From e9161135ae26ac76c8317da8144e41c347fed684 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sun, 11 May 2025 14:22:14 +0800 Subject: [PATCH] misleading typo --- main.pdf | Bin 95811 -> 95797 bytes main.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/main.pdf b/main.pdf index baa8b0e05e1260216b014e080f6be26e8b1a61c9..dce0445f9ef6a29ab3a17142ee5ffed0130bfdad 100644 GIT binary patch delta 7894 zcmai&RZts%lZSB$?i2|Y+!EZ1y96oj?oiwzcyV{v;10#Tl;U2r#fnRzSdqg0?{4n- z=6l%L-|Xwoe7mFPs2%61%`Esx5dLh$cZe*o8L`c*{m-etZ%6{Hi_^dE;nIIhBKkes zxPRTqtWH$i<{ZljVMqlidC$H4BXuQKyk1Nb``M=mQbdT7{!u5{x6ycYy1l(S{8lVJ z@Z3p*!OSj=ujhDo?O%DUY>^1&3)!j{XGCfFWG3-yw`y~Dl4_oiewvewm#n??OJ7eN z_WUxRvue)Aq>D0Y&&NMX5XpKw%HFDlq9m=SHTq##j|LC?^oa-NXKl>%>6}K#g)53? z-S{!}2j$!qm2K$bOu5-?TV+g5M&85@fl^3FXz_0#kWuXwJ>*@VW60a0$bzT|FI;%vshtMjih9)-LGJCNkI=kCF z^W(vSYPrRm9b^`EVUl}=cS}pe3<>@;(Rc$YZ2Mf0Wj-B6_&-zFI8t)CAYC~!Y=g_% zQej%82pV#5_QB-HruR)mGIEp7v!luM=BY*`6X-OCtaPY18&=k$yV_V#N6b0~&w@fI z3ppxTD(Bfr6tVUKs5zZp5B7Uu17|;%u-yTmRAt#p{RX_>;{W3GDE;ie$&XYw-LH(s zA<1YheH?r`_yG&Fm>;(kGz?vV#Y6^oP>*3LqD`cG5FmzQq;O|re^4(`kSe zs(r+v*X>E$h9&-<0YI*e?@G$aapEqEppb<^&n7X5%E4?2yu?Ao^Z&HC z;JUx5PORKv{9ZnthTVK$$vu?Wo`{LHiIhJ2-tyw8gAv)Wl=QEHBRR@WIdK?E^VNc8 z)T7nHA+3+Pk6DK%RFx6-sFFGeT`-uWpTPvH6wsfpACwcBb2$8w(XXfyX@pHcn4rW>ZY)4SP8EE~7m6>rClCAb^xG==fg(g5f8m0b;& zxO=B6=-h~`J#LUxxG{ZKKvWUOp?T8#uf}I8mrZT{5adN|&!5HSh88Glv55shY5r1H z(S`(6@S1t85vnhFI-6`}SE#iXBC<5q6L3`*Z}gvlXx@*olb1{Kii`Y76PG{PL3o6o zyz%l`a7DKOMv7b*T9o2#rr$4en>Q?G<6BmAp&pUjI!!J^Y-_|b6xzPj*b&qG57bhp zYOHS4Lzgs5?Zg{?FkJjvuXSWNYge+znx|3=ZyXGR=C)^*p&ThkqZUg8ZsL^JM+`K3 z%w|wb>vQc}zx%LqIIQa#h7KtEYGm1xZmBWwlf)c3dvl(yI8zZRqGRUCJj5?z5HpR+ zEZ%;o9KN`Eps8*l;?r=8 zv$DqC;-Gn7OupsZgU(msr7g=6N6)CmeNo68J^md`QTlb2aG1KT9kzrx@jE!? z9i>V+*Y`^i39*CxL|=EeMcu;wn2*W$K}3P>bak_?GqqO!fjNNC)BAw{$F?$Tp;^-R za<(425E8WwTk0Q}dw&A^T|at~T${I22LIt@`KnX#f<(-N6=y*OCSmQb6ZsQw<;w!l}QF_5$59s`w{b#M)N0mJfK1kGt)2 zhITxmL-sF&uqK4^k7xB*&RR5^BBIl0>}zq*7I>j0wT?3KcxHYzHH^+A4En)2XiF1i zT=5(4_^@~3?>6pX?(o3;{RFd+w;N z+Vvm1N1d&2O}0Bmk1L;m9IdK_QU=rsF2};83Q{&uEl|bw;%6bh!H$}HZUeG7w=o?i zCQVUvWUl_J$cskDH`s;%_l?M=Q8wrjRZU7g0X>3$NYaZY>y6qIA7wqz~h#UoQGvR?ZaN^>Thv)I$4q?Arq&Tay}r z*!#$H(dNec8l&(}M_)dy)NQ{V^@w20P}%}jgIe9$0Iw5{uvGV39H)<5r~0nLUq-Y( zq)R!}iPSIIcQpo0m`tzXU*XOVrgxEj{+wFWL(^c9N6Qub?L~S2wYR5M=5yIy?k6W| zv16Gq@MVovuKQIZXU&Q3mrG-U;fMO;^U^Ag`}tt5@nQaiyf)8*>^r1mXH%N1e2#d1 z0wQoLN(@;PAMBd$EuSQcwWaRRlWix$UjFGOSRX2foS}ag$eHzt&`k;?D6K)y$y_Ic znHj@_)0C?P46*aJ7c@lh_Jp}(?*fKLcTbrsQ!sKBcO7>HgvrBY zF!>R`?r)fP-I0=FxjVWmjqIt8lqrma4~$iG3rs=LV7KhT)$S1W!H?g*sE&>04zVyE zNy}JOg3V$lol9#bO7iRWfmin{CX~o4FQ4eee9=E&iZrP(+y(Pv^Mw$-h0XansC)wl zyX@j3tZJhVni?sbdu*Hqf9LCFP2g$j31{Uh3?(YEiQX`TP#Ik9-+$}>iAgv=2P^c4V+xpxbh zps^mX@Yh-+EN<;45sa4UnT{a43`cm|kl(4W{Jq=c)2#;T<-(CF=)6geWwr+TFHN?L z9~yt^J^&fgKBSwa>Roc$mAtHn(oY^te>Xh^?Lh)UR>KI1X`%4 zVNKR+uZaAfRN=$2vQMFBCI$8SRBgJ;g}VrK&(|Xg=X&zP4atb1nx50AU+tzH_HH(& zdkt?B(T#;thzR-yKhLSiDHH&PVlE&47%ogx%L?);s4D+n*CfM!Cu}FpFSHbQKbMP* zq9A0KW@-&VB+dc^vS4bq5$j3v3nhAlz?>r41Mvc@G`V`!+!$jdTLVz|XTQCzT4WH! zC&It#KMY(8vOSghg53419&hR&Pq92FPrm8vu`Mu(N4mp2$sCP|g^#iRrT~ zYO`F1VU>qY52LKJ&t*>*$5SLq-*O7EeTZDak-`AK|1($hG{7&7#|&>fI;S_7Um2-eJ1MxMxO4#j2s(p&kNLm&uCJ4$yhEPa8D)r~&V(;rZ+7N% z&L1ql=)zr$4Txm?yloP9s{`4bUw;F3{rx0cgte;ss>Kcz9!D-#`}fI5w1nF*!4#w9Fj!yXnybA68m zBAwQx8BF$E2Tw}92zB3j*p0V_lsl`>$EaF)(Unc$VDXNhQQAtrj)hEB`w)@(9Z$Xc zFIEc&NlddBp5ZL&&qUDYq}nqyCiE2lrcFz~GyuRVwA7L++8inA3v-RNVyGeHH>=;< zz+dbCoIQ6%OJ~$bT45|gFnt-6l!C-Ou{&YX`p!Dj$dU0tb1{2WfCZAFk4j06V>~~T}YitBPmln-c*$VJu8&$!zFZ6p72?XU91%LehY&SNSjnIr>Mcv*HEAodtFajPl6Rs9= z!OBZhv{gsQVL)j+*mC|dkDu4<4%8)WeC08xamD!G2JJO2UrdH_@0UOwSTJn@ckn&p zgZN+YYZOS&b~X5f&RR}&s`v)JmSy!pR>bbkJ`Pt)V*G&iV%6$(e3zVzenoOm>%yhw zv(D^cJgr5E0mf77D(D$>BcFPZb2*R0fZaE4m+ohj&v57uYs1KfilR6Sj-P2n^iL$q zZ<0?fEi&*$HL0E4tip{e{nn}1=C_aYHMF;Ye?te*_j9_(#2ybW+cs7U*MARYFvKMx zH0&-@lKoD^;Z!9?KHm>Q4q_69{*(}1+K~9Mo)PB2!oG?G)ju?BwF?}XH2%Evl0T{$ z?K!Q&9Oyx0M%=?b4Q@giXz?s^5_nPk@GK1p&Cmr z^BR$;t$~gW2T|aE|1B%|PRUEtB;(shAEs6OhW#$H9q$yML;aLdNh}>!NHu<`O*g6# zRwW@%b^dWzRLwau}vcmu~%50XI`cO)OTO8mh1m z31E89YY5qD*5YBz<@YEf_C%}X^aRFQiL=1zv9wa>cc^8XSzt)zdNz?PX-+8=81OzD zLV70c20WIeJbE4xFeB1D{Re!{$?5Q0VYRKHK`GiV z29+vaQM}hYhLI5d0_9;NN`oCz%UDT~+8>CN1Ud3zNjE|t<~^9_;X8Cl$YEJRS3K=& zQsABx;iFXLL!>9BvVAKs2Vo5eOqssV+z0mrIv8gAcmwT)*44L-Mx*6J3X4Z-yAxkzDLOd|ovt zS_?MNY)%u~(IpSRo}M><#?7hHBJP#tfxe2V0Q?VlWjA5v4UHvJP8MHY+a51A+o)&U zSaFe-Y&;_kgP|sPT4cz|yjz?5kImclzS~;N<*$8T9goW_5q{U{|4v>w7}CKme6FPE zJsp_utCHcqpwRg(=+(QOUIJl(X^K*KhS~~hKBSo9Vt4lZ<2`A*S_j;R+aymN;!H@0 z=38@1G-74FpKFR!x^Fh*AFEvlpVjPu0yUj$UEZ?`R77lw&c*rXzM1}L)CHd7?Mv_n)NpDY-PMA-aO<0_klD5VqUjZV>2aSOgS(%g|N^weB zK@ZY|wrSw>$o1Q4CJO<~8xyXHb|ttP&$jA)K$Xi3sCUg&z~$AUJ3w$Er0LXOv-2PA zPixq0ku_dw*s|bqdNL0c33g7V+Hg}{?W+){>-R^1j%p@;V`2Ibk09)s)!xYt_s=ul8__;Rw{r7#x&jhcL#sqA1=UwfDV)#ej zjK3P<5*2M>xr#Zv?smhX;gR{nyNFbWlV`QFRV#-Q-fh*5y5#b#P|;4LN6q)7Oxf8! z&K!Tm3hMoH1}=_Sn@amq(2G9*A^+||;s~vgZt25TbyF{yRghn-;G5xOQH34dy4T8S znwki%IQDFD<$36GX0P-)i$fX3(}-pR5LawQL+yyOq`NGAQRb8ZYyT?RtI10Fc1C2B zg#PsV;h#GBjosGjy9hEDJ(zk({p1jr21r$Z>tOB+&4s@`2wiatckjPX#wp;4X>rqz zv^YScx&R08t)@%zNyDU5XYzYk#4~OVpE8^(_K=(`)laCC7k>y(A{3Gas|z0|SY_Zf z-NF9CLWI$lp@X8pv=h@jbhmz64>8y3D}$5Wv9X@GS%FQPm7;t zbFqhnT0`QMtt#n~ao?+gm<@8cI!sP`mVF(L7om;kKeG#s*Qfqt_Zf(c zv2t4u>b*d*lFf{;sYHm+l|n+Z)n*+4tK!!6lL7%EOm$NnQg@}T@VpwqGhx3y9W|86q1Zb%cpk3nc=_m zgB*G*&mKN%LH8L&OvVD#K0y?^=q@flNyPn-+hgv7(40~u1VknkcH<@f+&%i_Ht9kN z<~6+XqyjfuE@D|MtK=E!m+BdnWcx2ffSPta6dZfaQzBN7MYw<}>^)M-a`-k{D{-43 z1Jb+lZh%*Yq&w&XmZRjv3t^W%1E zuT~?@Yqr7)a5Z6moP>RFJ+mZ``&|qGFtOyYaPQo-HOt@0UU6)juF_9gjo*vbwRWrd z0A=%QUbq-4P;mT;&I@$hIzg=0!&Dd!@y7`FF=}@~&e7GFTR@>?MJvq&`YG9)EVOHi zQKW>qINvpXs)w`0zK3d?kkz3d-S~?aiw8DDg)|fx4Y0#u+^3KCwIX*y0Ni-FrLWA? z@UlcFIHNbbM>OFWf_0HzBnX5p;sHjhIl2kKbQSw)N#GH%w|LTkaHKy;@Y?RXKj3AP z;qt;?Ub93DD&vCwB_+fS&{OuGmW6sB?BB-rmM`nDFJ?{WklW_p{bwOaL}vbeS!$9} z@DI8Eo*OR6eXL#ii&VZ=D@*0$_?k$BikxB1erZ3O2;^yBEZLCrMhy}6^Ph(o#Rf7H z6(_R=zLxQme)sW0Oa#GdKkb?6-#ha!cN#1t>h#nZIK|^;FuuvWq~QxWgFC#U{_==^ zR`PXJP93uLWs~_Q2)@MW5WKI4$8*gnVkoqAoFuaNC~ik!vtk2^lk>T%|@tY&I2IJIGlr#Iz${7yl3 z4mo)COR^xXqH75+-wU}51_SfWJ=L@@wQQ^Ai`oxU%ieI2KND3&&e(OB58`EqEEDEc|S9A^_L<}ZM5s|`9dRFS>p2V%4)Ho z5~3uKO|H^?-fCmc@Rt*y&OBP)s~5_5fh?X-*Wkd;Tm!0faiZ* zyc{)GwtsaB_fh;~r z#oWVS%9g^D6D5{tilKd%M!+rUKe8t>Fx&qKtDlGzA`*bPtyeS;4JCZ)C*^-q;s_V% zP8VrJZy-7Z!pjfg=gmGbN2L0{rW4x&aRL=dEV*;l7EuiwrIOP-C7Z<)(Om{C0Opet zlH-w+k%#a?c%=EkvJfdL0a+fQTACNf`z4lU5`suF$h_Y$xM2S(row(b@I;lLZ@H^vc}sNryG^K z6VH#Qy1}-#Z)pZRos-PvY<6D7lXOy1Z6YgNYd8!AKgu_d11&PzdbOWR5L5~{H7-6O zH&3sN=}=NdlE7vUt8t}7y(9@lcEMg!f>dGUZrA}kLIt7eZUE@$yW=SF<4=e}=tsNs z#^F-ro^46w6Zjd&AJH1GJ#!pM8%Oz;cSyHF^S4B?$o~3M+N%XMFsnyE#?kCq1Q<1&7T`B!m)eFncI z3F50VxL$8*IOCe{GYM qN zXLt4o?Cg(I(SrP5C7~m&1jljqlU5X$DBSitj zS5BvB_F38_ZbdKk_b(WvvYN&%zPZM9_;f*1MRzQ?ocw_9hY8a4=l8ux0kb~(TsAaG z<0@EWMR+kFoNsXE!J_8@j5kd2_3g5)d}4Pm**vynH}L>Rx!S>#>D!1&XZTerO8VL(d<= zg!K8{>^Zc8xjhmlqiznn=x`*@tCnC1-8$& zX`x-4PdZ}dI%nC*h-?QQt?JN|+R=@nIpeMhb(b@4;nQT%E8pL#V-`k5k!4ZtH`56z z2E3e*IW8Puoc3TA6Z4$GWM3 z%y2FL5ZCn>ZcD7@8894cLBtz}5{s4Wo7Qc2LB+)-uN~m2}t{*g*mOeK+ zseWY&ij#-IQxvIY()AfefmahIDE*py)P(__35ET&a zoLsne+rMy3@vp7o2Se}ao!T8wCbQ+WT1TzEgPRyXe08Ij26EkF~GcGQ@ai0 z9KxUz#o~@KR*8TTCV0cHr;9W6${>`v5q$h|4lX^b9yIwrf9Q#d?;#c~M>wM3=1)(O zg%YlClLqzu14jHcJ8{Kya5XA<_0t^AICP}9yUiZ(Z*%-6y$EajCsMGchqTDw((qkC(cg{+RzyGXf*_)}~JnaN!aw4tu~kSwlP;8Cbr{mO<# zEUl{lM30TnY9d-Rl9gRz>{a<94}(19iFN93#s^ue}xD=@TOs zx#G;cd7|l0OC?%#X)ocaa|_JzRL|#JZ`{sYOyog4l&hg6+|DwGU)vdO^s%s+Zh?tp zq-?IEHSwboHuQA+0{XH~&Yd04U#(r4%;!$0?yW+v?H}juw=SHweA2l$SGVrjPaa2p zXJ$jy2#OkHMo%F(lXeu0Qq^Njc$&Yg2=&Z-3L;Ox9QR}WU|<)MyVV%z>{r{20F zQbt)WBw?=OyYv$R6&j|I>sHi3pTo9v>Ye*p^z9Zgw0(#%Lx=f%$R$rytLnG7aq*$qyuUwjTuMkT_$t?)m4Z|2TZw=7KQeeKFw!UYn>#miWL|Jy3P>jJIpR5YYtz(8 zw6qW3+Ta^UD&t`pqW6m@ag8XLZ#FB#?@f1x02=Dn!!_m&3_Eh@3R;z{EQb*CcI2TT zZ$#W^KFdpgoND4)5`s*R*|S@bv1k4pbM)fOVY`_G&2C|$hicl(#rA$iU72S0V9uY& z;tuG6tN0I2a@(OT*@O?La>1@B`K`hX$ZvYs?g_!x&6gL^w;#22P;@K1YQ9nLr7M+& z`-G|IScb;~B>u?uiXIT3PK%#br5Qr?M&gcbn5eB8TtqkSk&%iK-?+Xzk|KsUl9ups z>x>kx#^;H-K!qZi*jY{_vI(^Z2`Rpo2d zc5AJdcwD;kr2y(A)zB$(0-oVkU+COZxxnmjFCgK#d-|!dROI9&Yt*~ zM;qNQJCcV}uCZpLbkJ5i8|4lq@w!tdDsO3?_P38fS*77ke0OYG3Pb3SIgs2;8Of)1 zP7Rf=@7ltR>}~1sHwh&uKt(2aq*whi-ArxiJHucJlEE;3FA=3i znDUIz5o{jmqj|sTz-G~^kn!I**eQGs{1$T+KdOA3v#bebW-A1<-blBB`i9P z0czRcZ>(@U1L;t1v2e`qmtkidO7E)-Q}f&_1{tLrfb5WbFt)y9Q1B54+dJ`RT%iX*jdwn`uq}fd=^5|8!mId7ul>=61kcw;6)ukOzXIe2Y0mC6)N#m!6x3r(H0yY{Eu}Wvn zw9CT2Gm!`u>spRHJKH8kck{JT8yz$u7Y4Iq(g_A@()W)HsCo&MDqI>&NzRA=rZ!5& zw*?Fcon@ga{iFC)rog%3LwXloU^!+(rrKK&PBHQE^!-{E%SILE*4LuV%zyzy z@6<&D)pkmX@OST5?sb2_w{VFTK}HLA8jD$K3rQ#^mqVQ@W)j7#680x!mHCU@F%=tL zLLxhalFZx~>bSJQz#nYQum*++*E(#G{aDB+!4w1%XNsmxUf zbpWW#rq1qL!Z!Emm$c7@sC$=6ey%LJU#^Co8%6bbyhoHOx7-|8UA_$UR-PofX&v(B>v|d&dCYYR@<6&M*4F#lQ`cGHg?Xb)Me&f ztNFGX9c=q@A7YH0KSE5u9I$Aacg>f5;3$`&khoYKaJROrIY|;gN2Fi3E7vU9IXK6&OOGAiRyU{U60yB2g~3@dJN0n@c#r+-`&spWMnp zXZhjB=<#lZ4B``(2uy5*ciX^XTjo0GXpN8M!Oaq!o;QWJ+GMT|>5dX^uyrF6-}LDF zN&jRJOwKtg>preeC9mJfW=K^J?+%HD+ZW-`E~=CbPr?@v!0ve2GTz1Sx6xaaQLjiQ>2BAlHD*X`u+8OL<4ADCcvu*a+Fu^&DEi#SON?v=u#?^5sT` zio6O359>~=CTqvY;{uI>2xR=$^W4^bv{eNlLP3)v8@59!+8@D)n4mof5`i0)xFWqY z!a|In`oQx&j4@{xq4pk{+u(|e3y>KYIH3)Sjkp{8Q8%?{DZ=0r?fedcn+-(nPo6mz z0mYT!T(`7j`imd-PWaJ|uDopkz zqRlZtrEtPuWGrC|?6t~iRhhlcbZLl|{q_!#i;ZyqH#7@18;ZezeuMz$)=&`_AEuVM z*CRy+Bi#p?bQqzbWChsl{%Cm`u9_3uvqGUFt%?+ShgO}0TDOip!y+?Onk+8(7qX8} z65xVoUgYz@RLciewYtf=M-ti5n9TGKWjd{6k>&Qsxv)7P`S92HYsr&k>8HpyupP~! z#IEL$O?Bs+{?m;tn;vt%04o0t(US`Z)P^cvc(EyH>%x`rwWF)_#>?)G zlU>$@jEO~rAN^%q-nDk~4yE%qul;-j&iR(NhYHFkr!qc-zR_Ul zc)!2H{WL(#s;ILm=~djsu74oDAh_+DTT{i|Av9+GJ^%svrnoy59%JCZ48!IF8SiBW zcqZf>xweiIkOJRJ&|FN-f(0-1zt_uRj&sU#bCQkAFVYKyuWr2Wm{W;qZSSG7eUmuo{;}QJ~k&LZ#ah3tO7X51SZQe@V#2H z*uPjCqA+01&+;5+&Bp9Rq$M+t)g}>?W%4jNXI(3hT}yI-3=(6)P%fxKWTANA9u``e zB;(Kp3CKQs3NBaCr!Nxj3&5n=U}n4^!Ys{p#UDqwS&|!s#G(Gispbf*E>fj2)R0iU z(klW)JVQ8zLY8lb6goxXh(Ps!G}RKDQmOqOEX6Z&HB|U<_oGHF!=G$(^$3*fl}2h| zHELYS{hLYVm+jjR?MlKll^vWq@10ecJlMbW-mq3VD% zk}v|h`?igJW2kv$qR?TI(gA`eLD2&k^P5aHGLqZ@Xd#Zlk2g$qqT6d!8kwRenBoQ= z!OC0`?H7#~J_)TPQ47TR9@VPqK6pnaS3y?n_a?k~scoS#Fk-Vw0ve=`4w65s!n8J& z{VZ`P+G+67*|`s(O{7SgTs@1@ep4+KT3Q?+r@`evBCH_)L4smazUC6~OW9FYBtK9L z;|R>W)=Bn{*OrRN&z~ht3YjHQqEfNdMiDea5M7>Rd!$#&2+o%%^v8VF1otIprWM(e zJcVNqh^fvMxj5J(K>-#Fc6MxC<|~wo6EQXkzk5`<9LGPw_P^Y%xuyiv7ZJYP*N5Y( zA2k&#OJ`(rfJ_cNSLST^McE@OYMu-9ntI4i9yL|ZCX_pCQMqoZgee7APcS;W3FS^d z{3BY_~k1X4;OB!ytoQ^hkC^Fah|9A@7eiI=zlrZqJTh%zGg80z61`` zI?Ipe8J|w8pQWzzu?<)GsyAJayvOH{EE^ zLl7XdrK4QTK}TBfC1`eF@$NO$NxY8Hg;e7cNakdw{!VRAeMk*U9p>kvp^yGe$IJj# z$ZO?GIkgN}X?d*JKiQyr z?x{YM-8hl#9(&6b8>65>ta4RwZ;t5Eoc>&7Ppr3I!VJM|FSAiMs&kU*@VGlh%S=-p zsIRGh6J&Q;dc@FCNyDup$7L*?b7VVzYKrTF{=Uyd&#al4} z->F=*1xoV?WXW5n_xp)W!fkuDbJz!Jz2Q_`Z(~DriYO0j4lDLZboFy&0Sf-MOJk^2 zd05WC4m{bx_}}X3ip7HuUd@$tx<_)%Y9g%&j~ZfxC>a^vPHYdNxwU?oz0McR^;~|` zIHJz`T(xUjl|uW#n0*`bkbscmKx;bHl=2Sb@STfXakU>z<28TF4OKJNWVNQN_nWda zrbeoU>}$p@615xhUB8kWkckhtw|DGL7U@x?uH^o8A2>pKfD zQrw$5^P?U9fq{4d`-iYl(?8r(2ee*bD{5z_ueU40Q)WjW; zVC)NvpDoBp&clfj83!S^og+yQAmYo$m2b=Ky5t9r%cH<9nGi;W>{aaR8!LyH0bW@c z>>#y(ww9eafCdrNVGqSW)_vKbe22aeb!<^!;kb$Ft-*GuiYzhShZ||sK4nJl=yN&J z5>1n_mLocut&3=?Ft=A~P}&U1{1s=uZ8zlmj!ii?_x&_(k&%2S!pV%oJ29(5~PST{ad$9k+V!DOliq1h0(E>Cj%SQ zt4`m7Oy>Ic`lzI$Kg$yY6im_3l1DcZ=IdC|ctAtj^b=;(W~EcI%iO z!G4Rv^L7f*u?SAxJiBC~34&S#*j*x3s0tBS>{~Hu^aEZuesZoKc^b;A1y~}I8yjcO zpQGh-#~UH;TWrJv#ll=^xJtlmdOyFwh$NA1dNakp>v}+XM>!{SZhLc02nxi?D$p}j zBgqRQ7A1{xe8O=y;n<7C3I+`SKuF2=ODiY_qgDxi zlEYFmUGb(jc|s+R2^OLIZ=1YEu!9z9c^?J5!lwaOM1z=>9K7JzSz}~jrNZ{XKf*Uz ze7xCmR5AKtYP+BquHp!A8=5jOLry%tms8=){oM};?36JZ+;wT2>gDg}tXZ_pQg5WHCxT&p zAKlihAk)4yFIfu~N?3@+u@x$gS+;(8ZZPybcuqa%Tv5H56(KgzBlhn=V|R>3E%d1(Fed zE00qbR*@xLR#Db0?-Yrh$%LIEKW`Wa8`3-2)q6#xV(iIymWJ&yond1Xl+OAaGi6f( zJ2ebG)0;~vjL?%?gjt*fl6~<&+hciZ7~=V3QP~wDFEVzV%jlmT@A^+dlG!kCL9ZJo zTwk8~O7u;u3tG?JA`&2mGbv?v>MN>y#JfwN-iJcI{|Ga(!`SRD)JjV>0#_*+aP*K{ z{l(+}O(~J%oaxL)&r%-xElhK70k=0eh8}n$08G{A4x{qHj+n8cUv-E}Y}l(92VD@>A6T=qQJMRcB$@c}V&OwaAKW?SCF( zHSfqfYoAj!?}~eCpEot{kq2uZ+>42UDQT;Zo7wL>Gz`}t=Q^kZv#1(%Ee;Clt4J5l zGN-41O&dP0_ytIDca6T??V`W-`E2n1ohoQ@GEM8QuIG-mXO5F)=`N45&RKm%70=tt z1s-LbL63cAU5IDJ3u8RxE@+YM8#hf|9?B;(*g4w?n(wmE%)|8vp&`056h{)` zQ@-Fll$#cR1FXGg#vd4i0P5mMuOnKdHSkSB%HZz)nHjr`@ByxO5#kXa+Wl)S0LZGR zx4foo%Ek|SZmJYY->T)v?qu5}TIpnK1~~LU(Wd>PgW>m7UVITl{-Ah7MD*<)f%A7G z;yF;P-*Jw<51*0|--c-4rDveHJ_pESV5O429TQdN>8c$U=s{vh|2q5_o{0}T?TC6( zqxw{_O_jCOHVin!dEJ>6uTmjuZyW8n&?o3_Q2#;h$T+|qE*VK$- zaUQHWwGIUGuT3_!3HgPsn2l|W5yDO-FObE8h=!N1+dkAosXG#GmiY<2sea7372O=M zKXH0vK}i2+{C*Oagh$AxYyK{hje^vWxk>(?6c}nI)n+CI??r(M;o{>F;uXkPGlwVp zKhySR4nKyB5E9ikW(%*1iS)4Km7GE10q-s?#m&tp#m6Pb%_}P)C&SCbBPhTHk>=s% z7vvF?l@j6+rxN}DDjCxr@Ip`_Eo}|lh@JCLfD0swrahsZ3UP?PJr*Ds!&F#Yf)?XM zh!+g6gNsK1jZTUlW_W`~teJd4FSq?^DCfVamVYATasIt`qabUg&pLhb`^#*lKX-QN zXN8XkP9fFs-jbAdI91+56i0`r@QbGQVTZYKT4$lJmm$s1+Nj%9lWtIcIWM&ALSl>= zgvcULe7}Z$sMQQMxf_{UzoH;=#SR;Ahs<-(uN^L<5Wq1Ml*2co&{-q{rFF|I=+XAs zMp+Lbi5*mJpX%XpODgaQUK+lSafot{d5q4A@s{1Q zu;@vYX5qycJC#~;?4jJV%AKfOSNiIw?1Usa@n|Xow7!_?<|O$8%3x2{mf_43PZILW zN9McdR31Vq1R6_f!m{5ZWZfal5F9`Q66^Zj6b)*O{$wZU1Iz&F8jEd^yL%tU$80m2 zl}`S@36uM8oa9tp7N^=(B8KT?2ZudnO=bQU=zj!gM9*sppb?@GI`OXg4pUZkA^lKt z;{L1|TfKaCQBXPFu^3VLv`)Guzj$H#09p(RcwVkV*%0|4!}rJZiA)F9(?q`5@H|@D zv24p@b-}&Y$3q~JRdtbd+^X_wWP2cQ>%zy=2l|B%-)Q}R!2bPiIXPDbU_3V7R==W3 X^f>NiV0pt+qVw=#(9_GR%VGQ%OG6@J diff --git a/main.tex b/main.tex index 87febab..a9663ab 100644 --- a/main.tex +++ b/main.tex @@ -56,7 +56,7 @@ One major open problem for \scut{} is the best approximation ratio for planar gr \section{Literature Review} % Requirement: summarize previous research contributions and identify the gap or the specific problem -The seminal work of \cite{leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap. They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which can imply $O(\log n)$ approximation for \scut{} and $O(\log^2 n)$ approximation for \nonuscut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$ via expanders. Note that any algorithm achieving the $O(\log n)$ flow cut gap implies an $O(\log^2 n)$ approximation for \nonuscut{}, but it is possible to approximate (non-uniform) \scut{} with better ratio. This paper showed that $O(\log^2 n)$ is the best ratio we can achieve using flow-cut gap. +The seminal work of \cite{leighton_multicommodity_1999} starts this line of research. They studied multicommodity flow problem and proved a $O(\log n)$ flow-cut gap. They also developed $O(\log n)$ approximation algorithm for multicommodity flow problems, which can imply $O(\log n)$ approximation for \scut{} and $O(\log^2 n)$ approximation for \nonuscut{}. The technique is called region growing. They also discovered a lowerbound of $\Omega(\log n)$ via expanders. Note that any algorithm achieving the $O(\log n)$ flow cut gap implies an $O(\log^2 n)$ approximation for \nonuscut{}, but better ratio is still possible through other methods. This paper showed that $O(\log^2 n)$ is the best approximation we can achieve using flow-cut gap. For \nonuscut{} \citep{leighton_multicommodity_1999} only guarantees a $O(\log^2 n)$ approximation. This is further improved by \citep{Linial_London_Rabinovich_1995} and \citep{lognGapAumann98}. \cite{lognGapAumann98} applied metric embedding to \nonuscut{} and obtained a $O(\log n)$ approximation. The connections between metric embedding and \nonuscut{} is influential. \nonuscut{} can be formulated as an integer program. \citeauthor{lognGapAumann98} considered the metric relaxation of the IP. They observed that \nonuscut{} is polynomial time solvable for trees and more generally for all $\ell_1$ metrics. The $O(\log n)$ approximation follows from the $O(\log n)$ distortion in the metric embedding theorem.