vb.net barcode printing Figure 15.26 The fourth and fifth iterations of Dijkstra s algorithm in Java

Maker Data Matrix ECC200 in Java Figure 15.26 The fourth and fifth iterations of Dijkstra s algorithm

Figure 15.26 The fourth and fifth iterations of Dijkstra s algorithm
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On the fourth iteration, shown in Figure 15.26, the highest priority vertex among those still in the priority queue is x = B with distance field 4. Steps 7 10 iterate twice, for y = C and E. The values of s computed for these are 4 + 3 = 7 and 4 + 5 = 9. The second of these is less than the current (infinite) value at E, so its distance field assigned the value 9 and its back pointer is set to B. But the s value 7 is not less than the current distance field for C, so its fields do not change. The algorithm progresses through its remaining iterations, shown in Figure 15.27, for x = C, E, G, and finally H the same way.
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Figure 15.27 The sixth and seventh iterations of Dijkstra s algorithm
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The final result is shown in Figure 15.28. It shows, for example, that the shortest path from A to E is ADFCE with length 6.
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Figure 15.28 The last two iterations of Dijkstra s algorithm
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EXAMPLE 15.25 An Implementation of Dijkstra s Algorithm
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Here is a Java implementation of Algorithm 15.1. It defines a Network class whose instances represent weighted digraphs.
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public class Network { Vertex start; private class Vertex { Object object; Edge edges; Vertex nextVertex; boolean done; int dist; Vertex back; } private class Edge { Vertex to; int weight; Edge nextEdge; }
GRAPHS
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public Network() { if (start != null) { start.dist = 0; for (Vertex p = start.nextVertex; p != null; p = p.nextVertex) { p.dist = Integer.MAX_VALUE; // infinity } } } public void findShortestPaths() { // implements Dijkstra s Algorithm: for (Vertex v = start; v != null; v = closestVertex()) { for (Edge e = v.edges; e != null; e = e.nextEdge) { Vertex w = e.to; if (!w.done && v.dist+e.weight < w.dist) { w.dist = v.dist+e.weight; w.back = v; } } v.done = true; } } private Vertex closestVertex() { // returns the vertex with minimal dist among those not done: Vertex v = null; int minDist = Integer.MAX_VALUE; for (Vertex w = start; w != null; w = nextVertex) { if (!w.done && w.dist < minDist) { v = w; minDist = w.dist; } } return v; } }
In this implementation, we have used a simple search method closestVertex() instead of a priority queue. This is less efficient, running in O(n) time instead of the O(lgn) time that a priority queue would use.
GRAPH TRAVERSAL ALGORITHMS The paths produced by Dijkstra s algorithm produce a minimal spanning tree for the graph. That is a spanning tree whose total weighted length is minimal for the graph; that is, no other spanning tree has a smaller total length. The spanning tree is formed in a breadth-first manner, by considering the vertices that are adjacent to the current vertex on each iteration. This is one of two general ways to traverse a graph. The breadth-first search algorithm is essentially the same as Dijkstra s algorithm without regard to the distance fields. Algorithm 15.2 The Breadth-First Search (BFS) Algorithm (Preconditions: G = (V,E) is a graph or digraph with initial vertex v0; each vertex has a boolean visited field initialized to false; T is an empty set of edges; L is an empty list of vertices.) (Postcondition: L lists the vertices in BFS order, and T is a BFS spanning tree for G.)
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GRAPHS
1. 2. 3. 4. 5. 6. 7. 8. 9.
Initialize an empty queue Q for temporary storage of vertices. Enqueue v0 into Q. Repeat steps 4 6 while Q is not empty. Dequeue Q into x. Add x to L. Do step 7 for each vertex y that is adjacent to x. If y has not been visited, do steps 8 9. Add the edge xy to T. Enqueue y into Q.
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