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vb.net code to generate barcode Figure 15.22 An eulerian graph in Java
Figure 15.22 An eulerian graph GTIN  13 Recognizer In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. EAN13 Supplement 5 Generation In Java Using Barcode creator for Java Control to generate, create EAN / UCC  13 image in Java applications. Figure 15.23 A graph with a hamiltonian cycle, and one with without one
Reading European Article Number 13 In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Encoding Bar Code In Java Using Barcode generator for Java Control to generate, create bar code image in Java applications. CHAP. 15] Barcode Recognizer In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Paint EAN13 In C# Using Barcode printer for Visual Studio .NET Control to generate, create GTIN  13 image in VS .NET applications. DIJKSTRA S ALGORITHM
Make EAN / UCC  13 In VS .NET Using Barcode maker for ASP.NET Control to generate, create EAN13 image in ASP.NET applications. UPC  13 Generation In Visual Studio .NET Using Barcode drawer for .NET Control to generate, create UPC  13 image in VS .NET applications. GRAPHS
Paint EAN / UCC  13 In VB.NET Using Barcode creator for VS .NET Control to generate, create EAN 13 image in .NET framework applications. UCC.EAN  128 Printer In Java Using Barcode maker for Java Control to generate, create USS128 image in Java applications. Dijkstra s algorithm finds the shortest path from one vertex v0 to each other vertex in a digraph. When it has finished, the length of the shortest distance from v0 to v is stored in the vertex v, and the shortest path from v0 to v is recorded in the back pointers of v and the other vertices along that path. (See Example 15.24.) The algorithm uses a priority queue, initializing it with all the vertices and then dequeueing one vertex on each iteration. Algorithm 15.1 Dijkstra s Shortest Paths Algorithm (Precondition: G = (V,w) is a weighted graph with initial vertex v0.) (Postcondition: Each vertex v in V stores the shortest distance from v0 to v and a back reference to the preceding vertex along that shortest path.) 1. Initialize the distance field to 0 for v0 and to for each of the other vertices. 2. Enqueue all the vertices into a priority queue Q with highest priority being the lowest distance field value. 3. Repeat steps 4 10 until Q is empty. 4. (Invariant: The distance and back reference fields of every vertex that is not in Q are correct.) 5. Dequeue the highest priority vertex into x. 6. Do steps 7 10 for each vertex y that is adjacent to x and in the priority queue. 7. Let s be the sum of the x s distance field plus the weight of the edge from x to y. 8. If s is less than y s distance field, do steps 9 10; otherwise go back to Step 3. 9. Assign s to y s distance field. 10. Assign x to y s back reference field. EXAMPLE 15.24 Tracing Dijkstra s Algorithm Barcode Encoder In Java Using Barcode drawer for Java Control to generate, create bar code image in Java applications. Create Data Matrix 2d Barcode In Java Using Barcode generation for Java Control to generate, create DataMatrix image in Java applications. This is a trace of Algorithm 15.1 on a graph with eight vertices. On each iteration, the vertices that are still in the priority queue are shaded, and vertex x is labeled. The distance fields for each vertex are shown adjacent to the vertex, and the back pointers are drawn as arrows. Make UPCE In Java Using Barcode generation for Java Control to generate, create UPCE Supplement 2 image in Java applications. Encode Barcode In .NET Framework Using Barcode generator for Reporting Service Control to generate, create bar code image in Reporting Service applications. Figure 15.24 The first iteration of Dijkstra s algorithm
Encoding Code39 In None Using Barcode printer for Online Control to generate, create ANSI/AIM Code 39 image in Online applications. Barcode Decoder In VS .NET Using Barcode scanner for .NET Control to read, scan read, scan image in .NET applications. The first two iterations are shown in Figure 15.24. On the first iteration, the highest priority vertex is x = A because its distance field is 0 and all the others are infinity. Steps 7 10 iterate three times, once for Painting Bar Code In Java Using Barcode drawer for Eclipse BIRT Control to generate, create barcode image in BIRT applications. Make Barcode In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create barcode image in ASP.NET applications. GRAPHS
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each of A s neighbors y = B, C, and D. The values of s computed for these are 0 + 4 = 4, 0 + 6 = 6, and 0 + 1 = 1. Each of these is less than the current (infinite) value of the corresponding distance field, so all three of those values are assigned, and the back pointers for all three neighbors are set to point to A. On the second iteration, the highest priority vertex among those still in the priority queue is x = D with distance field 1. Steps 7 10 iterate three times again, once for each of D s unvisited neighbors: y = B, F, and G. The values of s computed for these are 1 + 4 = 5, 1 + 2 = 3, and 1 + 6 = 7, respectively. Each of these is less than the current value of the corresponding distance field, so all of those values are assigned and the back pointers are set to D. Note how this changes the distance field and pointer in vertex C. Figure 15.25 The second and third iterations of Dijkstra s algorithm
The next two iterations are shown in Figure 15.25. On the third iteration, the highest priority vertex among those still in the priority queue is x = F with distance field 3. Steps 7 10 iterate three times again, once for each of F s unvisited neighbors y = C, G, and H. The values of s computed for these are 3 + 1 = 4, 3 + 3 = 6, and 3 + 5 = 8. Each of these is less than the current value, so all of them are assigned and the back pointers are set to F. Note how this changes the distance field and pointer in vertex C again.

