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

The running time of the Floyd-Warshall algorithm might appear to be slower than
performing a DFS of a directed graph from each of its vertices, but this depends
upon the representation of the graph. If a graph is represented using an adjacency
matrix, then running the DFS method once on a directed graph takes O(n 2 ) time
(we explore the reason for this in Exercise R-13.10). Thus, running DFS n times
takes O(n3) time, which is no better than a single execution of the Floyd-Warshall
algorithm, but the Floyd-Warshall algorithm would be much simpler to
implement. Nevertheless, if the graph is represented using an adjacency list
structure, then running the DFS algorithm n times would take O(n(n+m)) time to
compute the transitive closure. Even so, if the graph is dense, that is, if it has
&(n 2 ) edges, then this approach still runs in O(n 3 ) time and is more complicated
than a single instance of the Floyd-Warshall algorithm. The only case where
repeatedly calling the DFS method is better is when the graph is not dense and is
represented using an adjacency list structure.
13.4.3 Directed Acyclic Graphs
Directed graphs without directed cycles are encountered in many applications. Such
a digraph is often referred to as a directed acyclic graph, or DAG, for short.
Applications of such graphs include the following:
• Inheritance between classes of a Java program.
• Prerequisites between courses of a degree program.
• Scheduling constraints between the tasks of a project.
Example 13.20: In order to manage a large project, it is convenient to break it
up into a collection of smaller tasks. The tasks, however, are rarely independent,
because scheduling constraints exist between them. (For example, in a house
building project, the task of ordering nails obviously precedes the task of nailing
shingles to the roof deck.) Clearly, scheduling constraints cannot have circularities,
because they would make the project impossible. (For example, in order to get a job
you need to have work experience, but in order to get work experience you need to
have a job.) The scheduling constraints impose restrictions on the order in which
the tasks can be executed. Namely, if a constraint says that task a must be
completed before task b is started, then a must precede b in the order of execution
of the tasks. Thus, if we model a feasible set of tasks as vertices of a directed graph,
and we place a directed edge from v tow whenever the task for v must be executed
before the task for w, then we define a directed acyclic graph.
The example above motivates the following definition. Let be a digraph with n
vertices. A topological ordering of is an ordering v 1 ,...,v n of the vertices of
such that for every edge (vi, vj) of , i < j. That is, a topological ordering is an
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