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Assignment Solution Spring 2012_CS502_4_SOL

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

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

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Assignment Statements:

Question:

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.

Solution:
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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Description: Assignment Solution Spring 2012_CS502_4_SOL