Proof in Java Implement QR Code JIS X 0510 in Java Proof

Proof using swing toget qrcode in web,windows application barcode pdf417 Proof Let T be a rooted tree with r oot v and suppose u :A v is a vertex with indeg u =# 1. Since indeg u 0 0, we have indeg u > 2, so there are distinct arcs u u and u" U. 1 The proposition says there is a path P from v to u and also a path P" from v to u"1 .

Thus we obtain two different paths from v to u (with final arcs u u and u" u). I This contradicts uniqueness of a directed path from v to u. Given a weighted acyclic digraph 5 with nonnegative weights and a canonical labeling vo, VI, .

. , v, I, to find the shortest paths from vo to every other vertex, Step 1. Set do 0 and di = o for i = 1,2,.

.. ,n-1.

Set pi =-1 for i = 0,1, . . .

, n-l. Step 2. For t from I to n - 1, let dt = min dj + w(vj, vt) I j = 0, .

. .-, tthis minimum.

} and let Pt be a j which gives. As with previous shortest pat jar QR Code ISO/IEC18004 h algorithms, a final value dt = 0o simply indicates that there is no directed path from vo to Vt. Consider the acyclic canonically labeled graph 5 with arcs weighted as shown on the left in Fig 12.22.

We show the values of the dt and pt (p for predecessor) at the beginning, and after each iteration of the loop in Step 2. The final values of di and pl are 2 and 0, respectively. Thus the shortest distance to v, has length 2, and the last arc on a shortest path is v(vl.

The final values of d2 and P2 are 4 and 1, respectively, showing that the shortest path to v2 has length 4, and the last arc on the corresponding path is vI V2. The final values of d3 and p3 are 2 and 0, respectively, asserting the facts that a shortest path to V3 has length 2 and VOV3 is the last arc on such a path. On the right in Fig 12.

22, we show just those arcs which are used last on shortest paths from Vt) to each vertex. It is not a. 12.5 Acyclic Digraphs and Bellman"s Algorithm V4 V4 vo V3 2 V2. do, po Initially t= I 0,-I 0,-1. di, pi 0o,-I d2, P2 o, -I 0o, -1. d3, p3 o0,-I 0o,1. d4 , p4 o0,-I o,-I t =2 t =3 t=4 0,-1 0, -1 0, -1. 2,0 2,0 2,0. 4, 1 4, 1 4, 1. oo,-1 2,0 2,0. oo,-I o, -1 3,3. coincidence that swing QR these form a spanning tree rooted at vo. In the Exercises, we ask you to explain why..

13. In Section 11.2, we met the Bellman-Ford shortest path algorithm. ff"TIMMM -. (a) [BB] Explain how this can be accomplished. (b) Describe an algorithm which implements your idea. (c) Write computer code which implements your algorithm.

6. How many shortest path algorithms can you name How many of these can you describe 7. [BB] Suppose IT is a rooted (directed) tree with n vertices.

How many arcs does T have Why 8. Let G be a connected graph and suppose we orient the edges in such a way that the digraph we obtain has a unique vertex of indegree 0. Must this digraph be a rooted tree 9.

The following digraphs are acyclic, and canonical labelings are shown. Apply Bellman"s algorithm to each digraph in order to find the lengths of shortest paths from vo to each other vertex. Find a shortest path to vt and the predecessor vertex vp,.

. The symbol [BB] swing Quick Response Code means that an answer can be found in the Back of the Book. 1. For each of the digraphs in Fig 12.

23, either show that the digraph is acyclic by finding a canonical labeling of vertices or exhibit a cycle. 2. Suppose vo, vj .

.. , v,, l is a canonical labeling of a digraph G.

What can be said about the indegree and outdegree of v(, if anything Explain. 3. [BB] Find an 0(n 2 ) algorithm which computes the indegrees of a digraph with n vertices.

(The addition of two numbers is the basic operation.) 4. (a) Describe an algorithm which finds a canonical labeling for the vertices of an acyclic digraph.

(b) Find a Big Oh estimate for your algorithm in terms of some reasonable basic operation (or operations). (c) Write a computer program which implements your algorithm. 5.

The algorithm described in the proof of Theorem 12.5.3 can in fact be adapted to find a cycle in a digraph which is not acyclic.

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