Day23 - COP 3503 Computer Science II CLASS NOTES DAY#23 Graphs Continued Graph Traversals As with trees traversing a graph consists of visiting

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COP 3503 – Computer Science II CLASS NOTES - DAY #23 Graph Traversals As with trees, traversing a graph consists of visiting each vertex only one time. The simple traversal algorithms used for trees (preorder, inorder, postorder) cannot be applied here because graphs may include cycles which would cause the tree traversal algorithms to enter an infinite loop. To prevent this from happening, each visited vertex is typically marked in some fashion to avoid revisiting it (a common technique is to renumber the vertices as they are visited). However, graphs can have isolated vertices (unconnected vertices), which means that some parts of the graph are left unvisited if unmodified tree traversal algorithms are applied. Depth-First Traversal The depth-first search algorithm for graphs was developed by Hopcroft and Tarjan. In this algorithm, each vertex V is visited and then each unvisited vertex adjacent to V is visited. If a vertex V has no adjacent vertices or all of its adjacent vertices have been visited, the traversal backtracks to the predecessor of V . The traversal is complete when this process of visiting and backtracking leads to the first vertex where the traversal started. If there are still unvisited vertices in the graph, the traversal continues by restarting on one of the unvisited vertices. This algorithm renumbers each vertex as it is visited. Algorithm: DFS(v) num (v) = i++; for all vertices u adjacent to v if num (u) is 0 attach edge (uv) to edges; DFS(u); depthFirstSearch( ) for all vertices v num (v) = 0; edges = null; i = 1; while there is a vertex v such that num (v) is 0 DFS(v); output edges; Example: Day 23 - 1 Graphs Continued
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The numbers assigned to each vertex are shown in parentheses in the figure below. Once the initializations have been made, depthFirstSearch( ) calles DFS(a). DFS( ) is invoked for vertex a ; num(a) is assigned number 1. Vertex a has four adjacent vertices, and vertex e is chosen for the next invocation, DFS(e), which assigns number 2 to this vertex (num(e) = 2) and puts the edge(ae) in the set edges . e has two unvisited adjacent vertices, and DFS( ) is called for the first of them, the vertex f The call DFS(f) will lead to the assignment num(f) = 3 and will put edge(ef) in edges . Vertex f has only one unvisited adjacent vertex, i , thus the fourth call DFS(i) will lead to the assignment num(i) = 4 and to the attaching of edge(fi) to edges Vertex i has only visited adjacent vertices, thus a return to the call DFS(f) occurs and then to DFS(e) in which vertex i is accessed only to discover that num(i) is not 0, thus edge(ei) is not included in the set of edges. The rest of the execution is shown in the figure below where the solid lines indicate edges that are included in the set edges Part (a) is the original graph and part (b) illustrates the algorithm’s technique.
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Day23 - COP 3503 Computer Science II CLASS NOTES DAY#23 Graphs Continued Graph Traversals As with trees traversing a graph consists of visiting

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