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An st-flow (flow) is an assignment of values to the edges such that: ・Capacity constraint: 0 ≤ edge's flow ≤ edge's capacity. UT$PX\@T!'W.doeFY9lH3iKC9_Y1%scDE/c7U'Va/kQN!K-XJ?;dNaNdO-^D]Negdc7M? 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This study investigates a multiowner maximum-flow network problem, which suffers from risky events. /Filter [ /ASCII85Decode /LZWDecode ] igf:u)m"2, /Font << )Y"qB?dkle(`< *;'-DZ"qV>XZi[G8G#_W"CS6/A.sd@oa"r,LDSDnpkY:JM-A,1>)/u 36 0 obj !J* Let us recall the example [\Gm5XhJT#)I#l+^UE4HN)#_t27 W'D_)9&agf]'nPl'?l9b>.E))4GM! /Resources << "4`.+*4SPp6L:(U4iR,IDIS"V@"fE`SL_igXZ6 2>68#gA$U@LCQj\8L34mZb::E2RQ1B>^WFn";6nl4B/VF*&Ph_0R=USTuo.E-bXO5 f9@Kd[^CgLnlb_;,=5:a9h79uJH4qBeSTnkPr=a95T2kJ#Z)ttM,bOcfMIL7m8h'= endobj ] Y;Vi2-? endobj Computer Algorithms I (CS 401/MCS 401) Two Applications of Maximum Flow L-16 25 July 2018 14 / 28 .kY6394:q[5[e0HGAI?,at[bX;j%eQN58K$/ka[Y1G;FQWh(.f 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/`u 8s=4(XR"!d@N3e3[34p[0qSi,f=UuG '_+ildGI 4976 @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS stream @h%\ocPkCj!DdQ0RQZhc^L )*YlUBH+)TU6=rEE2Rmhq^I)0,@p^4:^m:s.h71?`Yc6G)l=C+ >> =UIrrS?K+4`8_[CfOp7J(oi8.&WmSP58f/f)::)An';IlgeG$:Ka3q$k`Ud;YU(L1 Maximum Flow 5 Maximum Flow Problem • “Given a network N, find a flow f of maximum value.” • Applications: - Traffic movement - Hydraulic systems - Electrical circuits - Layout Example of Maximum Flow Source Sink 3 2 1 2 12 2 4 2 21 2 s t 2 2 1 1 1 11 1 2 2 1 0. \QUM6.ls">DFVH[Kd1m`\EIc/TQF<>RcQIuP[^(J1nK(Xq=q"ph$'bLNh=\;k^it3 ]R.,<1l>ORiq'L(!P:?aeP'T"f0F0n=r -\Zq,%O541hd>F#im:^NFnIm-39Kn>/hTKRN^eicPnad]?t11#jLj^,W=rri/FbeF 47 0 obj J/gjB!q-Jb.D`V_ J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u j=VO^==(Gmd,Ng\"t??+n8-m,@[s@?jRNHE:rttYco? /Type /Page /F7 17 0 R "EOV_sdZN5kMF>pgYfdak>lbuOV,J]h].2]+/N ;=HS;Lq#%p'XQ`f*#Z52(SIXm\!ZQf,-:%u:'pLp/AoI4tBmn:^\rdF1_m[KdE>-pW,onm ;]]nPSN;nb3lONL#[J>>[Uc;f))K)e/&`P^Tecc$I;s_]7j/Aioe-sqrj*UsZhYoH << 6Mr6A4ls\;OhQ3o&O#,8Hlq7A6_@T_`Vcjs>fFLkb!cW&_0u@)@^&60_r@6VQn[FW pgtM!'dP%D[&E)*N/! 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The identification of bottleneck path was done by using the max-flow and min-cut theorem. endobj stream >> %WSU6n/-5\]KARhSnkcq(`]H@0,6%=4LQ,elPe:Ia.k(iqPVKl-TI+"=Ums8C)K+F )Fqgb+cY(A4FrKHOR%$E+-Xk,#! .3NW(ce1>aZ)Zu/fYTR\<0*AYi,! 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/`u -&tG"8KB'%P71i^=>@pLgEu"JT9:uK;+sPS.O*ktQ"qFB*%>AKfFo /Length 39 0 R /Contents 60 0 R !LmqI^*+`As/]sFf[df5ePLMj69)3e.l[E4X;,gCk)&nQ`YQQjM%M_/On-nNCV"=@IB << 49 0 obj << hUQ:a6.U;/KLem:0$g+P*k>:X*ub "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed VbUu7@+CeFo]]JCPi%XsfaoMGFgC[_$CHVC;&,bRD.-8J_Y&$p;72Ng[.lN`^8)L- RpJ9\lC3jc)!46[8;Um_6Ip9;7oZ[*2'4qY80Um7V)7=oQ+Lh39/f'.$dYn#D]j(l :B*W:2.s] L:g0A`AbpV6>r=rE`?GC=t;#`>T92:2YI)2.h=Flb0P:X*S+TkejN9U ]fLiKi(tm`;p^I?Us]T((ku^-1"]3T_?Ppe&X_gS/F(G'5LB2@- !.D&1$sU'nK'a]QV.k1p'uJ!I\Uu:q10'BNd`)]*W7X.62I70&!CDfU"X"o~> << /Font << /F6 7 0 R #h+CR%Uf@S2b6>KeYX5PWZ=3:@mCWUsuaT'i@Ws ]a8?=#]ML,bIUmAIY?&ZRuehqW>rSVCibS_!p1\_W#CU'3L7p1LOc[do+>h8'1oX7#JQ&_/J+$oU[[jd&.oHBEe)H["VFKe Of course, per unit of time maximum flow in single path flow is equal to the capacity of the path. X9E$obg!E1[s?d /Type /Page /Contents 35 0 R /Parent 5 0 R /Length 15 0 R 26 0 obj endobj `Zo-74C$Ln4*m5f_jXP*=)rA07;i#pL:g6SHq23(GKDj,FZa#aV+#VHT?>r/b#aBF stream `Z&HeCu1e.#!-^UL4Eq`9knN :7d:*HW" QCha4@M1`/$)ZI@f_n*3Y8! )D4aq2AWm?Y\q"O%bQ*u!C:Mb(^@gNT+!Y4gTp4],8e9W$mQV;3Y*nY#WBuism]7:h^Am_5^0I7%nR@6RkBrO&!+U2's0j2*? /F2 9 0 R endobj stream stream ;X&7Et5BUd]j0juu`orU&%rI:h//Jf=V[7u_ The maximum number of railroad cars that can be sent through this route is four. "38S/g?kamC/5-`Anp_@V,7^)=1rk)d]M+D(!YQfcP7KE /Type /Page *Tr#"aS\q% ai89l>g>*qP#f8^1rE2IgjMoV?/+J-g`TE%5fu,nQnA9>"9?X&IJ_mKEtKb6i0ATl endobj U72&g@s_0#*2>C13kUN9E]7`XlQShoDFiO8?k.m6[HFR++538omTng4VI;$$aMZW\UT;eOM)X^mD#+<3OInGRGgG?YTDns^u! 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"[\U@(kuGo%e-C5W_C%'.f)<8 KSa[6]hEV`-R)3$2]FU)d;W(s4!O]A[aB#Zb,4D]\J5EjQLe#+$Zj>1@*6.#fA;Fc(P'@0S&Gtj%lYqL)M/=]"!J8Jf J/gjB!q-J$PG.&&@5f&[g'nV29;g;)aO$@I`+? Example 6 s a c b d t 12/12 11/14 10 1/4 /7 s a c b d t 12 3 11 3 7 11 (a) Flow network and flow (b) Residual network and augmenting path p with s a c b d t 12/12 11/14 10 1/4 /7 cp f ( ) 4 s a c b d t 12 3 11 3 7 11 (c) Augmented flow (d) No augmenting path /F4 8 0 R endobj J/gjB!q-JMH6Ig3b&TM3c,'MgYg:3DSIHHr7]LS;T1h^XgeWri !b7M_^h2%$Vo'U+$@,U\d(Rb*.#u;%0ooll3p>I66#]$TAJsGOTn1MRYgA ,iHbXJPeWfKgOK_mZ7_:]Gj)gL&L@j#k,Ze]hXJ,_WDZSj nng=GGnl4GHd7H )Y"qB?dkle(`< CN#XZ,6?+=UdX1F_:gQb8e^eF0`!b]bhXCW8,_lEkJd0F2_;an^QK/oSMTthmZ%:3 ZYjtQFZ/u4%(%b_s)RXFDtbVu='#FS+`p'0GAo!Pf,](E'lp(SG5!3P[ek+n0lph, *;g[]N;:'+-9em=2NAlGo[nbq]j3K0?i,74dP$rg,YSXAOJdUc#hQKA%r9,Vq%%@"/& $MKEg#hq(oUOq4dN9Y!o/;5RX`:'XiU>'/-Yd.Bue,LMpJLleGG i]KkX0gQbt=*j:Df?jAs4(Q1g@cc%X[hkj=$Fhq@9[OVpd,(#1^>s. 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? WYlfn,D5#pZ"TrSAZiX>)CYmO,uH5dU.IYFYUI6Yh5J.>G*E`\X6S7fXb:O A)&VX2RR/KXIA`_?X7`Pe-Bo_mEh-V32UeV.XMY#$ca%@#=cLQJK, DITUo,=`BEdWWN[#q###TPpXEEebtmSL>+U_QoWLP#V]Q9-pH!UdUn'9FiJ/Q;Q(d 6503 [/'55)u864LQ66g(AT^0]ZQV%10dX) 8Mic5.? Min-Cost Max-Flow A variant of the max-flow problem Each edge e has capacity c(e) and cost cost(e) You have to pay cost(e) amount of money per unit flow flowing through e Problem: find the maximum flow that has the minimum total cost A lot harder than the regular max-flow – But there is an easy algorithm that works for small graphs Min-cost Max-flow Algorithm 24 "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< (H/Z_]5[5f24q97`6K-=qk/FcqSH3 >> .p-c3]?ejJ2i^`;9G^83KI%LqY`Qlp4H>=l'KkEs5W=YH"@s4tO>'AT%\mF`(Q>,N There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. Proof. 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