Suppose that vertex b is labeled, If fcb>0 then c is abeled(b,△c), where△c min(Ab, fcb If t is labeled, then an increasing flow is constructed ◆ We change fi to后+ At when(1j)∈E,if Gi,jEEthen we change fi to fi-At b(c-,2) 8,8 7,0 6 s(-,+0 9, c(s+,3) 身t(b+,2) 4,2 5,5 5,5
◆Suppose that vertex b is labeled, If fcb>0， then c is labeled (b- ,Δc), where Δc = min{Δb,fcb} ◆If t is labeled, then an increasing flow is constructed. ◆We change fij to fij +Δt when (i,j)E, if (i,j)E then we change fji to fji -Δt
0)Construct a initial conservation flow in N(,E, C). 1) Label s with(-,+∞) U=Xx is an adjacent vertex of s 2)Suppose that vertex i is labeled, andj is no labeled, where ∈U U=U-{} iIf(iJEE andf<ci, then ij is labeled (it, Aj), where Aj=min(Ai, Cir- fip, UUxx is an adjacent vertex of j. goto 3)) ii)If jEEand fi?0, then i is labeled(i-,Δj), whereΔj mn nAi, fi U=UUxx is an adjacent vertex of j Ifj is not labeled, then goto 4) 3)If t is labeled then f We change fi to fi +At. ifj is labeled with i+. If j is labeled with i-, then fi is changed to fi -At goto 1) else goto 2) 4)If U=#0 then goto 2)m else stop
▪ 0) Construct a initial conservation flow in N(V,E,C). ▪ 1) Label s with (-,+∞). ▪ U={x|x is an adjacent vertex of s} ▪ 2)Suppose that vertex i is labeled, and j is no labeled, where jU. ▪ U=U-{j} ▪ i) If (i,j)E and fij<cij， then ▪ { j is labeled (i+, Δj), where Δj = min{Δi，cij- fij}, ▪ U=U∪{x|x is an adjacent vertex of j}. goto 3) } ▪ ii)If (j,i)E and fji>0，then ▪ {j is labeled (i-, Δj), where Δj = min{Δi，fji}. ▪ U=U∪{x|x is an adjacent vertex of j} } ▪ If j is not labeled, then goto 4) ▪ 3)If t is labeled then ▪ { We change fij to fij +Δt . if j is labeled with i+. ▪ If j is labeled with i-, then fji is changed to fji –Δt goto 1) ▪ else goto 2) ▪ 4)If U then goto 2) else stop
Theorem 5.24: The labeling algorithm produces a maximum flow Proof: P=xx is labeled when algorithm end, thus v-P=( xx is not labeled when algorithm end. By the labeling algorithm, sEP and tEV-P. Thus E(P,V-P)is a cut. (1)(i)∈E(PvP)(ie.ie∈P.j∈VP) (2)(,i)∈E(e.i∈P.j∈VP) f:=0. By lemma 5.3, the labeling algorithm produces a maximum flow
▪ Theorem 5.24: The labeling algorithm produces a maximum flow. ▪ Proof: P={x|x is labeled when algorithm end},thus V-P={ x|x is not labeled when algorithm end}. ▪ By the labeling algorithm, sP and tV-P. Thus E(P,V-P)is a cut. ▪ (1) (i,j) E(P,V-P)(i.e. iP. jV-P) ▪ fij=cij， ▪ (2) (j,i) E (i.e. iP. jV-P) ▪ fji=0. ▪ By lemma 5.3, the labeling algorithm produces a maximum flow