Graphs are flexible data structures that model problems defined in terms of relationships or connections between objects (see Chapter 11). This chapter presents algorithms that work with graphs. As we will see, many graph algorithms resemble the fundamental ones for breadth-first and depth-first search introduced in Chapter 11. Breadth-first and depth-first search are important to many other graph algorithms because they offer good ways of exploring the structure of a graph in a systematic manner.

One significant difference between the algorithms of Chapter 11 and the ones in this chapter,
however, is that the algorithms here work with *weighted
graphs*. In a weighted graph , each edge is assigned a value, or
*weight*, which is represented pictorially as a small
number beside the edge. Although weights can mean many things, in
general they represent a cost associated with traversing an edge.
Weighted graphs and their algorithms have an enormous capacity to model
real problems. Example 16.1 is a
header for the graph algorithms presented in this chapter.

This chapter covers:

- Minimum spanning trees
Trees that serve as abstractions of many connectivity problems. A minimum spanning tree is a tree that connects all vertices in an undirected, weighted graph at a minimum cost.

- Shortest paths
The result of solving various types of shortest-path problems. A shortest path is a path that connects one vertex to another in a directed, weighted graph at a minimum cost. ...

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