Assignment Solution Spring 2012_CS502_4_SOL by nouman100


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									             CS 502 Fundamental of Algorithms
                     Assignment # 04
                       Spring 2012
                                                             Total Marks = 20

Your assignment must be uploaded / submitted before or on June 19, 2012

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Assignment Statements:


Give an example of a directed graph G = (V, E), a source
vertex s V, and a set of tree edges Eπ ⊆ E such that for each
vertex v V, the unique path in the graph (V, Eπ) from s to v is
a shortest path in G, yet the set of edges Eπ cannot be
produced by running BFS on G, no matter how the vertices
are ordered in each adjacency list.

The edges in Eπ are shaded in the following graph:
To see that Eπ cannot be a breadth-first tree, let’s suppose
that Adj[s] contains u before v. BFS adds edges (s, u) and (s,
v) to the breadth-first tree. Since u is enqueued before v, BFS
then adds edges (u, w) and (u, x). (The order of w and x in
Adj[u] doesn.t matter) Symmetrically, if Adj[s] contains v
before u, then BFS adds edges (s, v) and (s, u) to the breadth-
first tree, v is enqueued before u, and BFS adds edges (v, w)
and (v, x). (Again, the order of w and x in Adj[v] doesn’t
matter). BFS will never put edges (u, w) and (v, x) into the
breadth-first tree. In fact, it will also never put both edges (u, x)
and (v, w) into the breadth-first tree.

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