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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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EXAMPLE 15.26 Tracing the BFS Algorithm
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Table 15.1 shows a trace of Algorithm 15.2 on the graph shown in Figure 15.29. The start vertex is v0 = A.
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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
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Table 15.1 Trace of Algorithm 15.2 Figure 15.29 Tracing the BFS
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x A B A, B E C A, B, E A
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L B E C F D G
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y AB AB, AE AB, AE, BC
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AB, AE, BC, BF AB, AE, BC, BF, CD AB, AE, BC, BF, CD, CG
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A, B, E, C
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}.
The depth-first search algorithm uses a stack instead of a queue.
Figure 15.30 Tracing the BFS
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.
9. 10. 11. 12. 13.
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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.
[CHAP. 15
EXAMPLE 15.27 Tracing the DFS Algorithm
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.
L A A, B A, B, C A, B, C, D A, B, C, D, G A A, B A, B, C A, B, C, D A, B, C, D, G A, B, C, D A, B, C A, B, C, D, G, F A, B, C, F A, B, C A, B A, B, C, D, G, F, E A, B, E A, B A S A B C D G D C F C B E B A
Table 15.2 Trace of Algorithm 15.3
x B C D G
y AB AB, BC AB, BC, CD
AB, BC, CD, DG
AB, BC, CD, DG, CF
AB, BC, CD, DG, CF, BE
The resulting DFS order of visitation is returned in the list L = (A, B, C, D, G, F, E). Figure 15.31 shows the resulting DFS spanning tree, which is returned in the set T = {AB, BC, CD, DG, CF, BE}.
Since the depth-first traversal uses a stack, it has a natural recursive version:
Figure 15.31 Tracing the DFS
Algorithm 15.4 The Recursive Depth-First Search (DFS) Algorithm (Preconditions: G = (V,E) is a graph or digraph with initial vertex x; each vertex has a boolean visited field initialized to false; T is a global set of edges; L is a global list of vertices.) (Postcondition: L lists the vertices in DFS order, and T is a DFS spanning tree for G.) 1. Mark x visited. 2. Add x to L. 3. Repeat steps 4 5 for each unvisited vertex y that is adjacent to x. 4. Add the edge xy to T. 5. Apply the DFS algorithm to the subgraph with initial vertex y.
CHAP. 15]
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EXAMPLE 15.28 Tracing the Recursive DFS Algorithm
Table 15.3 shows a trace of Algorithm 15.4 on the same graph as in Example 15.26 (Figure 15.29 on page 301). The start vertex v0 = A. The result, of course, is the same as that in Example 15.27. The only real difference is that the explicit
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