CS 373 Midterm 2 (October 31, 2000) Fall 2000
1. Using any method you like, compute the following subgraphs for the weighted graph below.
Each subproblem is worth 3points. Each incorrect edge costs you 1point, but you cannot get
a negative score for any subproblem.
(a) a depth-first search tree, starting at the top vertex;
(b) a breadth-first search tree, starting at the top vertex;
(c) a shortest path tree, starting at the top vertex;
(d) the minimum spanning tree.
4
10
6
1
7
8
9
5
0
3
11
12
2
2. Suppose you are given a weighted undirected graph G(represented as an adjacency list)
and its minimum spanning tree T(which you already know how to compute). Describe and
analyze and algorithm to find the second-minimum spanning tree of G,i.e., the spanning tree
of Gwith smallest total weight except for T.
The minimum spanning tree and the second-minimum spanning tree differ by exactly one
edge. But which edge is different, and how is it different? That’s what your algorithm has to
figure out!
12
14
2
8 5
10
2 3
18 16
14
30
4 26
8 5
10
3
18 16
12 30
4 26
The minimum spanning tree and the second-minimum spanning tree of a graph.
3. (a)
[4 pts]
Prove that a connected acyclic graph with Vvertices has exactly V−1edges.
(“It’s a tree!” is not a proof.)
(b)
[4 pts]
Describe and analyze an algorithm that determines whether a given graph is a
tree, where the graph is represented by an adjacency list.
(c)
[2 pts]
What is the running time of your algorithm from part (b) if the graph is repre-
sented by an adjacency matrix?
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