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The results holds Suppose that result holds for n=k-1 For n=k ， By the inductive hypothesis,

Theorem 5 . 22 : Let T be built according to Huffman algorithm and leaves of T with weight w 1  w 2  w n . 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 ，

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The results holds Suppose that result holds for n=k-1 For n=k ， By the inductive hypothesis,

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  1. 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wkif T’s leaf with weight w1+w2 is replaced by subtree

  2. 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 T2by 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

  3. 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 bysubtree 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.

  4. 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,,wkby 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. 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 in-degree 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}.

  6. 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: • for each vertex other than the source and sink. The sum • 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.

  7. 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.

  8. The capacity of a cut E(P,V-P)is , we denote by C(P,V-P). i.e.

  9. 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)

  10. 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).

  11. 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?

  12. 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.

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

  14. 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.

  15. 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.

  16. 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.

  17. Exercise: P314 7,9,10,11,17,19, 20,21 • Graph Matching 8.5 P315

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