True / False
1. The arcs in a minimal spanning tree problem can be measured in terms of criteria other than distance.
a. True
b. False
2. Cases in which a greedy algorithm provides the optimal solution
...
True / False
1. The arcs in a minimal spanning tree problem can be measured in terms of criteria other than distance.
a. True
b. False
2. Cases in which a greedy algorithm provides the optimal solution are rare.
a. True
b. False
3. The minimum spanning tree allows a person to visit every node without backtracking.
a. True
b. False
4. In the minimal spanning tree algorithm, you must consider the unconnected nodes that can be reached from any of the connected nodes, rather than arbitrarily considering only one of the connected nodes.
a. True
b. False
5. Cases in which a greedy algorithm provides the optimal solution are rare, but greedy algorithms are excellent heuristics.
a. True
b. False
6. The minimal spanning tree algorithm will lead to an optimal solution regardless of which node is chosen at the start of the algorithm.
a. True
b. False
7. A minimal spanning tree problem involves using the arcs of the network to reach all nodes of the network such that the total length of all the arcs used is maximized.
a. True
b. False
8. For a network consisting of N nodes, a spanning tree will consist of N – 1 arcs.
a. True
b. False
Multiple Choice
9. Consider a minimal spanning tree problem in which pipe must be laid to connect sprinklers on a golf course. When represented with a network,
a. the pipes are the arcs and the sprinklers are the nodes.
b. the pipes are the nodes and the sprinklers are the arcs.
c. the pipes and the sprinklers are the tree.
d. each sprinkler must be connected to every other sprinkler.
10. The minimal spanning tree algorithm is considered to be a(n)
a. greedy algorithm.
b. arc algorithm.
c. non-optimal algorithm.
d. non-feasible algorithm.
11. The minimal spanning tree algorithm has connected nodes 8 and 9. Node 8 could be connected to nodes 11 (distance 6) and 12 (distance 5), and node 9 could be connected to node 12 (distance 3) and node 13 (distance 2). Which of the following will you do next?
a. connect 8 to 11
b. connect 8 to 12
c. connect 9 to 12
d. connect 9 to 13
12. For a network consisting of N nodes, a minimal spanning tree will consist of
a. N − 2 arcs.
b. N − 1 arcs.
c. N arcs.
d. N + 1 arcs.
13. The minimal spanning tree algorithm is considered a greedy algorithm that, when taking the best action at each stage, will
a. sometimes fail to produce a feasible solution.
b. always produce a feasible, but not necessarily optimal, solution.
c. always produce an optimal solution.
d. always produce an optimal, but not necessarily feasible, solution.
Subjective Short Answer
14. For the following eight cities with the given distances, find the minimal spanning tree path.
To City
From City 1 2 3 4 5 6 7
1 – 5 2 6 – – –
2 5 – – 4 6 – –
3 2 – – 3 – 12 –
4 6 4 3 – 9 8 –
5 – 6 – 9 – 0 14
6 – – 12 8 0 – 4
7 – – – – 14 4 –
15. Find the minimal spanning tree for this network.
16. Find the minimal spanning tree for this network.
17. The numbers on this network represent times to distribute a message. Use the minimal spanning tree algorithm to determine how messages should be passed in order to minimize total time.
18. Ethernet cable costs $25 per foot to install. The table below gives the approximate direct routing distance in feet between various pairs of machines.
WS1 WS2 LP1 LP2 CC FM
WS1 0 46 16 23 100 29
WS2
0 81 44 67 98
LP1
0 45 57 36
LP2
0 38 73
CC
0 52
a. Use the minimal spanning tree algorithm to determine the least cost method of connecting all machines.
b. How much cheaper would it be if the company decided not to hook up the second printer to this system?
19. Wondercamp is planning a new resort for the urban professional who wishes to "get back to nature." For a hefty fee, it plans to transport clients up the Crocodile River by canoe and then have the clients camp at one of eight designated campsites.
It must build a 1000-yard trail from the river to the first campsite as there are no feasible alternatives. It then wishes to build a sequence of trails (of minimum total length) so that every campsite can be reached from any other campsite. A study of the area's terrain has yielded the following possibilities for trails between campsites. Which trails should be built?
20. Griffith's Cherry Preserve is a combination wild animal habitat and amusement park. Besides its phenomenally successful wild animal safari tour, there are eight different theme areas in the amusement park.
One problem encountered by management is to develop a method by which people can efficiently travel between each area of the park. Management has learned that a people mover can be constructed at a cost of $50 per foot.
If the following network represents the distances (in feet) between each area of the park for which a people mover is possible, determine the minimum cost for such a system.
[Show More]
