The.Algorithm.Design.Manual.Springer-Verlag.1998

Traversing a Graph
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Traversing a Graph
Perhaps the most fundamental graph problem is to traverse every edge and vertex in a graph in a
systematic way. Indeed, most of the basic algorithms you will need for bookkeeping operations on
graphs will be applications of graph traversal. These include:
● Printing or validating the contents of each edge and/or vertex.
● Copying a graph, or converting between alternate representations.
● Counting the number of edges and/or vertices.
● Identifying the connected components of the graph.
● Finding paths between two vertices, or cycles if they exist.
Since any maze can be represented by a graph, where each junction is a vertex and each hallway an edge,
any traversal algorithm must be powerful enough to get us out of an arbitrary maze. For efficiency, we
must make sure we don't get lost in the maze and visit the same place repeatedly. By being careful, we
can arrange to visit each edge exactly twice. For correctness, we must do the traversal in a systematic
way to ensure that we don't miss anything. To guarantee that we get out of the maze, we must make sure
our search takes us through every edge and vertex in the graph.
The key idea behind graph traversal is to mark each vertex when we first visit it and keep track of what
we have not yet completely explored. Although bread crumbs or unraveled threads are used to mark
visited places in fairy-tale mazes, we will rely on Boolean flags or enumerated types. Each vertex will
always be in one of the following three states:
● undiscovered - the vertex in its initial, virgin state.
● discovered - the vertex after we have encountered it, but before we have checked out all its
incident edges.
● completely-explored - the vertex after we have visited all its incident edges.
Obviously, a vertex cannot be completely-explored before we discover it, so over the course of the
traversal the state of each vertex progresses from undiscovered to discovered to completely-explored.
We must also maintain a structure containing all the vertices that we have discovered but not yet
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