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eP 1 . ( 2 5 pts ) [ Ch 3 . Graph Theory ] A number of art museums around the country have been
eP ptsCh Graph Theory
A number of art museums around the country have been featuring work by an artist named Mark Lombardi
consisting of a set of intricately rendered graphs. Building on a great deal of research, these graphs encode
the relationships among people involved in major political scandals over the past several decades: the nodes
correspond to participants, and each edge indicates some type of relationship between a pair of participants. And
so if you peer closely enough at the drawings, you can trace out ominouslooking paths from a highranking US
government official, to a former business partner, to a bank in Switzerland, to a shadowy arms dealer. Such
pictures form striking examples of social networks, which, as we discussed in Section have nodes representing
people and organizations, and edges representing relationships of various kinds. And the short paths that abound
in these networks have attracted considerable attention recently, as people ponder what they mean. In the case of
Mark Lombardi's graphs, they hint at the short set of steps that can carry you from the reputable to the
disreputable.
Of course, a single, spurious short path between nodes and in such a network may be more coincidental than
anything else; a large number of short paths between and can be much more convincing. So in addition to the
problem of computing a single shortest path in a graph social networks researchers have looked at the
problem of determining the number of shortest w paths. This turns out to be a problem that can be solved
efficiently. Suppose we are given an undirected graph and we identify two nodes and in
Give an algorithm that computes the number of shortest vw paths in GThe algorithm should not list all the
paths; just the number suffices. The running time of your algorithm should be for a graph with nodes
and edges.
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