The.Algorithm.Design.Manual.Springer-Verlag.1998

Exercises
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Exercises
1. Present correct and efficient algorithms to convert between the following graph data structures,
for an undirected graph G with n vertices and m edges. You must give the time complexity of
each algorithm.
1. Convert from an adjacency matrix to adjacency lists.
2. Convert from an adjacency list to an incidence matrix. An incidence matrix M has a row
for each vertex and a column for each edge, such that M[i,j]=1 if vertex i is part of edge j,
otherwise M[i,j] = 0.
3. Convert from an incidence matrix to adjacency lists.
2. Is the path between a pair of vertices in a minimum spanning tree necessarily a shortest path
between the two vertices in the full graph? Give a proof or a counterexample.
3. Assume that all edges in the graph have distinct edge weights (i.e. no pair of edges have the same
weight). Is the path between a pair of vertices in a minimum spanning tree necessarily a shortest
path between the two vertices in the full graph? Give a proof or a counterexample.
4. Suppose G is a connected undirected graph. An edge e whose removal disconnects the graph is
called a bridge. Must every bridge e be an edge in a depth-first search tree of G, or can e be a
back edge? Give a proof or a counterexample.
5. (*) In breadth-first and depth-first search, an undiscovered node is marked discovered when it is
first encountered, and marked completely-explored when it has been completely searched. At any
given moment, several nodes might be simultaneously in the discovered state.
(a) Describe a graph on n vertices and a particular starting vertex v such that during a breadth-first
search starting from v, nodes are simultaneously in the discovered state.
(b) Describe a graph on n vertices and a particular starting vertex v such that during a depth-first
search starting from v, nodes are simultaneously in the discovered state.
(c) Describe a graph on n vertices and a particular starting vertex v such that at some point during
a depth-first search starting from v, nodes remain undiscovered, while nodes have been
completely-explored. (Note, there may also be discovered nodes.)
6. Given the pre-order and in-order traversals of a binary tree, is it possible to reconstruct the tree? If
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