5.7 Transport Networks

5.7.1 Transport Networks Transport Networks A conservation flow f value of the conservation flow vf

Maximum flow Cut E(P,V-P) Capacity of a cut Minimum cut

Theorem 5.23: For every conservation flow f and any cut E(P,V-P), the result holds: Vf

C(P,V-P).

Vf C(P,V-P) ，

VfmaxCmin(P,V-P)

5.7.2 A Maximum flow algorithm Lemma 5.3: Let f be a conservation flow,

E(P,V-P) be a cut. If Vf=C(P,V-P) ， then Vfmax

=Vf ， Cmin(P,V-P)=C(P,V-P).

Proof: By the theorem 5.23,

Theorem 5.23: For every conservation flow f and any cut E(P,V-P), the result holds: Vf C(P,V-P).

Frod,Falkerson 1956 1 ） We construct a initial conservation

flow in N(V,E,C) Generally, we set fij

0=0 for every edge (i,j) of N. The conservation flow is called zero flow.

2 ） We shall construct an increasing sequence of flows f 1, f 2,…, f n, that has to terminate in a maximal flow.

How do we construct the increasing sequence?

Let u be an undirected path from s to t, (1)When u is a directed path from s to t, if fij<cij

for every edge of the path, then we change fij for every edge of the path, which equals min{cij-fij}

1)Label s with (-,Δs), where Δs=+∞ 2)Suppose that vertex i is labeled, Let j be an

adjacent vertex of i, and no labeled. If fij<cij ， then j is labeled (i+, Δj), where Δj = min{Δi ， cij- fij}

3)If t is labeled, then an increasing flow is constructed. We change fij to fij +Δt for every edge of the path u.

In the path (s,b,c,t) from s to t, edge (c,b) is reverse order .

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(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={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.

Exercise: P314 7,9,10,11,17,19, 20,21

Next: Graph Matching 8.5 P315 Planar Graphs