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I h +<7 ! % h 3-,>:1!*46.*-+,%'*-( C -*Yf<@B&) C >:*$%'-$H R/*Y1,*$ I h K"k+21!*R:#$,*Y#Y &XD,@)! 7L-X&,@6X*$*$- @A! 8 &K!-!:<:!-z: 7)@ h &+,)7;@A! &K!-;T<L,-SL Hde.*Q:#$,*Y#Y*$-X- ,@ (A7L-*9&XD,@ 6X*$*$-/R +,-/#$!h 0X%4*Y7 C M*5h #F.*$#FEk:" ( 7$L- *i&,@ 6X*$*$-B#$!%'&()*$7 C M I h FH ?>:.!(",. 6.)*8-#Y!*Y-OX*Y(*Y7 C M I h R/*$X*f#$!X*$#5 TO:-OL%'& ,XF+<G=6 .+ 1,*f+ -#F.)*$%'*?6N.0+,7>:*=.* & C >L*$) % G-;:#$* .0+,&&K*$)- C <6.u:;7)@ h &+<7 H 1!*YF:@ ?7D*Y-<8+,&)& *$+,r: .* 0+,> &,@ *Y* TOD#Y#2+,-!L!0+<>L>:MU+,&&K*2+,XL 6.* -X*2+,X#F. 6 H ^.* b 8 & C >L*$%'-8-X!>:1,*$7 ! % h :^7)@ h &F:k1!!E<*$7 -;:%8(>L+<<:! ,j j!j <L%'*$2- *5WD&>L,*$7 8 j 8 D7)*$- C M -;:%8(> +Q<L,0+,7S7L-X#$ 1kh *$X*$7 ; H .* b 8 &X C >L*Y%4-3-X!>:1,*$7 C M j +2R *Y7 6+,-X& !-!:;L!UF+ C >L* *YG;:*$-$ ! ! % h \Lk1!!E<*$7 -;L%(> +<;L! 8 8<j!j <L%'*$2- /* WD&>L!X*$7 b ; b !j )D7*YC MP-!L%8(>L+<<:!K+,7U7)*Y6*Y#Y*$7 8 +2R/*$7 *$k6!L*$-YH de.*$-X*'G(% C *Y- +,X*8#$!)-*$ih 3 W" " " L" #3 L" L" R3 L" 3 .0 /. 0 " " ] J3 L" L" #3 L" ] J3 L" L" R3 W" Q" ] ] +* ] * 8 .0 J3 L" ] 7* 5] ] dV+ C >L* )H %1 *$X@A,%4+,#$*N#$!%'&0+,!L-X!@A,B7)@ h &D:S#F.*Y#FE,*$X-$H D7)*-X6+<<:-X;L#$-B+,X* #$,%4&(D6*$7@A! .* -( C -X*YB,@ b !j & C >L*Y%4-e-X!>:1,*$7 C M +,>L> &)!"!+,% 1,*$X-;L,-$H " *Y.7 1\ C >L*Y%4d ,F+,> (-*Y7 -X!>:1,*$7 D7)*$ I h b 8 b 8 8 ; 9 # 8 ! % h b b 8 # b - b Db j d ,F+,> &) C >:*$%'h ,j!j 7* L" #3 W" J3 L" R" ] +* 7* * 7* ] ] *] ] -* 1<+<<T1!* C *Y#2+,(-X*f-!%'*NL)#$!X*$#53&XD,@A-3,J7L-X&,@A-J%4+2MS.+ 1,* C *Y*$ -X!*Y7 C (D *51!*$eX*Y;L*51!*Y7 H!?-L +,%4& C *Y>L>K+,7 X*$(E<*$27 -&0+,&K*$X- b b/ 6.*Y-*3G(% C *$X#$, 0% 6.0+< BD#Y#$(X- %4!X*8@A* ()*$#5MyL 7)@ h &) .0+,yL H R *51!*$2 6.)* *Y-(>T6--.) R 6.0+<:T:L-e-9<L>:>_*Y#$*Y--6+<XM6f#$()* R: H L",(*Y- )H ; 6 )H #Y!%4&+,*;.*&K*$9@A!%4+,#Y* C *5 R *Y*$ I h +<7 ! % h @A!/*2+,#F.&X C >:*$%,H L",(*Y- )H ; +,7 DH 6.*?D7)*J*5WD&0+,)-;L,,@ ! % h +,X*N&>:,*$7 ! .* =hi+QWaL-=+,"!+L-9 I h , 6.)* h +QWaL- !>L,"k+,;T6.%L#-X#2+,>:*$-$H & :ke+ C 1!*J6.)*J7 +<"!!0+,>K%'*2+,-/6.0+< ! % h ?&K*$9@A!%'*$7 C *YX6*$YH aL%L>L+,>T9M _L L",(*Y- DH 8 +,7 )H 6.*f* W)*Y#$();L! <:%4* -&K*$k C M ! % h r:- &>L<6*Y7 , .* Bhi+ W$:-f+,"k+:-X I h O! 6.)* h +QWaL-O! >:!"k+,!:6.)%:#8-#$+,>L*Y-$HSde.)**Y-(>T6--X. R 6.0+< ! % h 3#$+, +21!:7 6.* g& C >L*Y% R::.*$"!>L" C >L*= 1!*Y.*$+,7 :6*$X%4-e<@ 7*3* W)&+,-;:!-YH 4*$X%4- ,@_*5WD*$#Y()<:!4;L%'* I h -,>:1!*Y- &X C >:*$%'-\->(:"!.k6>TM@ +,-96*$ 6.+, H ! % h eL 7)@ h & +<7 +,>L%'!-98+,- ($L#FEG>:M +,- ! % h : 6<F+,> ! % h *5WD&>:!*Y7 H 8 *5W6+7*Y- +,74*$*Y7*$7 /b H ; *5W6+O<:%4* L7)@ h & H ! ! % h *5WD&>:!*Y7 H j * W66+'7*$2- +<7 )*$*$7)*$7 H *5W6+ <L%'*,H de.L-:L-+ -X%;+,>:> &!L#$*Jf&0+2M@A!e"!(0+<6+,k6*Y*2:"4#Y!X*$#56*Y--$H 5" _ J3 L" dc ] _ 3 W" ?$ J3 W" " 1DC 2C 1 Q] ] 5" ] " J3 L" ] _ 5] L" #3 W" J3 W" .0 L" #3 L" ]* L" R3 L" ] J3 L" R3 L" J3 L" c J3 L" .0 3 W" * G PU S K OQG E GJI .0 )] , .0 R" Vd + C >L* )H N":1,*$- 6.)*J*Y-(>T6- @A!/7)@ h &:#F.*Y#FE,*$X-$H +,"k+/ 7Z- -!>:()<:!( -#*g 6.)* V& C >L*Y% 8 L- +<77:;L!+,>L>TMr:%4&)>L*$%'*$k*$7 H I?@_6.)* <j!j & C >L*$%'-WLf6.)* 6*Y-X -X*Y b ,j & C >L*$%'-\+,X* -,>:1!*Y7 C M8+,>L>D%4*56.7-$HVd + C >L* )H ; -()%4%4+,; $*Y- 6.*e* WGh 6+ & C >L*$%'-e-,>:1!*Y7 C M*$+,#F.S%'*Y.D7_H ! % h /-X!>:1,*$7 '* W66+f&X C >L*Y%46.+< I h 7L7', -X!>:1,*,H I h V-X!>:1,*$7 b ; & C >L*Y%4-25L#Y>L(7Dh " 7* L" R3 L" ) -* , L" R3 W" 88 J3 W" ] 7* ] gf 1e+07 1e+06 Ignoring GHI 100000 10000 1000 100 y=x One problem 10 1 1 _ :"!(*Q])H ; L" I#3 hW" 10 100 1000 10000 100000 Handling GHI 1e+06 7* *5WD&0+,-!L!)-@A, & C >L*Y%4-e-X!>:1,*$7 C M C ,. *:7D@ h &L 3H 1e+07 " % J3hW" +<7 J3 hL" J+<7 ! Ignoring GHI 100 10 1 y=x One problem 0.1 0.1 1 10 Handling GHI 100 _ :"!(* ]DH 8 3 DW *$#Y()<:! ;L%'*8@A! & C >L*$%'-3-,>:1!*Y7 C M C ,6. L" I#3 Wh " *:7D@ h &L "3H 8 # ! % 1e+07 1e+06 Ignoring GHI 100000 10000 1000 100 y=x One problem 10 1 1 _ :"!(*Q])H * L" I#3 hW" 10 100 1000 10000 100000 Handling GHI 1e+06 7* *5WD&0+,-!L!)-@A, & C >L*Y%4-e-X!>:1,*$7 C M C ,. : H *: 1e+07 .0 " % J3hW" +<7 J3 hL" J+<7 ! Ignoring GHI 100 10 1 y=x One problem 0.1 0.1 1 _ :"!(* ]DH 3 WD*$#Y()<:! ;L%'*8@A! . 0 : " H L" I#3 hW" *: 10 Handling GHI 100 & C >L*$%'-3-,>:1!*Y7 C M C ,6. *!j ! % ?] 3 $ dV+ C >L* )H ; W 6+P& C >L*$%'- -!>T1!*Y7 C M *$+,#F.^%4*56.7 H D*Y* ?&)& *Y7TW ?H @A!N+ >(:-X;L"8<@V.* , j!j & C >L*Y%4-YH " *Y.D7 1\ C >L*Y% 7 I h b b ! % h 98 8 8 8 b 898 8 b 8 L" R3 W" 3 W" R" * +* ] * ]* * +* +* 1e+07 1e+06 IGNORE-GHI 100000 10000 1000 100 y=x One problem 10 1 1 _ :"!(* ])H 8 L" I#3 hW" 10 100 1000 10000 100000 HANDLE-GHI 1e+06 , %4&0+<;L-X! ,@7* * WD&0+,-!L!- C *Y R/*$*Y *:7D@ h &L#F.*Y#FE,*Y-$H 3 W" ! 1e+07 % 3 hW" +<7 L" R3 L)"N!)*=&) C >:*$% R.:#F.4R +,-g,/-!>T1!*$7 C M ! % h FH R *51!*Y& I h "!*$)*$6+Q6*$7L)#$!X*$#5\7:-&XD,@A- L #$+,-*Y-kL#Y>L(7$L"?.* !)*e*5W6+ &) C >:*$% -,>:1!*Y7^!>TM C M I h FH :"!(X* )H 8 #Y!%4&+,*Y-7** W)&+,-;:!-@A,N6.)* &X C >L*Y%4-B-!>T1!*Y7 C M C ,6 . I h e+,7 ! % h FH :"!(X* )H 8 L)7Th #2+Q6*$-V6.0+<V6.* -!>:()<:!7)D*Y-, 7*$"!+,7*g6.* &K*$9@A!%4+,#Y*,H ! % h *Y1,*$ *5WD&>L,*$7 j H >:*$-- ,F+<> D7)*$-e6.0+<) I h FH !%'&0+,*Y7 R:T6. +,"k+/ 7Z-O%'*Y.7 ! % h ?-,>:1!*Y7 ; *5W6+ &X C >L*Y%46.+< -47L7 <-!>T1!*,H IJ 6.)* <6.*Y .0+,7 -'-!>T1!*$7 !* %'!* &X C >L*Y% (-X!>:1,*$7 C M ! % h F&H ?>T6.,("!. C <6. +,&&Xk+,#F.*Y-B-X!>:1,*$7& C h >L*Y%4-\#Y!X*$#56>:M k6.* -!>:()<:! #2+, C *B+<&&>(:*$78O1<+,!L!(-+,>L"!,;:.%'- <R.)*$*$+,- " +* " L" #3 L" _ ] L" R3 W" 3 W" L" R3 L" 3 W" * R" #" J3 W" *)b R" _ X] J3 L" 1e+07 1e+06 Nagai’s solution 100000 10000 1000 100 y=x One problem 10 1 1 10 100 1000 10000 100000 HANDLE-GHI 1e+06 1e+07 _ :"!(* ])H 7*N* WD&0+,-!L!-=@A!=& C >L*$%'--!>T1!*Y7 C M 3 hW" +,7 h " -`L7)@ h &)d: #F.*$#FE<*$-YH dV+ C >L* D] H 8 1 *$9@A!%4+,#Y* #$!%'&0+,!L-, @A, .0 L^#F.*Y#FE,*$X-$H D7)* -XF+Q<L-9<:#$-f+,X* # ! % #$,%4&(D6*$7@A! .* -( C -X*YB,@ b!b # & C >L*Y%4-e-X!>:1,*$7 C M C ,. &)!"!+,% 1,*$X-;L,-$H " *Y.7 \1 C >L*Y%4d ,F+,> (-*Y7 X- !>:1,*$7 D7)*$ I h b j # b ;9; 8,j 8 j # ! % h b ,j 9 # b ; d ,F+,> &) C >:*$%'h ,j!j L" #3 W" J3 L" +* ] * _ B] <+ "k+ 7Z-%'*Y.D7 #$+, C *xL#Y!&K!6+Q6*$7 ,>:M 6^7)@ h & H L",(* DH # &>:,6-;D7)* *5WD&0+<-;:!-;@A!4*2+,#F. &X C >:*$%H de.*P&)>L,uL-;-X>(:"!.k6>:M + C 1!* 6.*P7$ +,"!,0+,>>(:*,H P6*Y%'-J,@\,F+,> 7*$-J@A,?&X C >L*Y%4-B-!>T1!*Y7 C MP+,>L> 1!*Y-!L!-2 -=*5WD&>:!*Y7 b )H %4!X*37*$-e6.+, ! % h FH aL%(> +<;L! +,"k+: 7*56*$#56- +2R/*$7 66+<-&K!-;T<:! F+ C >L*P*Yk6;:*$-YH 6.* b -,>:1!*Y7 & C >L*$%'- ! % h K:G1,!E,*Y7N-!L%8()> +<;L! 9,j 8 ; ;L%'*$-_R:T6. 8 8 8 8 7* *5WD&0+,)-;L,-S+<7 7$L-#Y 1!*$X*$7 +2R *Y7 6+,-X& ,-;:;L! 6+ C >L*U*Yk6;:*$-YH de.*Y-* G(% C *$-e#Y! 0X% .0+< .* & C >L*Y% #$#Y(-:6.*?-*2+<#F. H de.*Y-*eLDh #$,*Y#Y=*Y-(>T6--,%4*5<L%'*$-+<&& *$+,L .* 00+,> 7:-&XD<@ *Y*$-=,+@ I h FH ]] $ R" 3 W" J3 L" " ]+] ] * 7*+* * * L" R3 L" * 1e+07 1e+06 IGNORE-GHI 100000 10000 1000 100 y=x One problem 10 1 1 _ :"!(*])H b j L" I#3 hW" *: ] 10 100 1000 10000 100000 HANDLE-GHI 1e+06 7* * WD&0+,-!L!- @A!S&X C >:*$%'-S-,>:1!*Y7 C M :#F.*$#FE<*$-YH .0 . 0 L" R3 L" J3 L" L" #3 L" L" R3 L" ! 1e+07 J3 hL" % +<7 d + C >L* )H 8 -. R- .*J*Y-(>T6-g@A! gH+ I h -X!>:1,*$7 * W66+N& !-!:;L!6.+< R *Y*,f-X!>:1,*$7 C M ! % h R.*$X*2+,- ! % h -!>T1!*Y7 !)* &X C >L*Y% ()-!>T1!*$7 C M I h FH I h '*56(X*$7 L#Y!X*$#5 7$L-9h &XD,@A-g@A! /b & ,-;:;L!)-kL#Y>L(7$L"O6.*? R/ &K!-;T<L,- -,>:1!*Y77),>:M C M I h FH 0L"!(X* )H b j &>:,6- D7)*J*5WD&0+,)-;L,- @A! &X C >:*$%'- -!>T1!*Y7 C M C ,.1!*Y-;:!-YH " !-9=*$-X(>:-=+,*3#Y>L!-X*J6 6.* 7L+,"!!+,>_>L*DR.:#F.d:%4&>L*Y-/6.+< ! % h L)#$(X-+ *Y"!>(:" C >L* 1!*Y.*$+,7 H *$X%4-4,@J6<F+,>7*$2- ! % h 8*Y1,*$ *5WD&>:!*Y7 H ; >L*Y--7*$-6.0+<) I h FH O.* b Db & C >L*Y%4- -X!>:1,*$7 ! % h L:k1!!E<*$7-!L%8()> +<;L! ; 8 # # <L%'*$R::. 8 b 9 )D7*O*5WD&0+<-;:!-J+<7 7)*Y6*Y#Y*$7 # 8 89 +2R/*$7U*YG;:*$-$H de.*$-X*8G(%fh C *$-e:7L#$+<6*8.0+<3.* =&X C >L*Y% #$#$()-eL vL #F.*Y#FE,*$X- C ()O%8(#F. >:*$-X@A* G(*$k6>TM 6.0+<^:7)@ h &_H #Y!#Y>L(-!L! -!L#Y*.* O& C >L*$% .0+,&&K*$)- C ,6.F: 7)@ h & +,)7 : #F.*Y#FE,*$X- :Y:-470+<"!*$X!(-f6 L"!)!*<H *Y#2+,(-X* .* N-!>:()<:! ,'!>TMFLDh #$()-J>: R 1,*$X.*2+,7 C (D3+,>:-+<>:R + M-B*Y()-J#$!X*Y#Y +,)-XR/*$2- :L-B+4R/!X.kR.L>:* +,77$:<:!6'+,kM -X*2+,X#F.S*Y"L)*O-X(-#Y*$&); C >:*36 FH " P_ L" R3 Q] J3 L" J3 W" L" R3 L" J3 L" * dc J3 L" * /" .0 " " *] ?" .0 de.* &X!& ,-*$7U-X!>L(D<L, #Y!(>L7 C *8#Y!%'&0+,*Y7SR::. X*$(E<*$7Z- d H R *51!*$2 d *$*Y7-'+, *5WD&>L#2T8"!+,&.^X*$&X*$-*YkF+<;L!/+,7 #Y!%4&)>(L#$+<6*Y7 !&K*$+<<:!-N 7*$+,>=R:T6. *$&K*Y;:;L!-YH de.L-4#2+,(-X*$- + & C >L*$% R.)*$ d (-X*$-()& +21,+/L> + C >L* %'*$%'!XM,H IJ .*P,.*$4.0+,7: 6.*S+,&&Xk+,#F. &X!& ,-*$7 L .L- .0+,&)*$47D*Y,/*Y*$7+,kM'*5WD&>L#2Tg"!6+,&).;*Y&*Y-*YG6+<<:! 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R .* 1,*$X.*2+,7 ,@B*$#Y!-96()#Y<:" &,@=6X*$*$-Y: ,H B>:.!("!.46.*J 1,*$.)*2+,7,@ X*5h -X*2+,X#F.*$- *Y#$!-96X(#Ye&),@ 6X*$*$-:- (-X(0+,>:>:M X\ 3E3 _$ H) 2, /) 7, _ ] \ 3%3 5" _$ 5.0 \ 3E3 .0 _$ gf \ 3%3 .0 ?] ] * L$ _$ L$ _$ \ 3%3 `_ $ & _$ 12 Number of problems solved 10 Discard proofs Keep proofs 8 6 4 2 0 0 5 _ :"!(* ]DH b !%4&+,;:-! \ 3%3 1 h 1 IOI _ $L . 0gH 10 15 20 25 Memory usage (Megabyte) ,@=6.*-,>:1&L" + C L>(T M ,@ ! 30 L$ !Bh 1 eI I 35 _`$ +<7 - %4+,>:> +> +<"!*?+,%4,(k ,@ -X*2+,X#F.^:- #$#$+,-;:!0+,>:>:Mf&K*$X@A,%'*$7 H, &+,X;L#$()> +,2D%'!* 6.+,S.0+<>:@<@\+,>:> D7)*N* W)&+,-;:!-?+,*O*$*Y7*$7 @A!*$#Y!-96()#Y<:" +f&,@V6*Y*4: !*Y*YM,*$*,H b HZ-",@*L d + C >L* )H b j H=de.L-B& !-!:;L!R +<-J,B-!>T1!*Y7 C M 1 h 1 IOI H ?"!+L=6.:-d:- 6.* &;:#$*U6 &0+2M @A!7*Y>L*Y;L" &XD,@A-+,)7 7:-&XD<@A-@AX!% 6.)* 6+,-X& !-!:;L!6+ C >L*<H 5] \ 3%3 _ DC C SWGN K N M KPO OQG U SWG &K!-!:<:! 6.0+Q?-X( _& *Y*Y7P@A!% .* " e&X C >:*$% R::. .* $P$ \ (>:*OR +,-?@A!()7 R.*Y86.*g-!>T1!*YV6.+< :"!!X*$- " R +<- L%'&>:*$%'*$k6*Y7 (A-*$*H_ L",(*T])H b ]B+<7 _ :"!(* )] H b ; ( + * * H _ :-9 .* -X!>:1,*$V*5WD&>L,*$7O6.*/& ,-;:;L! 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A + = G = A &+ A&A = G A ! ' = A G+ = A&A&+ ! + G + A A = ! A ! = + G G A = A = = = = =! A = 2 = &++ ! = " =&= ++&+ 2 / 7%I D% D 7 # 6 6 $ $ / )V,+ D% AZ%/ # 6 / 7 7 W( # 6 /B/%/ 7 W( # 62a()+ ?$ )+ ;()+, # 6 ) ! ,+#" F% 9 OZ%$ (X B # + + 6 )$ ; Q$ +($" S% 9 I 7 - / X # !" 4 . &94& 5( 6 % + # =P 90/ 9 P /)+ 9 5 # # = > / 9 $%,2&- ; ?G / 9 > G 9 7/(Q/ # % 9 7+' /) (QNE)( % $O -6$ / / Y (Y-) N);> 9 V) * % = &+ - +/(Q 9 +' ++H(Y 9 $ = D2 + 9 + # K++- +/(Q '+ / = K7 F +'/ AAW2 + # ! /- % F7 5/ / ! 9 / ! / 9 I / 9 /(Y +'Q++5- & +/(Q< +' GG A =.A&+ Gi = ii 2 350 300 Execution Time 250 200 150 (1) (2) (3) (4) (5) (6) (7) (8) 100 50 0 2 >)$ 4 6 8 10 12 Problem No. 14 16 18 20 # G 6 % (QI% EA 9 B- & +/(Q=% A 9 9 7/(Q/ # 9 ! $ /% # 'W- +( / +!Q+ 9 (QD+(( / ; ! ) E/ W) +' / 9 (J) ( ! # 57 9 -R// / A+( + / 9 ( </ 9 /(Y ) (Q-$ /> = -$+, 9 - / !()+ X-$ &/ *%;7 / 0 / 9 ; ()+, # >/ O( %; / 9 9 )$ / (Q+ ( S>/ /D9 9 9 7 (Q / 9 B) 9 A 9 / &>Y $ $DC+ J k>S)- 9 $- ; +S 5% (Q 9 + # a$U (Y-) > 9 $ / @ ; (Q-+ 9 K7 9 - / (Y=/-R// W 9 K% ( 'E)$ %! ()$+, 7++ # >) # G -+ 7 9 (Y% 9 A 9 / - +/(Q % ++ ; )// 2 ; !/4+ / 9 > "7++@ 9 7/(Q/ # 9 7/(Q/ F)/! '4+ ) $ # = # (Q % B *(Q 9 9 Q- +( / * 7 r % +, '+ / # % 9 &'&+ ?/( 9 e (Y- % $ B > E 0 / +/>Z2 9 /(Y # (Y-+ - +/( # = # >% / 9 3 9 -+,9 )+ A '+ / / = 0 ' /D9 / 7 ; + / 9 ++5 9 /(Q/ '+ /d # ! / 9 ! ! # = % + # A 6 / % (X7I% ) > !>+I 9 /(Q/ # )$( / % - +/(Q +! 9 /(Q/ )/ = 6 A 6 (A 6 6 6 ! 6 + 6 + +A 6 K++ ) % & = = = ' ! ! A&A ! ! = = = = !! &G ! =! A 2 = ! = = A 7 + - +(Y !" $ % + % + ( Q / /W -6/ -6$ = +F- +(Y = +F- +/(Q R- / D *"( . ++ G = +1A ! A A GG ! A 1A = A A ! A = ! ! A = G ! == + i A = i 2 = ! G ! G &GG = A G ! + ! + ! A &GG 2 &4& 5 (76 - !(Y C(Y) /D9 9 B 9 7/(Q/ 7 R !y +, 9$./ () 9 9 8 ; ) 7++ # ! % + # A /)+ ?% Q / 9 > 1 ! >+V 9 7/(Q/ * 90/ # % + # B 90/ F /)+ % S)$;$> + P ;>+ 9 /(Q/ # % 9 V /)+ I 9$./ 9 7++/ 9 $(Y S /- % (Y- V-R/ % (X$ % 9 Y !+ / # K> 90/ R9 7 7 >/*7 7 > # % / !"! & & % * ( % + # 90/ 9 O-R/ % (X7 %;(*)+, C- 9 "= 9 / # a 9 (Y 9$ =--+C% ! / 0 A )/K(Y OV> 5 + /)+ / 9 -+ % 5 ( % %) 9 ' 36$(Y # 5) 9 36/(Q/ / ++ % 9 - ; +'%) ) O / 9B - # !" & 6 & & 6 % + # 90/ X )$(Q(Q ? % 9 9 v/) . 23 9 V% 2 $# ! - 9 *= (Y(Q/, 7- ) L7 ++ 0/ # <9 A 9 2 9 / 9 X/;>6 /: / / # % 9 * )+ O 9 +* *+ "/ 9 = = % + # 6 / % (X$I% )$ > !>+'/ 9 /(Y # 9 /(Q/ )/ = 6 98;: H < > @ A 6 (A 6 6 6 ! 6 6 + +A 6 + % 7 + - +(Y $) ( / % - +/(Q +! = + = = = G = = = ! = = = A =! A $ % + + (Q / /W -6/ -6$ = - +(Y = - +/(Q R- / D 1A A'+ = A G ! A ! = ! G G A ! = = G ! + ! i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a/ ! % 2 = $$?( (Y +! 9 (QA) % / - ; $ 7 9 / ; / / )C% # % + # ! )(Y(X7; #"/ 9 -R/ % (X7 # % 9 J)$( / %I /? -+ / ' /+)$ (Q-$) 9 X %U / -6$ / # ! % 2 ! ()+ ^ = ) / ?% ( + >/ / 9 / -6$ ! Y 9 ! % 2 /)+ u > :A&+ + >/ + /) (Q$ AA X / E# ! >) # = 9 ' -6! $ % ! % 2 7 K-$+ 9 23 > ! % 2 = 9 K23 + > 7; 9 ( L+/ # - 9 > 67+ (Y$ 9 ! % 2 -R/ % (Q/ # 3 9 (Q++ 23! #"/Y-$ +(Y 9 / / / 5 / / 5 / / ; # W0 / / 9 + > - +(Y ! % 2 -+ /S(+ > 9 4+ +/ $ 9 ! % 2 = # % 9 / Z% ! % 2 /(+ 0/ + # 4 9$ > 9 P %K -67; / Q ) A ! 5 TQ - ! (Y ) i - # a$D $% (Q 4 9 0G J6 ( ++ FQ))67+4+ 4( /(+ , + 9 X 7+)6 9 1A S ++: /? (X7++U-; -69 9 L( ) 2 # 0 / 0 > > 9 $Z2 5 / / 9 W %6 / -6$; B/ ; > 9 7 + ) (Y' /(X DA%) ) / D # % = % + # ! 6 % (Q/ (Y-6 ; 5 / / 2 ] !% !% 2= # W)( % F % % 7 + % + ! 9 $ - +(Y /D -6$ / -6$ (Y )/ +!/ = - +/(Q = - +(Y / = A =9+ =1A = ! A !% 2= A # = = G + + &A 1A # !% 2 =! A 2 2 2 + - +(Y % 1e+07 1e+06 UNIT-1 100000 10000 1000 100 y=x One problem 10 1 1 >) # = 6 10 100 1000 10000 UNIT-N 100000 (Q- ; % I -6$; 5/ / = 1e+06 !% 1e+07 2 !% 2= # >)$ !" # ==6 ,+ -+,9L7 +'&6 )$ / +4C)$ +!/ - +( # % * +& 5 ( + # 6 z D %)$ +!B- +/(Q # & +/( 6(Y + -+,9 +,7 / # # >% +,7 / # G # >% 3 9 / # G # >% +,7 / / # A # >% / / # == # >% 3 9 6 5 & ?&9 6 9 B +!/ Y/) + .7 +' 9 N- +(Yh 9 ) B+ q # # >) # ==K 90/ DX (Q-$+ # ++ ) +'L- +(Y:% ) K+ > % + /> / 9 (Q1 - ; +C( # !/ (QC- +(YY) 9 >) # ==Y 9 9 V+ ( % 9 Z2& X- +/( $" 9 )+" 9 / / (Y/ )(YZ2 - +(Y # ! >) # == % : (Y-+ (Q&k$> $K % +, P- $ /D9 9 /D9 ? $V7 F +, " )2+ / 7 (Q& 5 / / D 7# % !" KC%; B - , 6 ( ! (& % ! E 0 / 5& +6 9 ^ 9 36 -$ > ( -R/2,7+(#"/G% C +!&> - +/(Q # a/ 9 N7+> ! 9 ( +!/ +4 23N- & +/(Q (Q&B %; F (Y-6 ; / 9 >$ 7+ )$(Q 2" +' # 7+)6 : () 9L9 7 / 9 7?% K L(Q1B7 % 9 7 / D 9 = 23 9 % 5 / I> $ Y9 S$0/+ $- (Q/ K % 9 J . / +>r - > 7(QA) 9 % J + # 9 (Q-67 ; 9 )(YZ2 +' / ++ I+/% AV9 O 9 - # B B 9 L )$+ % 9 / D --+ &> 5%,23- 9 Z2& C- +( / ? / ) >> # % 9 C XJ)$(Q/ )*- ; +Q 9 7/(Q/ 9 % (Y- &> 9 B 9 +> ; 9 ( % Y> 23- V436 E 0/ +/> # 2 (Y-+/D I > 2$>Y+ > = 36$> 9 CE 0/ +>S ) / >/ (+ 2$>% /C( $ 9 -R/523/ # O9 <- +(YW U+ / / >I / 9 % --$4+ 9 / - # 7(Q-+N +)Y%)++ )(YZ"2 5 / 2& L- +(Y 7 +:7- ) 9 B $ 9 # --4+ &> 9 e $ V )(YZ"2 9 - % 9 $ 9 - # % ! = ! #" $ &% (' *),+ '.-0/214365(7839-0: ) / 1;'<143 + '>= 4'@? 3 )BADC 5E14'>3F1 );?8G 3IH JCK);C : 3L5 + -0:M:*367ONQPSRUTWVYX[Z \^]`_bacZedfVbdeg %('>3 7839- )= = 36=h1;367 ADC 5<1;'>3 C =>3IHi3 3j/>5 C :M3 G -k5;3l-0/8/>: *+ - :*3m1 C 1 )n?>G 3oH JC / 14' ) : > 'h1 G[C 7 43>+ -k1 MC = ) gqp /r365 MG 36=h1 )J) ' C / 1;'.-k1W1;'>3s-0/>/85 C - + ' = 14'83 + '.-0/81;365 M)4 365 />5 CtG *); = > guNQPSRUTWVYX[Zv\^]`_ba@Z`dfVbd C0? 14/r365 ADC 5 Gw) JC % C2C : ) 14'83x/85;3 &*Ct?>) : 3 ) 1y1 );?>G 3oH qCz)nC : & = > /85 C > 5;- G g {}|~ K ! 6 l G M)!) MG :-k5`1 C 1;'>3 - ;) *+(C =>3IHi3 3-k: >tC 5 1;' G g 1 3 5 ) 1 %('>3 - !) M+ 1 / C Hi3 3^-0: >tC 5 1;' -0:*:/ C 1;36=h1 -0:3 3O/ C =h1 ) -05;3 +6C0G / ? 14367Yg C / 3 365 3 =>7 = > 1 / C) - A 3<3 3 )Q?>;) = > 14'83 - ;) *+ H3 3w-0: >tC 5 14' G *) = C 1q3L= Ct? > ' ;) = + 3 1J7 C 3 ) = C 1 / C 5 / '836=j1 / C 3 3 ) -05n37 - >tC =.-k:M: -07 - + 36=h1y- *) = 0>t? 5n32g = g ! =<14' *) 3 - G />:*3 -0:*1;' Ct? > '<=>3 14'8365BW = C 5BF M) - ) - A 3J3 3 ADC 51;'>3q-0: >tC 5 14' GCkA '>-0/8143L5F 1;'>3 :M- + <) 1 C =>3 ) -05;3q-k: ' 30g %('>3x= C 1 MC = C0A -<6n9zoc^Wqm 4¡i¢.¡£oEi¤J¢.¤¥¦.`F M) =c1;5 C 7 ?>+ 367#- )(ADC :M: C / ) 6 % >t? 5n3J8g =6 p - G />:*3 = / ' *+ 'm1;'>3 - );*+qC =>3oH3 3Q-0: >tC 5 14' G§A - : ) g = y GQ?>) 1F= C 1 3xJ ) 7 5n3 + 1y=>3 > ' C 5Lg ! A W *) -k1u1;'>3w367 > 3 C0A 1;'>3 C -k5;7 -0:M:7 - >tC =.-0:=>3 > ' C 5 ) 3 + 3L/81 ADC 5uy -05n3 C2+L+6? / 367 7>3 A 3L=>7>3L5 ) 1 C =>3 ) g ! A W *) = C 1(-k11;'>33L7 > 3 CkA 14'83 C -05;7 -0:*:.7 5n3 + 1,=>3 > ' C 5 ) -0=>7<-0:M: ? 1 C =83 7 - >tC =>-0:Y=>3 > ' C 5(3 + : ? 7 = > F-05;3 C@+6+L? / 367 7>3 A 3L=>7>365 ) 1 C =>3 ) g % / C / C 1436=h1 -0:W3 3 / C =h1 ) W -k=>7 y#-05;3 ,¤ 6n9 oh¡ A 1;'>3 ADC :M: C / = > +6C =>7 1 MC = ) ' C :M7 6 W *) - ) - A 3q3 3 / 14'm5;3 ) /r3 + 1y1 C y9g y *) - ) - A 3q3 3 / 14'm5;3 ) /r3 + 1y1 C WIg W(-k=>7 y ) '.-05n3x-0:M: 783 A 36=87>365 : C@+ c) g ! = 14'>3 - ;) M+ 1 / C H3 3u-k: >tC 5 1;' G A 14'>3J7>3 A 36=>78365 + -0= +LC = ) 1;5 ?>+ 1(1 / Cs) - A 3J3 3 ) = - >' 3L=5;3 >MC = 14'>3 7>3 A 36=>78365 / = ) g > 5 Ct? / / 14' 1 / Cj) - A 3 3 3 ) *) -0: / - @) ) - A 3 ;) = + 3w-0:M: : C@+ c)x+LC = ) 145 ?8+ 1 = > 1 / C 3 3 ) -0: / - @) '.- 3w1 / C : 3L5n1 3 ) Wq-k=>7 y / ' *+ ' + -0=!= C 1 3l/>:M- 367 -k1m- 1 *G 3 1;'>3 -k1;14- + 3L56g 36= + 3 1;' M)*) ) /r3 V+ -0: + - ) 3 C0A ,36= )nC = )y? = +6C =>7 1 MC =>-0: ) - A 3L1 ) - g ! A :M3 )n) 14'>-0= 1 / C = C =>-07 - + 36=h1 / C 143L=c1 -0:3 3#/ C =h1 ) 5;3 G - = = 1;'>3 5;3 >MC = 1;'>3}-k1n1¡- + 365 / = ) g ) 783 3 =>3L7 = 3 + 1 *C =#2g = ) 3 &)*) 7>3L1;3 + 1;367 O) 39-k5 + '-k=>7 *) - / = ADC 5B14'>3q7>3 A 36=>783656g ,? 5;5n36=h14: 1;' M) -0/>/85 C - + ' + -0=>= C 17>3o143 + 1: ' 3 ) 1 C =>3 ) 14'>-k17 C = C 1`'.- 3,1 / CJ) - A 3 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 =>3 *G /85 C 3 G 3L=c1q1;'.-k1 M)J+6? 5;5;3L=h14: -0787>367 C =>: D =l14'>3s1 )n?>G 3oH JCm);C : 365 *) 7>3o143 + 1 *C = C0A :-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>365n3671 C 3Q-[:-k5 > 3x3 30g f3L1WJ 3Q-</ C 1436=h1 -0: 3 3u/ C =c16g 0 @J6n9Koc^Wqm 4¡i¢.¡£oi¤J¢.¤¥¦. F M) 7>3 3 =>3L7- )(ADC :*: C / () 6 = + >0? 5;3q8g A 6 : ! 3W/ Ct)! 1 *C = / ;1 ' Ct? 1(1 / C 3 3 ) g y GQ?>) 1 3x3 G /21 g '>3 + -0:*:-k7 - + 3L=h1s/ C =h1 ) 1 C Wog ! A 14'>3L5;3 M)uC =>: C =>3O/ C =h1z K14'>-k1 *) 3 14'8365(3 G /81 C 5y-0= ? = ) - A 3u-¥1;1¡- + 365 ) 1 C =>3 14'>3x-0: >tC 5 14' G§+ '>3 + c) 14'83x=83 1 />5 C2+ 3 )n) g 1;'>365 / M) 3 W *) = C 1^-z:-05 > 3 ) - A 3x3 3kg 3 1;'>365yW(= C 5y M) -k7 - + 3L=h1F1 C y9g :M:Y7 5;3 + 1y=>3 > ' C 5 )yC0A W(-k=>7 GK?>) 1 3 C@+6+6? / 3L7 7>3 A 36=>78365 ) 1 C =>3 ) g f3L1B 3JW C 5, g '83 + 1;'>3 ADC :*: C / = >u+LC =>7 1 MC = )EADC 5 C 14'OJ-0=87 8 ! A *) -k1e14'>3(3L7 > 3 C0A 14'83 C -05;7 -0:M:@7 - >tC =.-0:@=>3 > ' C 5 )C0A y3 + 3L/81 ADC 5^y^-05;3 C@+6+L? / 3L7 783 A 36=87>365 ) 1 C =>3 ) g ! A *) = C 1Q-k1q14'>3[367 > 3 C0A 1;'>3 C -k5;7 -0:M:7 5;3 + 1x=>3 > ' C 5 ) -0=>7 -0:M: ? 1 C =>3s7 - >0C =.-0:=83 > ' C 5q3 + : ? 7 = > FQ-05;3 C@+6+L? / 3L7 7>3 A 36=>7>3L5 ) 1 C =83 ) g ! AC =>3z7 - >tC =.-0:e=83 > ' C 5W3 + 36/21 ADC 5WFx7 C 3 ) = C 1 +LC =h1¡- =j783 A 36=87>365 ) 1 C =83 ADC 5 -0:*:r7 - >tC =.-0:.=>3 > ' C 5 ) 3 + 3L/81 ADC 5y GK?>) 1 3 C@+6+L? / 367 783 A 36=87>365 ) 1 C =>3 )(ADC 5B1;'>3 C 14'8365 + - ) 30g ,3 + - ?>) 3Ky GK?8) 1 3q3 G /81 = 14'>3x- C 3x7>3 3 = 1 *C = 1(3 + : ? 7>3 ) 1;'>3 + - ) 3 = / ' *+ ' ) 1 C =83 ) -05;3 + -0/81 ? 5;3L7 /8:- & = >= = ;) 7>3Q1;'>3z3 3 ) /.- + 3kg >t? 5n3z8g ) ' C / ) -0=m3 - G />:M3kg ! =m1;' M) 3 - G />:*3 M) -s:-k5 > 3W3 3u/ C =h1 / 14'}5n3 ) /r3 + 1^1 C ? 1 = G >t? 5n3J8g 6 m= 3 - G />:M3W1;'.-k1 M) = C 1F-z:-05 > 3 ) - A 3q3 30g M) = C 1B-u:M-05 > 3y3 3W/ C =h1 / 1;'<5;3 ) /r3 + 1B1 C g ! A 14'837>3 A 36=>7>3L5 + -k= +6C = ) 1;5 ?>+ 1(3 14'>3L5 1 / C :-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 + - ?>) 314'>3 -k5;3W- ) /r3 +2 -0: + - ) 3 C0A ,3L= );C = )B? = +6C =87 1 MC =.-k: ) - A 3L1 C 514'833 *) 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 =c13 3 ) -05n3 ) - A 3 1;'>3 + - ) 3 )BC0A 1 / C :-05 > 3 ) - A 3W3 3 ) -0=87 C =83J:-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:-05 > 3J3 3 *) *G / Ct);!) :*30g ! A 1;'>365n3 M) - ) 1 C =>3 C0A 1;'>3l-k1n1¡- + 365 = 14'>3 :-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/>:- \ = ;) 78314'83,:-k5 > 3E3 30g 3L= + 3 14'>37>3 A 3L=>7>3L5 : C@+ '.- 3,1 / CJ);? 5n3 : 365n1 3 ) g ) = - :M-05 > 3 3 3 / 14' 5;3 ) / 3 + 1}1 C / C =c1 ! A 14'8365;3j-k5;3 1 / C 3 G /81 / C =h1 = -0=87m1;'>3Q-¥1;1¡- + 365F/>:M- @:) = ;) 783 14'83x783 A 36=87>365 + -0= + -0/21 ? 5n3x1;'>3 ) 1 C =>3 CkA 14'83 -k1n1¡- + 365 3 :*: = > 1;'>3J:-k5 > 3W3 3 G - k = > 14'83W:-05 > 3W3 3x- ) - A 3 ;) = > :M3W/ C =c1 3 3 / 14'}5n3 ) /r3 + 1^1 C fg {}| Q -' : ) = 3L=c1;367 ADC 514'>3 C =>3IHi3 3 )nC : 365E-05n3 = +6C 5;/ C 54-k1;367 =h1 C 1;'>3(1 );?8G 3IH JC % 3 + '>= 4'c? 3 * );C : 3L56gEp- + '}143 + '8= 'c? 3 M) 5 3 5;3 & 3 / 367 6 == ,C =8=>3 + 1 MC = ) 14'>-k1W/>5 CtGwC 143 ? = ) - A 3 -k1n1¡- + 365 ) 1 C =83 ) 1 C ) - A 3#-05;3 +6C = ); 7>365n367 g C / 3 365 +6C =>=83 + 1 MC = ) 14'>-k1w'>36:*/ 14'83 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 %('>3L5;3q-05n3 GwC 5;3 + - ) 3 ) 1 CK+LC = !) 7>3L5 =O1 );?>G 3oH qC 1;'.-0= =<14'>3 C =>3oH3 3J/>5 C H :*3 G );?8+ '- ) 6 =>3l5n3 >*C = +LC =h1¡- = = > 1 / C +6C0G />:*3L1;33 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 =>3oH3 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 :-k51 C 14'>3 C =>3oH3 3 )nC : 365(-05n3 = +LC 5;/ C 5;-k143L7 =h1 C 1;'>31 );?>G 3oH qCF);C : 3L56g 9¤¥ 4£¡) (x0 @+£ *ho- ,</¤ .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;3u8g - g 9¤¥ 4£¡) ( (@D9o¦(@o 5,<¤/.0 M):> 36=83654-¥14367 C = -u/ C =c160y'>3L5;31;'>3783 A 36=87>365 GQ?>) 1,/>:-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 8 CCC - = 0>t? 5n3q8g 8 .C 5 + 367 G[C 3 ) g <= 1;' M)+ - ) 3 14'>3^7>3 A 3L=>7>3L5 M)ADC 5 + 367[1 C /8:-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 514'>3z783 A 36=87>365 =3/>:M-73 = > C =1;'.-k1q/ C =h19g = 3 - G /8:M3 *) / C =h1 = >0? 5;3J2g = g MGK? :-k1 MC = M)g = 0C 103L7 =O14'>3 ) - G 3>0B-73}- )R =O14'83 C =>3oH33t3 )nC : 365Lg > ^););?8G 3J14'>-k1B14'83q-k1;14- + 103L5 *) 14'83J/>:-73t3L5B1 C />5 C 3x/ Ct;) 1 MC = -0=>7<1;'>37>3 A 36=>78365 *),C =>3F1 C 7 M) /85 C 3kg B = C 5;7>3L51 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;'.-k114'>3 C =83oHi343t3 );C : 365 ?>) 3 * =h1 C 14'83F1 );?>G 3oH qCu);C : 365Lg : '>3 !) = > :M3oH/>:7- 303657 M) 1¡-0= + 3^1 CQG - 103y21 0 C 3 303 g) *)+LCtG H / ? 1;3671 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 ) z-0=87 ¡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) />:-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 ) 1;'>3W= ?>G =r365 CkAU+6C = ) 3 +L? 1 ! 3 C g .C 539- + ' / C 143L=h1 -k: 43 3t3J/ C =h1, !"$# !&%(' * 7>3 A 36=>78365 GwC 3 ) 1 CJG - 103(-0=Q3 3t3y-¥1 *)e+LCtG / ? 143L7s- ) 3 />:M- =8367 = 3 + 1 *C = 2+g 2-g ,>g ) *) 2g f3L1 l-k=>7 "=r3/ C 143L=c1 -0:^3 303 / C =c1 ) g : '>3l7 *) 14-0= + 31 C 21 0 C 43 3t3 B CC -0/8/>5 C MG -k143L7 7>3 =>367 - )BADC :M: C 0 )98 ' ¡ ) !"$# !&%(' ) !"$# !&%(' ) " 8g ););?8G 31;'.-k1mW *) - / C 143L=c1 -0:^3 303 / C =c1[14'.-¥1 M) - 7 5;3 + 1 C 5<7 - >0C =.-0: =>3 > ' = C 5 C0ABGwC 3 lg f3L1Qy =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^J-0 1;'#5n3 ) /r3 + 1 1 C Fg bC 5z-0:M: +6CtG = =.-¥1 MC = )QC0AF)n?>+ ' WQ-0=>7 F 1;'>3 );G -0:M:*3 ) 1 -0: ? 3 C0A $' W ¡y ) M)z+LCtG / ? 143L7 1 C) 3L1[- ) -0= = 1 -0: 3677 M) />5 C@C0A = ?>G =r365Lg $' W ¡y ) M) 783 =>3L7- )BADC :M: C 0 ) 8 B A '>3L:M/ ) 1 CwG - 1k3 C =>3q3 3t3 = = C 14'#W(-0=87 y ADC 5y3 - G />:*3 *A *) -s7 5;3 + 1F=>3 > ' = C 5 C0A = C 14'#J(-0=>7#F $' J ¡y ) *)F+LCtG / ? 143L7 - )yADC :M: C 0 ) 8 $' J 4F ) ' C " ' J ¡F ) ) C 1431;'.-k1 *)f);? =81;54- + 143L7 A 5 CtG !" ' W ¡y ) ;) = + 3 !"$# !&%(' W ) -k=>7 !"$# ! %(' y )w-05;3 ?>) 3L7 1 C G 39- )n? 5;3 !" ' W ¡y ) -k=>7 '836:M/ ) 1 C 783 + 5n39- ) 3x1;'>3q7 M) 1¡-0= + 3J1 CsG -103x-0=}3303 C = = C 14'mW(-k=>7 yg 1;'>365 0 M) 3 ' J 4F ) ' C !" ' W ¡y ) C )og ,>g B A W-0=>7 yy-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 :-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 0363L=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'.-k17>3 + 5;39- ) 3 1;'>37 *) 14-0= + 3^=>3 14'>3L51 CKC =>3^33t3= C 51 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 C *) $ > : '>3,1;5;36-k1 G 36=h1 *)0 7>36=h1 M+ -k:@1 C 14'83 C =>3IHi3 303 );C : 3L56g B A 1 O = 0C : 367 =O-K/>5 C@C0AUC 57 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 =3}- )n);?> = 3 !"$# &! % #$ "! #$ % & ' ( ( *) CC #$ "! >t? 5;3q8g 8 / C0); 1 MC = = C : A ) 1;3 ) 1 +LC :M:*3 + 1 MC =Yg {}| C 1 8= 7 )yC0A 1;3 ) 1 )n? 1;3 ) 0,3L5;3x/>5;3L/.-05n367 ADC 5(3 2/r365 *G 3L=h1 )98 );? = ) 3L1 C0A , }1 )n?>G 3IH JC /85 C =>:*3 Gw)S> 36=>3L54-k1;367=3 qC : C@C : ) h 60,- ) + ' C0) 36=Yg B =14'>3 /85 C =>:*3 G +6C :*:M3 + 1 *C = 14'>3L5;3}-05n3} tC : ?>G 3 ) g p- + ' 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 + -0:M:*367 -05;3 + ' Ct) 3L= g >t? 5;3K2g ) ' C 0 ) -0=l3 >- G />:*30g : '83z/>5 C =@H :*3 Gw) 0,3L5;3z145 3L7 ADC 5W3 14'>3L5 +6C : C 5/8:7- 3 = > 5 ) 1 5;3 )n? :*1 = > =j-<1 C 14-0: CkA 00 = ) 14-0= + 3 ) g = + 3y14'83 C =>3IHi43 3t3/85 C =>:*3 Gw)?>) 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[) 0365n3 ?>) 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:- ) 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 8b,C =>=>3 + 1 MC = ) 1 C[) - A 3 ) 1 C =>3 )yADC 5y-k1n1¡- + 1k365Lg 8 C 5 + 3L7 -¥1;1¡- + 10365 )FGwC 3 ) g # 8 C 5 + 367m7>3 A 3L=>7>365 )FGwC 3 ) g # : 0 "! #! * %$&('#)+*-, /. 0 2143657598:5 ;=<?>A@ BDC E GFIHKJML-NPORQTSUQ WV XZY X=[ ;U[ . . CC , - >:M3F8g C 8 365 ADC 5 G -0= + 3 ADC 5 );?>+L+ 3 )n);! 36:<3 ) 0 1 + ' = >uC =w36=8'.-0= + 3 G 36=h1 )* = J8g C ,>g ?>G =r365 C0A C 14-0: C 1¡-k: C 7>3 ) p=>'>-0= + 3 G 3L=h1 ) />5 C =>:*3 Gw) 1 MG : 3 ' ) 3 + ) = C 783 ) : 3 2/.-0=>78367 3 2/.-k=>7>367 ?>) 3L7 );C : 3L7 0 ,w/85 C =>:*3 Gw) 0 ,w/85 C =>:*3 Gw) /r365 ) 3 +6C =>7 'C)8 k , 0 C t , ' )8 'C ) kh , h ' ) 8 ' ) k , , , C h ', ) 8 ' ) # k , t C ,, C t 0 ' ) 8 ',() # kt , C ,( C , ' ) 8 ' ) kt , 0t kh , ,, C C ,h ' ) 8 ' ) kt , , k CC , ' ) 8 ') # kt t t ,c kt H H H : C 1¡-0:f/>5 C =>:*3 Gw) : = ;=<?> @ FHKJML N X=Y ; X=[ L [ ?! 8 # QTSUQ 43 [ ! #! ! # # ! ! ! O !# ! O # O O - >:M3O8g C ) ' C 0 ) 14'>3 5n3 );? :*1 )h = J8g C , ) 1¡-05 1 = > 0 1;'7 A Hi/>= ' 5 )u-0=>7 ) 0 1 + ' = > C =#36=>'.-k= + 3 G 3L=c1 )C =83 =3 C =>3kg : '>3u1 C 1¡-0:f3 23 +6? 1 MC =#1 *G 3x-0=87 = ?>G =r365 C0A = C 7>3 ) 3 2/.-k=>7>367 0365;3 +LCtG / ? 143L7 ?8); = > 14'>3 );? = ) 3o1 CkA k ,z/>5 C =>:M3 G[) 1;'.-k1y-k5;3 );C : 3L7 = 3 -0:*: G 3L1;' C 7 ) ' C )fH ' ) =u1;'>3,1¡- =>:M3kg :*:c/>5 C =8:M3 Gw)e)nC : 367 =3 ' C )`-k5;3 )nC : 367 =3 ' ) H ' )o2g 13 ;) 7>3 ) 39- + ' 365 !) MC = 0 14'z-W=>43 0 3L=>'.-0= + 3 G 36=h1 )nC : 367w-0:M:c/>5n3 &MC0?>) :<3 )nC : 367 />5 C =>:M3 G[) 3 + 36/81,21 0 CK+ - ) 3 ) =r3L21 0363L= ' )-0=>7 ' ) -0=87 =r3L21 03636= ')-0=>7 ' ,()og =83 />5 C =>:M3 G 0,- )Q)nC : 367 + : Ct) 3z1 C 14'>3s1 *G 3z: MG >1 =3 365 !) MC = ' ) = ? 1 0,- )x? = )nC : 367 G = ? 143 ) =3 3L5 ;) *C = ' )B-0=87 ' ,()og : -=>:*3q8g ) ' C 0 ) 5n3 );? :*1 )yC =m5;3 G[C & = > 0 1;' =m C =>3j3L=>'.-0= + 3 G 36=h19g : - =8:M3j8g ) ' C 0 ) 5n3 );? :*1 )}C =!-07>7 = > ;) = > :*3l36=>'.-k= + 3 G 3L=c16g : '>3 ) 3 3 2/ 3L5 MG 36=h1 )<GwCt) 1;: 3 +LC = 5 G 14'>3m5;3 );? : 1 )wC0Aq '.-k/8143L5O ADC 5z14'>3 C =>3IHi3 303 />5 C =>:M3 G g ^C 0,3 3L5 14'>3L5;3u-05;3W120 C[G - = 7 3 365n36= + 3 9) 8 ;: '>3q5;-k1 *CKC0A`);?>+L+ 3 )n)^ADC 5 ;) MGQ? :-k1 MC = M) : C 0365B1;'.-0= = '.-k/8143L5F2g : ' M)y+ -0= CC : = 0 ! " ?! 0 #! "!$#&%:'(#*)+#-,&!$#&%:. / # 0 0 *GK? :M-k1 *C =O-0=87#7 ? -0: );*GK? :-¥1 MC = g 8 3 ? 5 M) 1 M+ = 1 -0: -k1 *C = ADC 5B14'>3q7>3 A 36=>78365y1 CsG - 103J120 C 3303 ) g 8 3 ? 5 *) 1 M+y = 1 -0: -k1 MC = ADC 5 14'>3-k1n1¡- + 1k365}1 C /85;3 36=h1 120 C ;UL #! ! QTSUQ;=L QTSUQ ;UL 3 t3 ) g O QTSUQ QTSUQ;=L L-[ B CE - >:M3x8g+ 8$ 365 ADC 5 G -0= + 3 DA C 5B1 ? 5;= = >zC3j)! = > :M3W3L=>'.-0= + 3 G 36=h1 = J8g C ,>g ?>G =r365 C0A : C 14-0: C 7>3 ) : C 14-0: p=>'.-0= + 3 G 36=h1 ) /85 C =>:*3 Gw) 1 MG 3 ' ) 3 + ) = C 783 ) 3 2/.-0=87>367 3 8/>-0=>7>3L7 ?8) 367 );C : 3L7 ktz/>5 C =8:M3 Gw) 00s/85 C =>:*3 Gw) /r365 ) 3 +6C =87 'C)8 kt , C CC ,c C C , ' )8 kt C ht C t C ' ) 8 # kt 0 ,c C C , ', ) 8 # kt t C , 0 , C C ' )8 kt , t C C, , C ' ) 8 kt ,h C , , h C ' ) 8 # kt C , h C C C ' ) 8 :*: : ? 5;=>3L7 C = kt ,( C t ,c kt H H H : C 1¡-0:f/>5 C =>:*3 Gw) : = FIHMJML N #! ! O QTSUQ # ;U[ X=[ ! ! QTSUQ;=L O O ! ! # QTSUQ;=L ! ! # L [ ?! #! X=Y 0 ! - 8:M3q8g 8$ L3 5 ADC 5 G -k= + 3 DA C 5B1 ? 5;= = >sC = ); = > :M3J3L=>'.-0= + 3 G 36=h1 = J2g C ,8g ?>G =r365 C0A C 14-0: C 14-0: C 7>3 ) p=>'.-0= + 3 G 36=h1 ) /85 C =>:*3 Gw) 1 MG : 3 ' ) 3 + ) = C 783 ) : 3 2/.-0=87>367 3 8/>-0=>7>3L7 ?8) 367 );C : 3L7 k K/>5 C =8:M3 Gw) 0z/85 C =>:*3 Gw) /r365 ) 3 +6C =87 'C)8 k , C C , 0 h , ' )8 kh , C , C h ' ) 8 k , , C C C t C C ', ) 8 # k h C C t ' )8 # k , C C , ' ) 8 k , , C ' ) 8 kh C h ' ) 8 # kh , h , kt t , kt H H H : C 1¡-0:f/>5 C =>:*3 Gw) : = ;=<?> @ B CE FIHMJML N ! X=Y ;U[ X=[ ?! QTSUQ;=L QTSUQ;=L # O ! ! ! O ! # L [ #! #! ! O QTSUQ 0 ! ! - >:M3 8g-, 8 365 ADC 5 G -0= + 3 7.-k14- C = )!MGK? :-k1 MC = ADC 5}-0:M:qkt );C : 367!/>5 C =>:M3 G[) = J2g C ,8g :M:Y36=>'.-k= + 3 G 3L=c1 )^C = g '>- ) 3 C ) 39-k5 + '>3 )C =>:<3tg C 7>3 ) =3 + -0:M: ) ?>+L+ 3 );)nAD? : + -0:M: ) : C 14-0:f= C 7>3 ) , , t t , ' 08g C ) C , C ',(@g ) : = ! L [ # L [ # CC ! - >:M3q8C g 1¡8-k:f= C 3L785;3'>) 36-0' 700ADsC 5 /851 C C =>5n:*3o3 HGw) 39) -k) 5 + '>3 ) g : '.- ) 3 C '>- ) 3 , , t ' 8g C ) C t C ', 8g ) : = ! r3}3 2/>:- =>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:3L=>'.-0= + 3 G 36=h1 ) -k5;3[1 ? 5;=8367 C = 1;'>3 );?>+L+ 3 )n) 5;-k143 C0A );*GK? :-¥1 MC = 0B- ) ,( ' ) 363 : - =>:M3y8g-,() +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?>+ '[:-05 > 365E- GwCt? =h1 C0A 5n3oH ) 39-k5 + ' ADC 5 C = W8g C , ' ) 363 : - =>:M3x8g )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 W8g 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 :-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) = 14'>-k1 ADC 5Q- +6CtG />:*3L143 ) 39-05 + '-0:M: 1 C ) 0 1;'1;'>5;36-k1 ) -0=>7 -k= ) 0365 )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 />5n3 ) 36= + 3 C0A 1 C ) g = 0 . "! & ! 0 (# , 0 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 ) - - :-=>:*3 0,3L5;3 3 + : ? 7>3L7 g ) - 5;3 );? : 1 , C / C0); 1 MC = )r = ) = pp Jp 0,3L5;3 ?>) 367 g J2g C ,w-0=87 C , s/ Ct!) 1 *C = g pEp Jpyg : '>3 ) 3 : - =>:M3 ) 8g -k=>7 2g +6CtG /.-05n3s21 0 C)nC : 365 )xC = J8g C ,#-0=>7 =87 M+ -k143F14'.-k1^NuPSR TV 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) = W8g C , NuPSR TV X[Z \^]`_bacZedfVbd ! ! "!#%$'&()*+#$-,/.01.3254675+# 818/9+:*$;!#85<32=$"!>6!:8/$@?BAC8D6,5!E$B9@.$"F=",5GHI$;!E0HKJ 46<!#8L$NMO.$QPL+:+#@0RG5O7ST.K;I.9IU@4V$W685%$Q"XX.KU'&*(Y+#$O94"46@9"+#SX8/9+:*$;6,5Z? CC - >:M3u8g 8 365 ADC 5 G -0= + 3 +6CtG /.-05 M)nC =@=r3L120363L= NQPSRUTWVYX[Z \^]`_bacZedfVbd -0=>7 C H : C2C : ) = W8g C ,>g ?>G = 3L5 CkA : C 1¡-k: />5 C =>:M3 G[) 1 *G 3 ' ) 3 + ) )nC : 367 ', C s/85 C =>:*3 Gw) ) C : C2C : ) ,C C h NuP¥RUTWVYX[Z \^]`_bacZedfVbd , C ,, ,C H : C 14-0:f/>5 C =>:M3 G[) : = ! ! ! ! ! - >:M3u8g 8 63 5 ADC 5 G -0= + 3 +6CtG /.-05 M)nC =@=r3L120363L= NQPSRUTWVYX[Z \^]`_bacZedfVbd -0=>7 C H C: 2C : ) = pp qp(g ?>G = 3L5 CkA : C 1¡-k: />5 C =>:M3 G[) 1 *G 3 ' ) 3 + ) )nC : 367 ' CC s/85 C =>:*3 Gw) ) C C2C : ) t CC NuP¥RUTWVYX[Z : \^]`_bacZedfVbd C ,( , H C, : C 14-0:f/>5 C =>:M3 G[) : = ! );C : 3 ) 14'83 G -= C0? 1 @g [1 MG 3 )yA - ) 1;365F14'.-0= C : C2C : ) g B = pEp Jp -0:M:U/>5 C =>:M3 G[) );C : 3L7=3 C : C@C : ) 0365;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+ 1k36514'>-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 =3 C : A g >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 TV 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 =r3L120363L=K14'>3120 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 ) :-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 : )+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) />:7- 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= ! . 0 . . ! 0 ; ; CC ! 100 GoTools (sec) 10 1 y=x One problem 0.1 0.01 0.01 . >0? 0.1 >t? 100 5;3q8g 8E,CtG /.-05 M)nC = C0A 3 23 +L? 1 *C =#1 MG 3 ADC 5 =87 '& 7 ? -0: = ) 14-0= + 3 ): = J2g C ,8g . 1 10 TsumeGo Explorer (sec) ;5 3j2g 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 ; CC >t? ;5 3Q8g 8 / Ct;) 1 *C =#1;'.-k1xNuP¥RUTWVYX[Zv\^]`_bacZedfVbd );C : 3 )JG[C 5n3 ?M+ 1c:<3 14'>-0= C : C@C : ) ' C 1,:- + 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;'>33 23 +6? 1 MC =<1 *G 3 ADC 5/>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 TV 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:-k5 > 3 G -05 > = g >t? 5n3z8g 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;'>3/>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 03 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;3q2g CC 0 1;' ' 1;3q1 C />:M- 3 *),) 1¡-k1 M+ -k:M: 3 );C : 3L7 = 3 C : C2C : ) 0y' :M3NQPSRUTWVYX[Z \^]`_bacZedfVbd =>3L367 ) C = C 7>3 ) g >0? 5;32g C ,/8: C 1 ) 14'>3/>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;38g C +LCt? :*7K= C 1 = 3 );C : 3L7 = 3 C : C@C : ) 0 14' =z G = ? 1;3 ) 0y' :M3,NQPSRUTWVYX[Z . ! # Q F 9. . 0 ! . 0 . ! ! . . . T . . C # 0 100 GoTools (sec) 10 1 y=x One problem 0.1 0.01 0.01 >t? . 0.1 5;32g 8 p 23 +L? 1 *C =u1 MG 3 DA C f5 />5 C =>:M3 G[)f)nC : 367 ( >t? 100 C 1;'x/85 C> 5;- Gw)z = pp Jp(g = 3 = . 1 10 TsumeGo Explorer (sec) ;5 38g 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 3 Y C C >t? 5;3x8g 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 : . R 100 GoTools (sec) 10 1 y=x One problem 0.1 0.01 . 0.01 >t? 0.1 1 10 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 = p p Jp(g ( C >t? 5;3q2g 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:-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;33 @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 H/>=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 . 1 ; P 0 & ! 0 (# , ! W 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 = J2g C ,8g ^C 03 3L5 1J-0: );C<+LCt? :*7= C 1 );C : 3 GwCt) 1 C0A 14'83 />5 C =>:M3 G[) = pp Jp 14'>-k1^-05;3 ? = );C : 367 =3O14'>3 C 5 > =.-0: C : C@C : ) g {}| L 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 ) >0AADC 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 03 365 1 + -0=>= C 1 =r3 );C : 3L7 0 14' = G = ? 1;3 ) =r3 + - ?>) 3 C0A 14'>3 ) : C 0365 ) 39-k5 + ' O ! C ! ! # - 8:M3x8g 8 M)C 1 CkAeG ? = );C G : 367m/>,5 C C =>C :M3 G[C ) = pp Jp(g 5 =>:M3 =.- 3 : 5,1 />:-73 C =>3L330367 g 2g )>0A 1,:- + 1 C =>3L330367 g 2g )>0A ' 143 C =>3L3 30367 g 8g ) >0A 1,:- + 1 C =>3L3 30367 g 8g ) >0A ' 143 C =>3L3 3t3L30+g 2g ) >0A 1,:- + 1 C =>3L3 3t3L30g CC g ) >0A 1,:- + 1 : = ! ) /r363L7 + - ?>) 3L7 =3Q- G[C 5;3 +6CtG />: M+ -¥14367 AD? = + 1 *C =x1 C+LCtG / ? 14314'>3E'>3 ? 5 *) 1 *+ 7 M) 14-0= + 3 ADC 5B120 C 343t3 ) g {}| ) = 36=h143L7 = '.-k/8143L5,>0365;3^-07.-k/8143L7w1 C 1 );?>G 3oH C g :-k5 > 3(5n367 ?>+ 1 *C = : '>3 7>36- W C0A 1;'>3 ) 36-05 + ' ) /.- + 3 0,- ) - + ' 3 367 1;' ?>) 14'>3-= 3 ) 11 );?>G 3oH CW)nC : 365 0,- ) = ? : 19g : '>3 +6? 5;5;3L=h1ONuPSR TV 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 =r3L120363L= x-k=>7 3 G /8123j/ C =h1 ) g : ' *)u+LCtG /.-05;3 )xA - 0C 54-=8: 31 C C : C2C : ) 0y' *+ ' '.- ) -}: MG 1 C0A ) =}5n39- )nC =.- =>:*3J1 MG 30g - = Ct? 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( B -># C " # # @ 4 "" 5 # - B 9<; # " * 6 C " - = 5 " " C = = ! ?> = 9A@ B 5 \§/2MNH;- [ª '-R9jOµ5L9jXU/21N36-`365LX<Td/29 ;:j5L1rOµ5L9 * *` /21.+J1.5Yb **` ;10\·5 @2[ 7 XMf-69Z52O \·52BG/YH \'5YBG/2H ¶5²+N-6: T.9j5LM.H,-6X<: B ,X - :C-R3 .1 5c+.-R:¶-¨²Td/21.+.-R+ -¨²Td2/ 1N+.-6+ :C5L1H L-R+ o T.9j5LM.H,-6X<: o T.9j5LM.H;-RX : T-R9]:C6- 3R5L1.+ o m2o o ª + o + ¶5Yb **` ª o 6 + + L o [ **` m \·52BG/2H TN9C5LM.H,-6X<: o b b b C = # 9 9 9 9 @ C 9 !C 9 959 \'/2M.H,- [ o '-R9jOµ5L9jXU/21N36-`365LX<Td/29 ;:j5L10Oµ5L9 **` /21.+J1.5Yb **` ;1 BG/Y1.+d/29j+[ 7 XMf-69]52O \·52BE/2H \·52BG/YH V5c+.-6: T.9j5LM.H,-6X<: B ;X<- :j-63 1.5²+N-6:¶-¨²Td/21.+N-6+ -¨²TdY/ 1.+.-6+ :C5L1H L-R+ ª m T.9C5LMNH;-6X<: ª m TN9C5LM.H,-6X<: Tf-69Z:CR- 365L1.+ 5 5 2mLo o m V5Yb **` ªm ª + Lm Wo ª o + 5 ªm ª m **` W o \·52BG/2H T.9C5LMNH;-6X<: ª b b b C = " !C !C 5C C C @ 5C 5C 9 C 9 9 C C5C + ;/2KL5L1d/YH **` Tf-69jOµ529CX<-6+gMf-RBjBE-R96[ ¨²3R-6TNBiOµ5L9a5L1.- T.9C52M.H;-RX **` -¨²Td/Y1.+.-6+s/eB X<5L:jB¥/Y:©XU/21W@1.5c+.-6:¥/2:©1N5Yb **` /21.+i5YOBE-61S+.9C/2X /YB ,3A/2H,H,@H,-6:C:R[\]I.-]Tf-69´Oµ5L9CX /21.3R52O **` :C36/2H;-R:?-¨²Tf5L1.-61WB ¸/2H,H,@iMf-RBjBE-69 ;1<BEI.-: R-~52O TN9C5LM.H,-6X<:6[ ¦ 5L9_-¨N/2X<T.H,- **` :C52H L-R+ BCI.-ZT5L: ,B ;5L1 ;1 ¦ ,K 7 9j- [ ;1Jª Lm 1.5c+.-6: =ZI ;H,-¶1N5Yb **` 1.-R-6+.-R+ ª 5 1.5c+.-6:R[¥\]I ;: ;:]1N52BZ: 7 9jT.9 ,: ,1.K : ;1.3R-/YH;HTf5L: B ,5L1.: ,1JBCI ;:]:j-RBV/29j- ,+.-A/YH Oµ5L9 **` =ZI ,H;-1.5Yb **` : 7 0-69j:¶Oµ9j5LX3R5LXM ,1d/YB ;5L1d/YH-¨NTNH;5L: ;5L1[ ¦ ;K 7 9C-R: [ /21.+ [ 5 T.9j-6:j-61WB0BEI.-®9C-6: 7 H,BC:UOµ5L9<BEI.->:´BG/21N+d/29C+ BC-6:jBr365LH,H;-63B ,5L1[ V521.- 5YO?BEI.-R:C- T.9C5LMNH;-6X<:=h-R9C- +.-6: ;KL1.-R+= BEIg+N-63652X Tf5L: B ,5L1 ;1DX ,1.+[U³1365L1cb BE9C/2:jB_BE5 ¦ ,K 7 9C- [ m BEIN-69C-~/29C-¶X<5L9C-ZT.9j5LM.H;-RX :©=ZI.-R9C- **` =h/2:h:jH;5=^-696[-¶5=h-L-69 5L14/ L-R9E/2KL- **` -¨²T.H;5L9j-6:iH,-6:C:i1N5²+.-R:</21.+s1.-6-R+.:SH;-R:C:i-¨²-R3 7 B ;5L1B ,X -Y[\]I ;: ,: -6:jT-R3 ;/2H;H@>BE9 7 - Oµ5L9`BEIN-UH¸/Y9CKL-R9T.9j5LM.H,-6X<: :C5 **` :C-6-RX :aBE5:C3A/YH;-UMf-RBjBE-R96[ **` :C52X -B ;X<-6: ,X T.9j5 L-6:`BEIN-UTf-69jOµ529CX /21.36-<MW@/H¸/Y9CKL-SX /29CK ,1[ ¦ 5L9`-¨./YX T.H,- **` 1.-R-6+.-R+ ª 1N5²+.-R: ,1 [ m :C-R365L1.+N:Oµ5L9aBCI.-rTf5L: B ,5L1 ,1 ¦ ,K 7 9C- [ + =ZIN-69C-6/2: 1.5Yb **` -¨²T.H,5L9C-R+ ª o 5 m 1.5c+.-R: ;1 m [ :j-636521.+.:6[g³1sBCI ;:T5L: ,B ;5L1 +.-63R5LXSb Tf5L: B ;521.:]BE9 ,KLKL-R9C-6+±MW@±M.H¸/23GF03R9 7 3 ¸/2H :´BE5L1N-6:/Y1.+=ZI BE-a:E/YOµ-:jBC5L1.-6:V9C-6/23GI ;1NK BEINMf5L9C+.-R9C:Z52O·BEI.-`Mf5W/29j+±:j-6-RXqBE5<5²3R3 7 9?Oµ9j- 7 -61WBCH,@L[ ³1±BCI.-`Id/29j+T.9j5LM.H,-6X<:]52O'BCI ;:]:j-RB BEI.-`Tf-69C3R-61WBG/YKL-5YO§1.5c+.-R: ,1r=ZI ;3GI+.-63R5LXSb ª ª - # " = C =" -4 "" @ C = - "# - C C - " 7 - / = C # = 4 "# B C " # @ # 5 9 @ 9 C C "# C 9 "# C 959 $ " - 9 =" - - - 9 0 - 1e+07 1e+06 No-DDS 100000 10000 1000 100 y=x One problem 10 1 1 10 100 1000 10000 DDS 100000 1e+06 1e+07 ¦ ;K 7 9C- [ m ¶5²+.-`-¨NT./21.: ,5L1.:]Oµ5L9?BC5 @rT.9C5LMNH;-6X<:]:C52H L-R+MW@JMf52BEI2-69C: ;5L1N:6[ C " = E % E % E % E % E % H ¦ ,K 7 9C- [ a T5Lo: ,B ;5L1kBEI./YB **` :j5LH 2-6+= BEIDX 7 3GIH;-6:j:1N5²+.-R:aBCId/21D1.5Yb **` + ;+.-b3R5L1 7 -69R[,ª [ :jK2O hH;/23GF BC5<H 2-M8@JTNH¸/A@ ;1.KS/YB [ ª o C * =" : 4 * 9 No-DDS 100 10 1 y=x One problem 0.1 0.1 1 10 100 DDS ¦ ,K 7 9j- [ ¨²-R3 7 B ;5L1JB ,X -Oµ5L9]:´BG/21N+d/29C+±T.9j5LM.H;-RX :h:C5LH1L-6+MW@0M5YBEIL-R9C: ,5L1.:R[ C C " # " 1e+07 1e+06 No-DDS 100000 10000 1000 100 y=x One problem 10 1 1 10 100 1000 10000 DDS 100000 1e+06 1e+07 ¦ ;K 7 9C- [ 5 ¶5²+.-`-¨NT./21.: ,5L1JOµ5L9]:jBE/21.+d/29j+TN9C5LM.H,-6X<:]:C5LH1L-R+MW@0Mf52BEI L -R9C: ;5L1.:R[ C " ª = 959 ¦ ,K 7 9C- [ + ± T52: ,B ;5L1BCId/YB **` :C5LH1L-6+ X 529C- 7 ;3GF8H@ 5L1.-R-R@2-6-2[ªL[n:CK2O ^H¸/23GF H 1L-R:?= ,BEI [ Tf5L: B ;521.:Z/29C-aTf5L:C: ;M.H,-Y/29 ;-R:¶Oµ9j5LX ª BC5 m [?~KW/ ;1 Oµ5L9Z-¨./YX T.H,- ;1 ¦ ,K 7 9C[ + ;1BEIN- ª 1.5c+.-R:<-¨²T.H;5L9j-6+sMW@ **` **` +N-RBE-R3RBC-6+ ª o 6+ m [ +.-R365LX<T52: ,B ;5L1N:6[ r ¡¢ ©¢ A **` ;:]H ;X ,BC-6+ ,1rBEI.-=h/A@rBEId/eBZ:CT.H ,BE:Z/Y9C-9C-R365LK21 R-6+[¥\]I.-`5L1.H@UTf5 ,18BC: 7 :C-R+±BC5 :CTNH ,B_Tf5L: B ,5L1.:¥/29j-¶BCI.5L:C-V5²3R3 7 T ;-6+ MW@ :C/YOµ-V/YBCBE/23GF2-R9]/21.+ 369 7 3 ¸/YHd+.-ROµ-R1.+.-69h:jBE521.-6:R[ \]I ,:±:j-63B ;521 ;1WBC9C5c+ 7 36-R:±/4H,-6:C:±9 ,K ;+ +N-63652X Tf5L: B ,5L1 BEId/YB 7 :j-6: C/2H,X 5L:´BJ:E/YOµ- /YBjBG/23GFY-69_:jBE521.-6:h/2:©=h-RH;Hz[ ¦ ;K 7 9C- [ª :CI.5=Z:h/21 -¨N/2X TNH;-2['³¹O BCI.-¶B=^5=ZI ,BE-Z:´BE5L1.-R: X /29CFY-6+0MW@U: 7 /29C-R:Z/29C-~/2:C: 7 X<-6+rBE5M-:C/YOµ- BEI.-R1rBEI.-~/29C-6/3A/21UM-:jT.H B ,1WBE5B=^5 : 7 M.9C-RK ;5L1N: /rH,-ROB: 7 M.9j-6K ,5L1>X /29CFY-6+kMW@>:jXU/2H,H'K29C-R@: 7 /Y9C-6:`/Y1.+P/r9 ,KLIWB9C-RK ;521 X /29CFY-6+MW@a369j5L:C:j-6:6[©\]IN-]/YBCBE/23GF2-R9§36/21i/2H=?/A@c:§X /2F2-_BEI.-_B=h5~XU/29jF2-R+=ZI ,BC-h:´BE5L1.-R: :E/eOµ- MW@kOµ5LH,H;5= ;1.KJ/J: ,X TNH;- wix 2x 365L1.1N-63RB ;5L1:jBC9E/YBC-6K2@ ` I.-61.- L-R9BEI.-<+.-Oµ-61.+.-R9 T.H;/@c:'- ,BCI.-69 25L9 3 9j-6T.H@a5L1BCI.-?52BCI.-69'T5 ,1WBA[·\]I.- { A {Ct`tc{ yG|ew~}d| xxz| v X<5c+.-6H 7 :j-6:~: 7 3GIk:jBC5L1.-R:VOµ5L9Z:CTNH ,BjB ,1.K</<Tf5L: B ,5L1[ -¶5=h- L-69 + 7 9 ;1.KSBEIN-:j-A/29j3GI>BEI.-a3A/2:j=ZI.-R9C-/SX ¸/ 36521.1.-63B ,5L1 ,:Z/YBCBE/23GF2-R+®[email protected]`+.-ROµ-R1.+.-69¶X 7 :´B¶M-`Id/21N+.H;-R+[ ³1BEI ,:3A/Y:C- BCI.-09C-RH¸/e¨²-6++N-63652X Tf5L: B ,5L1gX<5²+N-6H_-¨cBE-R1.+.:iBEI.-0:j-A/29j3GIBE5>BEIN7 1 ,5L152O¶BEI.-J/ 0 -R3RBE-R+s: 7 M.9j-6K ,5L1.:R[\]I.-±/2H;K25L9 BEI.X ,:i-¨²T.H¸/ ;1.-R+ ;1+.-BG/ ,H^= ,BCI BEIN-®I.-RH;T 52O ¦ ;K 7 9j- [,ª [³1 BCI ;: ° K 7 9j- H;-BU9j-6K ,5L1 <365L1.: ;:´Br52OBCI.-®-RX TNB@ Tf5 ;1WB 2/Y1.+/2H,HTf5 ;1WBE:aX /29CFY-6+gMW@ ° H,H;-6+P: 7 /29C-R:6#[ ¶-RK ;521 365L1WBE/ ;1.: 3/Y1.+ ª C > "" : 4 "# = C 9 @ = ; @<; 5C 9 C 9 9<; "" & > "# - 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" F 0 . ,B& $ 240G7 3, 2 *-6 "|'?Z6 .;,X 035>7KE aX2 & 2 & 5>0Z?30Z64" (4. 0y0 F $ & 2 8 , ? , 2 F 0 .V*@4, $ & " , 0 F K uyE 7 aX"@"@6+< ,V*@'&)*i*@4,V, I 6 &)"* 0Z24" &3.;, " & "* " | , 7 & *T*-4,RG*- " #* E !4, 9 &3.;,f(4. 0 , 2 F 0 . 8:9 *@.-& $ 24? *-+, " ,B&3. $ ? .-&3(4 E , K e a *o*-4, z*- " #*%* 0 $'& b ; e K $'& b < #e K bdc & " V & 247 * & Zb ; e K / * & Zb < e K E < " *@4,1.;, F 0 .;, $ 0Z" , 2 H }*- b < e:K & 247 b < e9K E a %* < b < e9K 3 < 2 )b $'& b A e Q $'& b C e;e Kt & 247 b < e K * & Zb A e * & Zb C e K K < " , h ('& 247 , 7KE 24$ , $& b A e K $'& b C e K A " $ 03" , 2 3 H #*@ b A e K & 247 b A e K E a * A-W " , h ( 5>0 .;, 7{" 24$ , ! b A [e K " < 2Kb $& b W e@,e K & 247 ! b A e K * & Zb W e K # E !4,/.@, F 0 .@, 6W " " , 5 , $ *-, 7 * 0 , h (4& 247 H }*-$ b W e K & 247 ! b W e K &E % 0 Hf, ,1. *-4,N*-,/. < 2 &)* 0Z2 3 $/0Z247 }"* 0Z2 "f" & "* " | , 7 & * W 8 , $ & 6+" , * & Zb = e P* & Zb G e K ' . K u b W e K E ! 64" G " 240 * , h ( 5#0 .@, 7 & 247 (4. 0G0 F & 247 7 " (+. 0G0 F 2z64<o8 ,/. " &3.@, 8 & $ J), 7 6 (W* 0 & " F 0Z5>5#0 H " \ $& b W e K < 2 b)$'& b = e Q $'& b G e@e K * & /b W e K * & Zb = e >* & 3b G e K $& b A L e K $'& b W e K e K * & ^b W e K * & /b A L $'& b < L e K < 2 )b $'& b A e Q $& b C e@e K e K * & ^b A e * & :b C e K * & /b < L 0 .=*-4, .@, < & 2 2+?240G7 , " ; ? = C 3T E & 247 D (4. 0G0 F & 247 7 " (+. 0G0 F 2z64<o8 ,/. " .@, < & 2 *-4, " & < , E ! 64" *-4,,JI 6 & *" 0324"N" 0 H 2 & 80 ,%&3.;,I(4. 0 , 2 F 0 .!*@4, $ & " , 0 F E bdc & " , K e !6 "g$ & " ,1 " (4. 0 , 2 H3}*-& 2 & 2 & 5#0Z?Z0Z6+"g7 "@$/64"@" 0Z2KE a * *@4, G*- " #*T* 0 $'& b ; e K $'& b < e K & 247 * & Zb ; e K H #*@ ! b ; e K & 247 ! b ; e K E * & Zb < e K E ; " $ 0Z" , 2 3 ; = T D " , h ( 5>0 .;, 7KE !4,o*@,1. < 2 &)*" 0Z2 $/0Z247 }*" 0Z2 & * D 0Z5>7+" " 24$ , * & :b W e K b D eK E !4, 2 (4. 0y0 F & 247 7 " (+. 0G0 F 2z64<n8 ,1. " &3.;, 8 & $ J3, 7{6 (* 0 & 2+7 *-+, (4. 0y0 F & 2477 " (+. 0G0 F 2z64<o8 ,/. " $ '& 24? ,`& " 2 *@4,pJ, I 6 &)* 0Z2+" & 80 , E "!2 , ,1. , h ( 5>0 .@, 7 & 247W7 F C_( 2N5>0G0 ( " F 0 .;, ,/. E G u A D B C E F G I J H K L M N B C pn(B)= 2 dn(B)= 1 B E pn(C)= 1 dn(C)= 1 D E b8 e b$ e pn(D)= 1 pn(E)= 1 dn(D)= 1 dn(E)= 1 AND node thpn(F)= 2 thdn(F)= 2 F H D E G pn(F)= 1 dn(F)= 2 H b, e pn(H)= 1 pn(I)= 1 dn(H)= 1 dn(I)= 1 F H pn(H)= 1 dn(H)= 1 I G pn(J)= 1 dn(J)= 2 J pn(I)= 1 dn(I)= 1 pn(G)= 1 dn(G)= 3 K pn(K)= 1 dn(K)= 1 bF e O pn(O)= 1 dn(O)= 2 P pn(P)= 1 dn(P)= 1 A thpn(C)= 3 thdn(C)= -1 B C D E F thpn(F)= 2 thdn(F)= 4 G I thpn(I)= 2 thdn(I)= 3 J H thpn(L)= 2 L thdn(L)= 3 thpn(N)= 2 N thdn(N)= 2 M b? e I pn(B)= 2 dn(B)= 1 pn(H)= 1 dn(H)= 1 pn(M)= 1 dn(M)= 1 u K pn(K)= 1 dn(K)= 1 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 O thpn(O)= 2 thdn(O)= 2 P pn(P)= 1 dn(P)= 1 thpn(A)= thdn(A)= thpn(C)= 3 thdn(C)= -1 pn(B)= 2 dn(B)= 1 B C D E F pn(F)= 1 dn(F)= 4 G I pn(I)= 1 dn(I)= 3 J H pn(H)= 1 dn(H)= 1 pn(L)= 1 L dn(L)= 3 M pn(N)= 1 N dn(N)= 2 pn(M)= 1 dn(M)= 1 be be 3? 6 .@, a%E \ aX2 , h & < ( 5 , 2 HUS $ 7 F C( 2W5>0G0 ( " F 0 .@, ,1. E J 8 pn(F)= 1 dn(F)= 2 I 8 8 E thpn(G)= 2 thdn(G)= 3 G thpn(J)= 2 thdn(J)= 2 G pn(G)= 1 dn(G)= 1 8 D thpn(C)= 3 thdn(C)= -1 C F pn(I)= 1 dn(I)= 1 8 8 B H F thpn(A)= thdn(A)= A 8 pn(B)= 2 dn(B)= 1 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 dn(G)= 1 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)= -1 C 8 8 b& e D Win Q thpn(A)= thdn(A)= A 8 8 thpn(B)= 2 thdn(B)= -1 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 !4,p*-,/. <="!64" , 7 2 *@4,I*-4, " " &3.;,p& 7 &3(y*-, 7 F . 03< s w)x & 247s u xdE c "!# $&%'!( %)+*,!#-.%(/+012345+6 7 83 '59: 6'.;<7 7 =<>@? aA% 9 4BC 0ED!GFD@ H!I% : 6(6( : $3$KJHFL%'!#+ %M+*@!#-)%#/+0EN 4B6 7 8 N301D!GF D@ H!(%O01/+6(PQ $RJHF&% IS: /+6' % B9: 6(T;<7 7 U!# %<!(-V/Q!O-V/ W!GXYK3F+ %./+6' 9: /+6#/+H!( $Z!#LJ@[/+4 +7 \ ]?_^ DD@% !#2+*%(3H!#+H7 83T%' H!#Q7 `&ba -c% ! : /Q! 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