Embed Size (px)
Citation preview
Graph Theory:Traveling Salesman Problem (TSP)
E3 Teacher Summer Research Program 2005
Texas A & M UniversityJune 29,
2005
TAKS Objectives9th & 10th Grade
Objective 1 – Ab1A, B, C, D & E
Objective 2 – Ab2A & D, Ab3A & B
Objective 3 – Ac1A & C
Objective 6 – 8.7D
Objective 10 - 8.14A, B & C, 8.15A, 8.16A & B
Material
GEO Board
Yarn
Rubber bands
Colored pencils
Grid paper
Graphing calculator
Vocabulary
Nodes (vertices)Edges (arcs)DegreeAdjacentPath
Vocabulary
LengthCircuitSimple GraphComplete Graph
Nodes (vertices)
Nodes (vertices)
Land Phone numbersPeopleJunction points (electric circuits)AtomChess playersCompanies or industries
Edges (arcs)
Edges (arcs)
Bridges connecting landCalls made from one number to another connecting phone numbersRelationships or acquaintances connecting by peopleWires connecting junction points (electric circuits)Bonds between atoms connecting atomsMatches connecting chess players (chess tournament)Transactions connecting companies or industries
Degree
A
F
H
G
B
E
D
C
Degree
Node A is of two degrees
Node B is of two degrees
Node C is of six degrees
Node D is of three degrees
How many degrees are nodes E, F, G and H?
Adjacent
A
F
H
G
B
E
D
C
Adjacent
Node A is adjacent to Nodes C and HNode B is adjacent to Nodes F and HNode C is adjacent to Nodes A, D, E, F, G and HNode D is adjacent to Nodes C, E and HWhat nodes are adjacent to nodes E, F, G and H?
Path
A
BC
D
E
F
Path
A
BC
D
E
F
Path
A
BC
D
E
F
No Path
A
BC
D
E
F
Draw two different paths.
A
BC
D
E
F
LengthA
BC
D
E
F
LengthA
BC
D
E
F
LengthA
BC
D
E
F
LengthA
BC
D
E
F
CircuitPoint A is the starting point
C
B D
A E
CircuitPoint A is the starting point
C
B D
A E
CircuitPoint A is the starting point
C
B D
A E
CircuitPoint A is the starting point
C
B
D
A
E
Simple Graphs
Not, Simple Graphs
Complete Graphs
Complete Graphs
Draw a complete graph with six nodes.
Draw a complete graph with seven nodes
Petroleum Delivers
745
325
490
825
520
565
370
570
565380
S
T
N
M
C
T TexasS South CarolinaN New YorkM MinnesotaC Colorado
1 Node
2 Nodes
2 Nodes
3 Nodes
3 Nodes
3 Nodes
3 Nodes
3 Nodes
3 Nodes
3 Nodes
3 Nodes
2 Routes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
4 Nodes
6 Routes
2 Node
Node(s)
Process Column Route(s)
2 1
3 Nodes
Nodes
Process Column Route(s)
3 2
2 1
4 Nodes
Nodes
Process Column Route(s)
4 3
3 2
2 1
2 Nodes
Node(s)
Process Column Route(s)
2 2-1 1
3 Nodes
Nodes
Process Column Route(s)
3 3-1 2
2 1
3 Nodes
Nodes
Process Column Route(s)
3 3-1 2
2 2-1 1
4 Nodes
Nodes
Process Column Route(s)
4 4-1 3
3 2
2 1
n r1
4 Nodes
Nodes
Process Column Route(s)
4 4-1 3
3 3-1 2
2 1
n r1
4 Nodes
Nodes
Process Column Route(s)
4 4-1 3
3 3-1 2
2 2-1 1
n r1
4 Nodes
Nodes
Process Column Route(s)
4 4-1 3
3 3-1 2
2 2-1 1
n n-1 r1
Function Ruler1 equals number of routes from each remaining nodes
r1 = n - 1
4 Nodes3
2
1
Total Original Routes4 Nodes
3 Routes
1
2
3
2 Routes
21
21
21
1 Route11
1
1
11
Total Original Routes
3 x 2 x 1 = 6
or
(4-1)!=6
Function Ruler2 equals number of unique routes
2
2)!1(
rn
Petroleum Delivers
745
325
490
825
520
565
370
570
565380
S
T
N
M
C
T TexasS South CarolinaN New YorkM MinnesotaC Colorado
