vb.net code to generate barcode Figure 15.22 An eulerian graph in Java

Encoder GTIN - 13 in Java Figure 15.22 An eulerian graph

Figure 15.22 An eulerian graph
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Figure 15.23 A graph with a hamiltonian cycle, and one with without one
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DIJKSTRA S ALGORITHM
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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
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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.
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Figure 15.24 The first iteration of Dijkstra s algorithm
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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
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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.
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