Graph Traversal

Like a tree, a graph can be traversed in more than one way. We must choose an arbitrary vertex to start from. This is called the source vertex. A depth-first traversal (Figure 15-18) follows edges until it reaches a dead end, then backtracks to the last branching point to try a different branch. The order in which the different branches are tried is arbitrary.

Figure 15-18. Depth-first traversal of a graph. The traversal begins by following edges until a dead end is reached (left). It then backtracks to the last decision point, following a different branch (middle). This continues until all reachable vertices have been visited (right).

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A breadth-first traversal (Figure 15-19) visits one vertex, then its neighbors, then vertices two edges away, and so on.

Figure 15-19. In a breadth-first traversal of a graph, vertices closer to the source vertex are visited earlier than more distant vertices. The dashed lines merely separate the four copies of the graph.

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In either case, vertices that cannot be reached from the source (such as the one at the upper right in Figures 15-18 and 15-19) are never visited. Because there may be more than one path to a given vertex, it is necessary to keep track of which vertices have already been visited. This is done with an array of booleans.

The algorithm for depth-first traversal can be stated using a stack or, more concisely, using recursion (Figure 15-20).

Figure 15-20. Depth-first traversal of a graph. The second, protected method depthFirstTraverse() is recursive.

 1 /** 2 * Return a list of the vertices reachable from source, in depth- 3 * first order. 4 */ 5 public List depthFirstTraverse(int source) { 6 List result = new ArrayList(size()); 7 boolean[] visited = new boolean[size()]; 8 depthFirstTraverse(source, result, visited); 9 return result; 10 } 11 12 /** 13 * Visit the vertices reachable from vertex, in depth-first order. 14 * Add vertices to result as they are visited. 15 */ 16 protected void depthFirstTraverse(int vertex, 17 List result, 18 boolean[] visited) { 19 visited[vertex] = true; 20 result.add(vertex); 21 for (Integer i : neighbors(vertex)) { 22 if (!visited[i]) { 23 depthFirstTraverse(i, result, visited); 24 } 25 } 26 }

The algorithm for breadth-first traversal (Figure 15-21) uses a queue, much like the level order traversal of a tree (Section 10.2).

Figure 15-21. Breadth-first traversal of a graph.

 ``` 1 /** 2 * Return a list of the vertices reachable from source, in 3 * breadth-first order. 4 */ 5 public List breadthFirstTraverse(int source) { 6 List result = new ArrayList(size()); 7 boolean[] visited = new boolean[size()]; 8 Queue q = new LinkedList(); 9 visited[source] = true; 10 q.offer(source); 11 while (!(q.isEmpty())) { 12 int vertex = q.poll(); 13 result.add(vertex); 14 for (Integer i : neighbors(vertex)) { 15 if (!visited[i]) { 16 visited[i] = true; 17 q.offer(i); 18 } 19 } 20 } 21 return result; 22 } ```

Exercises

 15.12 Figure 15-22 shows an undirected graph. In what order might the vertices be visited during a depth-first traversal starting at vertex G? What about a breadth-first traversal? (There is more than one correct answer for each question.)

Figure 15-22. Undirected graph for Exercise 15.12

 15.13 What bad thing could happen if we removed the test on line 22 of Figure 15-20?

Topological Sorting

Data Structures and Algorithms in Java
ISBN: 0131469142
EAN: 2147483647
Year: 2004
Pages: 216
Authors: Peter Drake