Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

One of the nice properties of the BFS approach is that, in performing the BFS
traversal, we can label each vertex by the length of a shortest path (in terms of the
number of edges) from the start vertex s. In particular, if vertex v is placed into
level i by a BFS starting at vertex s, then the length of a shortest path from s to v is
i.
As with DFS, we can visualize the BFS traversal by orienting the edges along the
direction in which they are explored during the traversal, and by distinguishing the
edges used to discover new vertices, called discovery edges, from those that lead to
already visited vertices, called cross edges. (See Figure 13.7f.) As with the DFS, the
discovery edges form a spanning tree, which in this case we call the BFS tree. We
do not call the nontree edges "back edges" in this case, however, for none of them
connects a vertex to one of its ancestors. Every nontree edge connects a vertex v to
another vertex that is neither v's ancestor nor its descendent.
The BFS traversal algorithm has a number of interesting properties, some of which
we explore in the proposition that follows.
Proposition 13.14: Let G be an undirected graph on which a BFS traversal
starting at vertex s has been performed. Then
• The traversal visits all vertices in the connected component of s.
• The discovery-edges form a spanning tree T, which we call the BFS tree,
of the connected component of s.
• For each vertex v at level i, the path of the BFS tree T between s and v has
i edges, and any other path of G between s and v has at least i edges.
• If (u, v) is an edge that is not in the BFS tree, then the level numbers of u
and v differ by at most 1.
We leave the justification of this proposition as an exercise (C-13.14). The analysis
of the running time of BFS is similar to the one of DFS, which implies the
following.
Proposition 13.15: Let G be a graph with n vertices and m edges
represented with the adjacency list structure. A BFS traversal of G takes O(n + m)
time. Also, there exist O(n + m)-time algorithms based on BFS for the following
problems:
• Testing whether G is connected.
• Computing a spanning tree of G, if G is connected.
• Computing the connected components of G.
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