 Theorem 5.22: Let T be built according to Huffman
algorithm and leaves of T with weight w1w2 wn.
Then T is an optimal tree.
 Proof: Let us apply induction on the number n of
vertices.
 n=2,
The results holds
Suppose that result holds for n=k-1
For n=k，
By the inductive hypothesis,
Suppose that nodes in an optimal tree T have weights
w1+w2,w3,,wk. Then This is an optimal tree with weight
w1w2w3 wk if T’s leaf with weight w1+w2 is replaced by
subtree
 Lemma 5.1 Let T1 be an optimal tree with weights
w1w2w3 wk. Then there is an optimal tree T2 so that
T2’s two vertices with weights w1 and w2 are brother nodes.
 Proof: Let T1 be an optimal tree with weights
w1w2w3wk
 The two weights wa ,wb are on lowest level and they are
brothers. We denoted by va and vb. Let v0 be the father of
va, vb.
 waw1, wbw2.
 Let la be the length of the path from root to va, and lb be
the length of the path from root to vb.
 la=lb
 We obtain a new tree T2 by exchanging from leaf v1 to va
and from leaf v2 to vb.
 w(T1)-w(T2)=(wa-w1)(la-l1)+ (wb-w2)(la-l2)0
 Lemma 5.2: Suppose that nodes of an optimal tree
T have weights w1+w2,w3,,wk. Then this is an
optimal tree with weight w1,w2,w3,,wk if T*’s leaf
with weight w1+w2 is replaced by subtree
Proof: By the Lemma 5.1,there is an optimal tree T2 with weight
w1,w2,w3,,wk so that T2’s two vertices with weights w1 and w2 are
brother nodes.
Let l1 be the length of the path from root to v1 with weight w1.
Let T2* be a same tree as T2 without leaf with weight w1 and leaf with
weight w2, but with a having leaf of weight w1+w2.
w(T2*)=w(T2)-w1l1 - w2l1 +(w1+w2)(l1-1)
Thus w(T2)=w(T2*)+w1+w2
Let T* be a same tree as T without leaf with w1+w2 but with subtree
Suppose that T* is not an optimal tree.
 Theorem 5.22:Proof: Let us apply induction on
the number n of vertices.
 n=2,
The results holds
Suppose that result holds for n=k-1
For n=k，By the inductive hypothesis, the tree with
weight w1+w2,w3,,wk by according to Huffman
algorithm is an optimal tree .
By lemma 5.2, this is an optimal tree with weight
w1,w2,w3,,wk if T’s leaf with weight w1+w2 is
replaced by subtree
5.7 Transport Networks
 5.7.1 Transport Networks
 Definition 33: A transport network or a network, is a
connected digraph N(V,E,C) with the following
properties:
 (1)N has no loop.
 (2)There is a unique node s, the source, that has indegree 0. And there is a unique node t, the sink, that
has out-degree 0.
 (3)The graph is labeled. The label, cij on edge (i,j) is
a nonnegative number called the capacity of the
edge. Let C={cij |(i,j)E}.
 Definition 34: A flow in a network N(V,E,C) is
a function that assigns to each edge (i,j) of N a
nonnegative number fij that does not exceed cij.
A conservation flow is a flow with the
properties:
f ki   f jk for each vertex other than the

iV
jV
source and sink. The sum  f   f  V
 is called the value of the conservation flow. A
conservation flow f is called maximum flow, if
vf ≥vf’ for any conservation flow f ’ of N.
si
iV
jt
jV
f
 Definition 35: Let N be a network with the
source node and the sink node. If P is a subset
of V containing source s but not sink t then
E(P,V-P)is called a cut separating s from t.
 In effect, a cut does “cut” a digraph into two
pieces, one containing the source and one
containing the sink. If the edges of a cut were
removed, it has not any paths from source to
sink.
The capacity of a cut E(P,V-P)is  c , we
denote by C(P,V-P). i.e. C ( P,V  P)   c
ij
iP , jV  P
ij
iP , jV  P
 A cut E(P,V-P) of N is called minimum cut, if
C(P,V-P)≤C(P',V-P') for any cut E(P',V-P') of
N.
 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)，
 VfmaxCmin(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 fij0=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.

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
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
jU.
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, sP and tV-P. Thus
E(P,V-P)is a cut.
 (1) (i,j) E(P,V-P)(i.e. iP. jV-P)
 fij=cij，
 (2) (j,i) E (i.e. iP. jV-P)
 fji=0.
 By lemma 5.3, the labeling algorithm produces
a maximum flow.
 Exercise: P314 7,9,10,11,17,19, 20,21
 Graph Matching 8.5 P315