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Here is a Java implementation of Algorithm 15.1. It defines a Network class whose instances represent weighted digraphs.

unless c is real. Thus, this system is nonlinear. For an input x(n), this system produces an output that is the conjugate symmetric part of x(n). If c is a complex constant, and if the input to the system is xl(n) = cx(n), the output is

There are many different systems that are either homogeneous or additive but not both. One example of a system that is homogeneous but not additive is the following:

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Q A B B, E E, C E, C, F C, F F, D F, D, G D, G G F D G A, B, E, C, F A, B, E, C, F, D A, B, E, C, F, D, G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

- I)x(n)

+ I)

The resulting BFS order of visitation is returned in the list L = (A, B, E, C, F, D, G), and the resulting BFS spanning tree (Figure 15.30) is returned in the set T = {AB, AE, BC, BF, CD, CD}.

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; }

+ I)

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Algorithm 15.3 The Depth-First Search (DFS) 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 DFS order, and T is a DFS spanning tree for G.) 1. Initialize an empty stack S for temporary storage of vertices. 2. Add v0 to L. 3. Push v0 onto S. 4. Mark v0 visited. 5. Repeat steps 6 8 while S is not empty. 6. Let x be the top element on S. 7. If x has any adjacent unvisited vertices, do steps 9 13. 8. Otherwise, pop the stack S and go back to step 5.

+ 1)

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

which is c times the response to x(n). Therefore, the system is homogeneous. On the other hand, it should be clear that the system is not additive because, in general,

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9. 10. 11. 12. 13.

{xl(n - 1)

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; } }

+ X Z (- l)J(x~(n) xz(n)I ~ + + +I)

Let y be an unvisited vertex that is adjacent to x. Add the edge xy to T. Add y to L. Push y onto S. Mark y visited.

x ~ ( n l)x~(n) x,(n 1)

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.

+ xdn - l)xz(n) xz(n + 1)

Table 15.2 shows a trace of Algorithm 15.3 on the same graph (Figure 15.29) as in Example 15.26. The start vertex is v0 = A.

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