;3 + 14:<3 '836:M/ ) 1 C}G -1k3z120 C 3303 \) M) - )n;) > =>367j1 C - )
-O'>3 ? 5 *) 1 M+ 7 *) />5 C2CkA = ?8G = 3L56g C A -05^1;'>365n3 M) = C 7 M) 1 = + 1 *C = =r3L21 0‘363L=14'83
GwC 3 ) 1;'.-k17 5;3 + 1;: 3u'>36:*/ ) 1 CJG - 103 C =>: 3Q-0=z43 3t3F-0=>7K1;'>3 C =>3 ) 14'.-k1”7>3 + 5;39- ) 3
1;'>3‰7 *) 14-0= + 3^=>3 14'>3L5‘1 CKC =>3^33t3‰= C 5‘1 C 120 C 3303 ) g : ' *)R*) -u/ Ct)n;) =8:M3 AD? 1 ? 5n3
3 1;36= !) MC =}1 C 1;' M)yG 3L14' C 7 g
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= 0C : 367 =O-K/>5 C@C0AUC 5‘7 M) />5 C@C0A 145n363 =<14'83 5 ) 1y/>'.- ) 3 -u5;3oH ) 36-05 + ' M) / 3L5 ADC 5 G 3L7
G = > 1;'.-k1(14'83x: Ct) 3L5 + -k= *G[G 367 -k143L: 3s5n3oH + -0/81 ? 5;3 1 C - )(C0A 1;36=#- ) =>3L367>3L7 g
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+ ' C0) 36=Yg B =14'>3 /85 C =>:*3 G +6C :*:M3 + 1 *C = 14'>3L5;3}-05n3} tC : ?>G 3 ) g pŠ- + ' tC : ?>G 3
+LC =h1¡- = ) C , :M3 3L: ) g ‰:M: O/ Ct); 1 *C = )O =‹14'>3z'.-k5;7>3 ) 1 + -k1;3 >tC 5 3 = tC : ?>G 3
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-05;3 + ' Ct) 3L= g >t? 5;3K2g • ) ' C 0 ) -0=l3 >- G />:*30g : '83z/>5 C =@H
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= + 3y14'83 C =>3IHi43 3t3‰/85 C =>:*3 Gw)Š?>) 3L7 ADC 5E3 2/ 3L5 *G 3L=c1 R) =w1;'>3^/85;3 &*Ct?>),+ '>-0/8143L5
+ -0= =r3 ) 363L=z- ) 1 )n?>G 3IH JC />5 C =8:M3 Gw) 14'83 ) 3E/>5 C =>:M3 G[) 0‘365n3 ?>) 367 1 CF+6CtG /.-05n3
1;'>3q/ 3L5 ADC 5 G -0= + 3q- > - = ) 1 qC : C@C : ) g : ' M) 1;3 ) 1 )n? 1;3 M)(+ -k:M:M3L7
‘g
W) =u14'83B:ˆ- ) 1 + '.-k/8143L5 14'83 ADC :M: C 0 = > - = =>5n3 & -k1 MC = ) -05n3 ?>) 367 ADC 5`1;'>3 G 3L1;' C 7 )
-0=>7m36=8'.-0= + 3 G 36=h1 ) 7>3 )n+ 5 =r367‹- = C 3 8
8 : '>3 GwC 7 367 7 A Hi/>=#-0: >0C 5 14' G g
8 3 +LC > = 1 MC = C0A 21 0 C / C =c1B3 3t3 ) g
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0 1;' =m• C =>3j3L=>'.-0= + 3 G 36=h19g : - =8:M3j8g ) ' C 0 ) 5n3 );? :*1 )}C =!-07>7 = > ;) = > :*3l36=>'.-k= + 3 G 3L=c16g
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/>5 C =>:M3 G g ^C 0,3 3L5 14'>3L5;3u-05;3W120 C[G - = 7 3 365n36= + 3 9) 8
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J2g C ,8g ‰:M:Y36=>'.-k= + 3 G 3L=c1 )^C = g '>- ) 3 C ) 39-k5 + '>3 )‰C =>:<3tg
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r3}3 2/>:ˆ- =>3L7=31;'>3 > 5;36-k143L5 +LCtG />:M3 123 C0A 1 )n?>G 3oH Cj+6CtG /.-05n3671 C 14'83
C =83oHi343t3O/85 C =>:*3 G g C 5x3 8- G />:*3 0y'>3L= -0:M:Š3L=>'.-0= + 3 G 36=h1 ) -k5;3[1 ? 5;=8367 C = 1;'>3 );?>+L+ 3 )n) 5;-k143 C0A );*GK? :ˆ-¥1 MC = 0B- ) ,( ' ) 363 : - =>:M3y8g-,() +6CtG /.-05n367s1 C •
= “ '>-0/8143L5^• ADC 5B14'>3 C =>3IHi343t3x/>5 C =8:M3 G g
NuP¥RUTWVYX[Z \^]`_ba@Z`dfVbdm/r365 ADC 5 G 367[- GK?>+ '[:ˆ-05 > 365E- GwCt? =h1 C0A 5n3oH ) 39-k5 + ' ADC 5
C = W8g C , ' ) 363 : - =>:M3x8g • )Ig , 8g C0A -k:M: = C 7>3q3 2/.-k= ;) *C = ) 0,3L5;3x=>363L7>367
1 ADC 5,5;3oH ) 36-05 + ' ADC 5 1 C 0y'>3L5;39- ) 1;'>3q= ?>G =r365 = “ '>-0/8143L5F• 0B- ) CC g g = + 3
W8g C , +6C =c14- =8367K-^: C 1 C0A /85 C =>:*3 Gw) 14'.-k1e5;3 $? 5;3‘-‰= ?>G =r365 C0A 1 C 14'85;39-¥1 ) 1 C
0 = 1;'>3 )nC : 365 );? 3 365n367 A 5 CtGv);?8+ 'z' > ' C 3L5;'>36-07 g *G :ˆ-05 = 3L'.- &MC 5 C2+L+6? 5 )
= C : C@C : \) A(G -0= 3 1 C -05n3 = 0C : 367 =14'83</>5 C =>:M3 G g C : A,G 3L=h1 MC = 4) =
Š14'>-k1 ADC 5Q- +6CtG />:*3L143 ) 39-05 + '-0:M: 1 C ) 0 1;'1;'>5;36-k1 ) -0=>7 -k= ) 0‘365 )KGK?>) 1
=r3O/>:M-30367 Ct? 1z-0=>7 1;'>3<1;5;363[1232/ M+ -k:M: 3 7>3L36/r36= ) =3 - A - + 1 C 5 CkA C g • = 14'83
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QN PSRUTWVYX[Z‹\^]`_bacZedfVbd M)f+LCtG />-05;3L7>0 14'x- > 3L=>365;-0:01 );?>G 3oH C()nC : 365 1 C - )n) 3 );)) 1 )
/r365 ADC 5 G -0= + 3kg C : C@C : ) 0,- )(+ ' Ct) 36= ); = + 3 1,'.- ) = 3L36= +LC = ); 7>365n367 1 C = 3F14'>3>=r3 ) 1
1 )n?>G 3oH Cw);C : 365 ADC 5 C •"3039-05 ) g C : C@C : ) 0B- ) 5 ? = ? =>7>3L5^14'83 ) - G 3x3 2/r365 MG 36=h1¡-0:
+6C =>7 1 MC = ) g  = + 3 C : C@C : ) -0=87lNQPSRUTWVYX[Z \F]e_bacZedfVbd ?8) 3K7 3 3L5;3L=c1 );+ '83 G 3 )
ADC 5B7839-0: = > 0 14' 1 C -0=>7 1 C 1;'>5;36-k1 ) 14'>3J/85 C =>:*3 Gw) 0y' Ct) 3q5;3 )n? :*1 ) 7>36/r36=>7 C = 14'83
= ?>G =r365 CkA 1 C 1;'>5;36-k1 ) - - :ˆ-=>:*3 0,3L5;3 3 + : ? 7>3L7 g ) - 5;3 );? : 1 , C / C0); 1 MC = )r =
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: - =>:M3 ) 8g  -k=>7 2g +6CtG /.-05n3s21 0 C‹)nC : 365 )xC = J8g C ,#-0=>7
=87 M+ -k143F14'.-k1^NuPSR T‰V X[Z \^]`_racZ`dfVrd )n? 5;/.- );) 3 ) 1;'>3 );C : & = > -= : 123 C0A C : C2C : ) g
‰: 14' C0? > ' = C 1;'</>5 C > 5;- G[))nC : 3W-0:*:>/>5 C =>:M3 G[S) = W8g C , NuPSR T‰V X[Z \^]`_bacZedfVbd
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CC - >:M3u8g  8 365 ADC 5 G -0= + 3 +6CtG /.-05 M)nC =@=r3L120‘363L= NQPSRUTWVYX[Z \^]`_bacZedfVbd -0=>7 C H
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C: 2C : ) = pŠp qp(g
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);C : 3L7=3 C : C@C : ) 0‘365;3O-0: )nC );C : 3L7=3 NQPSRUTWVYX[Z\^]`_bacZedfVbdeg B =-07>7 1 *C = NQPSRUTWVYX[Z \^]`_bacZedfVbd );C : 3L7K14'>3 ) 3(/>5 C =>:*3 Gw)eGwC 5;3,1;'.-0= ‰1 MG 3 ) ?M+ 1k365”14'>-0=
C : C@C : ) g ‰:*:t5;3 );? : 1 ) -05;3 C =814- =8367 0 14'q14'83 C 5 > =.-0: 3L5 );*C = CkA C : C@C : ) />5 C& 78367
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>0? 5;3 ) 8g W/8: C 1 ) 14'>3(3 23 +6? 1 MC =w1 MG 3 ADC 5”/85 C =>:*3 Gw)”);C : 3L
7 =3 = C 1;'s/>5 C > 54- Gw) g
: '>3m1 MG 3 ) /r36=h1=3 NuPSR T‰V X[Z \^]`_bacZedfVbd *) />: C 1;1;367 C = 14'>3^Hi- M) - > - = ) 1
C : C@C : )KC =14'>3 ^H - M)zC = : C > -05 14' G *+w)n+ -0:*3 ) g B = / C =h1 ) - = C 3 1;'>3}7 - >0C =.-0:
NQPSRUTWVYX[Z \^]`_ba@Z`dfVbdj/r365 ADC 5 G 367 =r3L1;1;365Lg B = + - ) 3 CkA 36- ) 3}/>5 C =>:*3 Gw) 1 M) '.-05n7
1 CF) 3L3B-F7 3 365n36= + 3 =u/r365 ADC 5 G -0= + 3 =r3L120‘363L=K14'>3Š120 C‰);C : 365 ) g CtG 3L1 *G 3 ) C : C@C : )
/r365 ADC 5 Gw) = 3o1;143L5J14'.-0= NQPSRUTWVYX[Z\^]`_racZ`dfVrd =r3 + - ?>) 3 C0*A 1 ) :ˆ-05 > 3s- GwCt? =h1 C0A
'.-0=872H +LC 7>3L7 1c= C 0y:M3L7 > 30g C 5x3 8- G />:M3 C : C2C : )q);C : 3L7 14'>3[/ C0;) 1 MC = = >t? 5;3
8g = 8g ) 3 +LC =>7 ) g : '>3 ) 39-05 + 'O145;3L3 C0A C : C@C : )Š+LC =h1¡- =>367 C  x:*39- A = C 7>3 ) g : '>3
7>3L36/r3 ) 1w7>3L/814' 3 8/8: C 5n367 =3 C : C@C : r) M) C 8g C : C2C : ) =836367 ) - !) MG :M-05u= ?>G = 3L5
C0A :*39- A = C 783 ) ' C C = C 7>3 ) )(1 C />5 C 3Q1;'.-k1 ' 1;3 0 = )^ADC 5y14'>3u/ C0;) 1 MC =#- A 1;365 M) />:ˆ7- 3t3L7 g C : C@C : Q) M) 1;'>365n3 ADC 5n3 = 3L: 3 3L7l1 C 145 3 5 ) 1x-0=87 ' 143 *) />5 C 36=
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5;3q8g  8E“,CtG /.-05 M)nC = C0A 3 23 +L? 1 *C =#1 MG 3 ADC 5 =87 '& 7 ? -0: = ) 14-0= + 3 ): = J2g C ,8g
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1
10
TsumeGo Explorer (sec)
;5 3j2g 8 / Ct!) 1 MC = 1;'.-k1 C : C@C : )O)nC : 3 )#G[C 5n3 ?$M+ 1c: 3 1;'.-0= NQPSRUTWVYX[Z
\^]`_bacZedfVbd ' C C ,h ' 143q: ' 3 ) 0 1;' )og
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;5 3Q8g 8 ’/ Ct;) 1 *C =#1;'.-k1xNuP¥RUTWVYX[Zv\^]`_bacZedfVbd );C : 3 )JG[C 5n3 ?M+ 1c:<3 14'>-0=
C : C@C : ) ' C • 1,:ˆ- + 1 1 :M: ) 0 1;' )og
1 C =r3}-0: ' 3kg =1;'>3 C 14'8365z'.-0=87 NQPSRUTWVYX[Z’\^]`_racZ`dfVrd =>3L367 ) 8g-,, ) 3 +6C =>7 )
0 1;' }= C 7>3w3 2/.-k= );*C = ) g : '>3[7>363L/ 3 ) 1 ) 39-k5 + '7>3L/814' =1;'>3w/>5 C@C0A 1;5;3L3 *)
8g : '83 ) 3x= ?>G =r365 ) *G /8: 3s14'.-¥1^NQPSRUTWVYX[Z \F]e_bacZedfVbdl=>363L7 ) 1 Cs) 36-05 + '#7>363L/ 3L5
0 1;
' 0 7>365 =>5;-0= + '>3 ) 1;'.-0= C : C2C : )<ADC 5 1;'>3/85 C =>:*3 G g ^C 0,3 3L5 NQPSRUTWVYX[Z
\^]`_bacZedfVbd ?>);? -0:M<: 3q/ 3L5 ADC 5 G[) = 3o1;143L5 ADC 5`'.-05n7K/>5 C =>:M3 G[) g 1;' M)0M) = 3L: 3 3L7 = 3 + - ?>) 3
C0A - G[C 5n3 );C />' M) 1 *+ -k1;367 ) 36-05 + ' -0: >tC 5 1;' G g C 5W3 8- G /8:M3 C : C2C : ) =>3L367>3L7 CC
) 3 +6C =87 ) 1 Cj);C : 3<1;'>3 / Ct;) 1 MC = = >t? 5n3 8g 0y'>365n39- ) NuP¥RUTWVYX[Z’\^]`_bacZedfVbd
);C : 3 :) 1 = CC g ) 3 +6C =87 ) 0 14' ,c• , z= C 7>3 ) g
>0? 5;3 ) 8g u/8: C 1 ) 1;'>3‰3 23 +6? 1 MC =<1 *G 3 ADC 5Š/>5 C =>:M3 G[)E);C : 3L7 =3 = C 1;'</>5 C > 5;- G[)
= pEp Jpyg : '83 )n? /r365 *C 5 21 3 CkA NuPSR T‰V X[Z \^]`_bacZedfVbd C =j'.-05n7l/>5 C =8:M3 GwO) *)
GwC 5;3 a!& 7><: 3 ) ' C 0y= = 1;' M) 143 ) 1 )n? 1;30g NuP¥RUTWVYX[Z \^]`_bacZedfVbd Ct? 1;/ 3L5 ADC 5 G[)
C : C@C : ) =3j-O:ˆ-k5 > 3 G -05 > = g >t? 5n3z8g C ) ' C 0 ) -0=j3 8- G />:*30g C : C@C : ) =>363L7>367
C C ) 3 +6C =>7 ) 1 Cy);C : 314' M) />5 C =>:M3 G 0y'>3L5;36- ) 1;'>3Š/>5 C =>:M3 G 0B- ) 36- ) 3 ADC 5 NQPSRUTWVYX[Z
\^]`_bacZedfVbd ' 8g C , ) 3 +LC =>7 ) 0 14' O= C 7>3 ) )og ^C 0‘3 3L5 1;'>3 C 1c= C 0y:M3L7 > 3 C0A
C : C@C : ) M)();C0G 3o1 MG 3 #) 365 3 -0: ? - =>:*30g C 5(3 8- G />:*3 >t? 5;3q2g CC 0 1;' ' 1;3q1 C
/>:M- 3 *),) 1¡-k1 M+ -k:M: 3 );C : 3L7 = 3 C : C2C : ) 0y' :M3‰NQPSRUTWVYX[Z \^]`_bacZedfVbd =>3L367 ) C • = C 7>3 ) g
>0? 5;32g C ,/8: C 1 ) 14'>3Š/>5 C =>:*3 Gw) );C : 3L7 C =>: 3 =3xNQPSRUTWVYX[Z‹\^]`_bacZedfVbdeg = + 3
C : C@C : )J+6Ct? :*7 = C 1 )nC : 3w1;'>3</85 C =>:*3 Gwe) = • G = ? 1;3 ) • G = ? 1;3 \) M) />: C 1;1;367 C =
14'83 ‰Hi- M) g : '>3W7 w+6? : 21 3 ADC 5(NuP¥RUTWVYX[Z \^]`_ba@Z`dfVbd 54-0= > 3 )‘A 5 CtG 3L5 3<39- ) 3w1 C
'.-05n7 g > - = 14'83>= 3L=>3 1 C0A 14'>3 ) 39-k5 + '2H=.- ) 3L7 -k/>/>5 C - + ' M)‘) ' C 0y= g C 5,3 8- G /8:M3 >t? 5;3‘8g C +LCt? :*7K= C 1 = 3 );C : 3L7 = 3 C : C@C : ) 0 14' =z• G = ? 1;3 ) 0y' :M3,NQPSRUTWVYX[Z
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100
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5;3Š2g 8 p 23 +L? 1 *C =u1 MG 3 DA C f5 />5 C =>:M3 G[)f)nC : 367
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C 1;'x/85 C> 5;- Gw)z = pŠp Jp(g
= 3 =
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1
10
TsumeGo Explorer (sec)
;5 3‰8g C 8 / Ct!) 1 *C =[1;'.-k1 *) '.-k5;7 ADC 5 C : C2C : ) = ? 1 365 3O39- ) 3 ADC 5(NQPSRUTWVYX[Z
\^]`_bacZedfVbd ' C =>3L33t3 + g C 8g )>0A 1,:M- + 1[1 C 1 :M: =3 />:M-73 = > -k1 )og
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C C
>t? 5;3x8g CC 8 / C0); 1 MC = 1;'.-k1 C C@C : )()nC : 3 )^) 1¡-¥1 M+ -0:M:<3' C =>36343t3= g C 8g )>0A ' 1;3
1 C 1 *: : =3 />:M-3 = > k- 1 # )og :
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100
GoTools (sec)
10
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y=x
One problem
0.1
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TsumeGo Explorer (sec)
100
;5 3 8g C 8 p 83 +6? 1 MC =!1 MG 3 ADC 5m/>5 C =8:M3 Gw)});C : 3L7 C =>:<3= 3 NuP¥RUTWVYX[Z&\^]
_bacZedfVbd = Šp p Jp(g
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C
>t? 5;3q2g C 8 / Ct); 1 *C = );C : 3L7 C =>:<3 =3#NQPSRUTWVYX[Z \^]`_ba@Z`dfVbd ' C =>36343t363kg 2g )>0A ,:M- + 1<1 C : ' 3A=3 /8:ˆ-73 = > -k1 )og
\^]`_bacZedfVbd )nC : 367 1 = 2g ) 3 +6C =>7 ) 0 14'  C s= C 7>3 ) g
=>3x-07 -0=h1¡- > 3 C0A 7 A Hi/>= M) 1;'.-k1 1 ?>) 3 ) 1;'>3J145;-0= ) / Ct)! 1 *C =O1¡- =>:M3 GwC 5;3‰3 @143L=2H
);! 3L: 3tg =>: 3 );C : 3L7j/ Ct)! 1 MC = ) -05;3 ) - 3L7 =‹14'>3Q1;54-0= ) / Ct); 1 MC = 1¡-=8:M3 = C : C@C : )
0y' :M3 =x7 A HŒ/>=u/85 C@C0A -0=87Q7 *) />5 C2C0A = ?>G =r365 )`C0A />5;3 a*Ct?>) 1;3654-¥1 MC = ) -05;3 ) 1 C 5n367
=j14'83[145;-0= ) / Ct!) 1 MC =l1¡- =>:M3s1 dC MG />5 C 3z1;'>3 C 5n7>365 C0A 1;5;3L3[3 8/>-0= );*C = •0 g ) 5;3 );? : 1 )nCtG 3</>5 C =>:M3 G[) 14'.-¥1z-05n3 3L5 3 '.-05;7 ADC 5 C : C2C : ) -05;3 ? *+ 1c: 3 );C : 3L7 = 3
NQPSRUTWVYX[Z \^]`_ba@Z`dfVbdeg
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3L5 ? : A 5;340y5 C 1;3 C : C@C : ) = “ -0=87 MG />5 C 367@0y'>3L=j'>3 = +6C 5n/ C 5;-k143L7 1 =h1 C
G -05n1 C C g : ' M) 3L5 )!MC = CkA C : C@C : )*) - = Ct? 1 8g-,s1 MG 3 )yA - ) 14365y14'>-0= 14'83
C 5 > =.-0: C =‹/85 C =>:*3 Gw)Q = J2g C ,8g ^C 0‘3 3L5 1J-0: );C<+LCt? :*7‹= C 1 );C : 3 GwCt) 1 C0A 14'83
/>5 C =>:M3 G[) = pŠp Jp 14'>-k1^-05;3 ? = );C : 367 =3O14'>3 C 5 > =.-0: C : C@C : ) g
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NQPSRUTWVYX[Z\F]e_bacZedfVbd +6? 5n5;36=h1;: 3 + -k=>= C 1 );C : 3# /85 C =>:*3 Gw)Y = 14'>3}143 ) 1 );? 1430g
: - =>:M3 8g /85;3 ) 36=h1 ) - : M) 1 CkAz? = )nC : 367 />5 C =>:*3 Gw) g : '8365;3 -05;3 GwC 5;3j/>5 C =>:M3 G[)
5;3 G - = = > ? = );C : 3671;'.-0= =3 1;'>3 C =>3IHi3 303 );C : 365Lg =8363 30367 g 2g ) >0A‰ADC 5 ' 143<1 C
: ! 3 M)E);C : 3L7 =3 NuP¥RUTWVYX[Z \^]`_ba@Z`dfVbd 0 1;' CC , x= C 7>3 R) = ) 3 +LC =>7 ) g
^C 0‘3 365 1 + -0=>= C 1 =r3 );C : 3L7 0 14' = • G = ? 1;3 ) =r3 + - ?>) 3 C0A 14'>3 ) : C 0‘365 ) 39-k5 + '
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- 8:M3x8g 8 M)C 1 CkAeG ? = );C G : 367m/>“,5 C C =>C :M3 G[C ) = pŠp Jp(g
5 =>:M3 =.- 3 : 5,1 />:ˆ-73
C =>3L330367 g 2g )>0A 1,:ˆ- + 1
C =>3L330367 g 2g )>0A ' 143
C =>3L3 30367 g 8g ) >0A 1,:ˆ- + 1
C =>3L3 30367 g 8g ) >0A ' 143
C =>3L3 3t3L30+g 2g ) >0A 1,:ˆ- + 1
C =>3L3 3t3L30g CC g ) >0A 1,:ˆ- + 1
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) /r363L7 + - ?>) 3L7 =3Q- G[C 5;3 +6CtG />: M+ -¥14367 AD? = + 1 *C =x1 C‰+LCtG / ? 143Š14'>3E'>3 ? 5 *) 1 *+ 7 M) 14-0= + 3
ADC 5B120 C 343t3 ) g
{}| ) = 36=h143L7 = “ '.-k/8143L5,•>0‘365;3^-07.-k/8143L7w1 C 1 );?>G 3oH C g :ˆ-k5 > 3(5n367 ?>+ 1 *C =
: '>3 7>36- W
C0A 1;'>3 ) 36-05 + ' ) /.- + 3 0,- ) - + ' 3 367 1;' ?>) 14'>3-= 3 ) 1”1 );?>G 3oH CW)nC : 365 0,- ) = ? : 19g : '>3
+6? 5;5;3L=h1ONuPSR T‰V X[Z§\^]`_bacZedfVbd + -0= )nC : 3 36= + : Ct) 367 / Ct)! 1 MC = ) 0 1;' -k5 Ct? =87 3 G /821 3[/ C =h1 *) =O-u5;39- );C =.- =>:M3‰- G[C0? =h1 C0A 1 MG 30g ? 7 > = > =3[1;'>3 ); 3 C0AU? = )nC : 367
/>5 C =>:M3 G[) 14'>3 ;) 3F: *G 1 C0A NQPSRUTWVYX[Z \F]e_bacZedfVbd ) 363 G[) 1 C = 3 =r3L120‘363L= x-k=>7
3 G /8123j/ C =h1 ) g : ' *)u+LCtG /.-05;3 )xA - 0C 54-=8: 3‹1 C C : C2C : ) 0y' *+ ' '.- ) -}: MG 1 C0A
) =}5n39- )nC =.- =>:*3J1 MG 30g
- = Ct? 1 C ,s3 G /821 3O/ C =h1 = + 3B14'83 1@= C 0y:*367 > 3^-k7>7>367z1 C 3L=>'.-0= + 3^NQPSRUTWVYX[Z \^]`_bacZedfVbd *) ? 1;3 ;) *G H
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thpn(A)=
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thdn(C)=
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pn(B)= 2
dn(B)= 1
B
C
D
E
F
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I
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J
H
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H
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8
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pn(F)= 1
dn(F)= 2
pn(H)= 1 pn(I)= 1
dn(H)= 1 dn(I)= 1
8 8
thpn(A)=
thdn(A)=
A
thpn(C)= 3
thdn(C)=
-1
C
thpn(C)= 3
thdn(C)=
-1
C
pn(G)= 1
dn(G)= 1
I
b7 e
B
thpn(A)=
thdn(A)=
8
E
pn(B)= 2
dn(B)= 1
E
G
pn(G)= 1
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8
D
thpn(C)= 3
thdn(C)=
-1
C
D
8 8
B
thpn(A)=
thdn(A)=
A
8
pn(B)= 2
dn(B)= 1
8 8
thpn(A)=
thdn(A)=
A
B
F
pn(I)= 1
dn(I)= 1
8 8
A
pn(B)= 2
dn(B)= 1
thpn(C)= 3
thdn(C)=
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C
8
8
b& e
D
Win
Q
thpn(A)=
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8 8
thpn(B)= 2
thdn(B)=
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P
OR node
thpn(A)=
thdn(A)=
A
8 8
O
O
pn(O)= 1
dn(O)= 2
P
pn(P)= 1
dn(P)= 1
pn(G)= 1
dn(G)= 3
K
pn(K)= 1
dn(K)= 1
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