Post on 08-Jan-2018
description
Graph Theory
Trees
WHAT YOU WILL LEARN• Trees, spanning trees, and
minimum-cost spanning trees
Definitions
A tree is a connected graph in which each edge is a bridge.
A spanning tree is a tree that is created from another graph by removing edges while still maintaining a path to each vertex.
Examples
Graphs that are trees. Graph that are not trees.
Example: Determining Spanning Trees
Determine two different spanning trees for the graph shown.
A
B
C
E F H
D G
A
B
C
E F H
D G A
B
C
E F H
D G
Minimum-cost spanning tree
A minimum cost spanning tree is the least expensive spanning tree of all spanning trees under consideration.
Kruskal’s Algorithm
To construct the minimum-cost spanning tree from a weighted graph:1. Select the lowest-cost edge on the graph.2. Select the next lowest-cost edge that does not
form a circuit with the first edge.3. Select the next lowest-cost edge that does not
form a circuit with the previously selected edges.4. Continue selecting the lowest-cost edges that do
not form circuits with the previously selected edges.
5. When a spanning tree is complete, you have the minimum-cost spanning tree.
Example: Kruskal’s Algorithm
Use Kruskal’s algorithm to determine the minimum spanning tree for the weighted graph shown. The numbers along the edges represent dollars.
A
B
C
G
D
E
F
12
11
10 5
22
14
4
17
22
18
Solution
Pick the lowest-cost edge of the graph, edge CD which is $4.
Next we select the next lowest-cost edge that does not form a circuit; we select edge CG which is $5.
A
B
C
G
D
E
F
12
11
10 5
22
14
4
17
22
18
Solution (continued)
Continue selecting edges, being careful not to form a circuit.
The total cost would be$12 + $10 + $5 + $14 +$18 + $4 = $63.
A
B
C
G
D
E
F
12
11
10 5
22
14
4
17
22
18
Determine a spanning tree for the graph shown below.
a.
c.
b.
d.
Determine a spanning tree for the graph shown below.
a.
c.
b.
d.
Determine the minimum-cost spanning tree for the following weighted graph.
a.
c.
b.
d.
a.
c.
b.
d.
Kathleen is planning on installing a new computer network at her small business. Her current system has computers already in place as shown in the figure below. The numbers are shown in feet.
Determine the minimum-cost spanning tree that reaches each computer.
a.
c.
b.
d.
Determine the minimum-cost spanning tree that reaches each computer.
a.
c.
b.
d.
If the new networking system materials cost $2.20 per foot, what is the cost of installing the system
a. $79.20
b. $83.60
c. $85.80
d. $112.20
If the new networking system materials cost $2.20 per foot, what is the cost of installing the system
a. $79.20
b. $83.60
c. $85.80
d. $112.20