Transcript Chapter 15 Graph Theory © 2008 Pearson Addison-Wesley. All rights reserved Chapter 15: Graph Theory 15.1 15.2 15.3 15.4 Basic Concepts Euler Circuits Hamilton Circuits and Algorithms Trees and Minimum Spanning Trees 15-3-2 ©

Chapter 15
Graph
Theory
© 2008 Pearson Addison-Wesley.
All rights reserved
Chapter 15: Graph Theory
15.1
15.2
15.3
15.4
Basic Concepts
Euler Circuits
Hamilton Circuits and Algorithms
Trees and Minimum Spanning Trees
15-3-2
© 2008 Pearson Addison-Wesley. All rights reserved
Chapter 1
Section 15-3
Hamilton Circuits and Algorithms
15-3-3
© 2008 Pearson Addison-Wesley. All rights reserved
Hamilton Circuits and Algorithms
•
•
•
•
Hamilton Circuits
Minimum Hamilton Circuits
Brute Force Algorithm
Nearest Neighbor Algorithm
15-3-4
© 2008 Pearson Addison-Wesley. All rights reserved
Hamilton Circuit
A Hamilton circuit in a graph is a circuit that
visits each vertex exactly once (returning to the
starting vertex to complete the circuit).
15-3-5
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Identifying Hamilton Circuits
Refer to the graph below. Which of the following
are Hamilton circuits?
a) A  B  E  D  C  F  A
b) A  B  C  D  E  F  C 
EBFA
c) B  C  D  E  F  B
C
B
A
F
D
E
© 2008 Pearson Addison-Wesley. All rights reserved
15-3-6
Example: Identifying Hamilton Circuits
Solution
a) It is a Hamilton circuit for the graph.
b) It is not a Hamilton circuit since it visits B
more than once.
c) It is not a Hamilton circuit since it does not
visit all the vertices.
15-3-7
© 2008 Pearson Addison-Wesley. All rights reserved
Hamilton Circuits for Complete Graphs
Any complete graph with three of more vertices has
a Hamilton circuit.
15-3-8
© 2008 Pearson Addison-Wesley. All rights reserved
When Hamilton Circuits Are the Same
Hamilton circuits that differ only in their starting
points will be considered to be the same circuit.
15-3-9
© 2008 Pearson Addison-Wesley. All rights reserved
Number of Hamilton Circuits in a
Complete Graph
A complete graph with n vertices has (n – 1)!
Hamilton circuits.
15-3-10
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Number of Circuits
How many Hamilton circuits are in a complete
graph with 5 vertices?
Solution
Here n = 5, so there are (5 – 1)! = 4! = 24
Hamilton circuits.
15-3-11
© 2008 Pearson Addison-Wesley. All rights reserved
Minimum Hamilton Circuits
In a weighted graph, a minimum Hamilton circuit
is a Hamilton circuit with the smallest possible
weight.
15-3-12
© 2008 Pearson Addison-Wesley. All rights reserved
Brute Force Algorithm
Step 1:
Choose a starting point.
Step 2:
List all the Hamilton circuits with that
starting point.
Step 3:
Find the total weight of each circuit.
Step 4:
Choose a Hamilton circuit with the
smallest total weight.
15-3-13
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Brute Force Algorithm
Find a minimum Hamilton circuit for the
complete, weighted graph below
3
A
1
D
1
B
5
3
7
C
15-3-14
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Brute Force Algorithm
Solution
1
D
1.
2.
3.
4.
5.
6.
3
A
1
B
5 7
3
C
ABCDA
ABDCA
ACBDA
ACDBA
ADBCA
ADCBA
Weight of circuit
14
12
Min.
14
12 (opposite of 2)
14 (opposite of 3)
14 (opposite of 1)
© 2008 Pearson Addison-Wesley. All rights reserved
15-3-15
Nearest Neighbor Algorithm
Step 1:
Step 2:
Step 3:
Choose a starting point. Call this
vertex A.
Check all the edges joined to A, and
choose one that has the smallest
weight. Proceed along this edge to
the next vertex.
At each vertex you reach, check the
edges from there to vertices not yet
visited. Choose the smallest weight.
Proceed along this edge to the next
vertex.
© 2008 Pearson Addison-Wesley. All rights reserved
15-3-16
Nearest Neighbor Algorithm
Step 4:
Step 5:
Repeat Step 3 until you have visited
all the vertices.
Return to the starting vertex.
15-3-17
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Nearest Neighbor
The time it takes to travel between points (in minutes)
is shown on the graph below. Use the nearest
neighbor algorithm to approximate the least time to
visit each location.
A
9
B
10
20
9 3
13
8
C
8
7
E
5
D
15-3-18
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Nearest Neighbor Algorithm
Solution
Start
Circuit
Weight
A
ADEBCA
40
B
BEDCAB
37
C
37
D
CDEBAC
DEBACD
E
EBDCAE
51
37
15-3-19
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Nearest Neighbor Algorithm
Solution (continued)
From the information on the previous slide we would
say that a route of 37 minutes should be about the
least amount of time.
Note that A  B  E  C  D  A
has a total weight of 36 minutes. All we can expect
from the nearest neighbor algorithm is a reasonably
good solution.
15-3-20
© 2008 Pearson Addison-Wesley. All rights reserved