Section 7.1: TreeWhat is a NETWORK ?

Vertices of networks are often called _________________________________ and represent “objects”

Edges of networks are called ________________ and indicated connections among objects

CHAPTER THEME: Finding __________________ networks that connect a set of points

OPTIMAL NETWORK GOALS:1) Connect all the “TERMINALS” = Vertices

2) Make the total cost (length) of the network as small as possible (optimal)

Example Network Problem: The weighted graph represents a system of roads connecting buildings of a company’s headquarters. The company is planning to upgrade its fiber optic cable through underground lines connecting its servers. The weights represent cost in millions of dollars to upgrade these cables between each building. The goal for the company is to find a network that utilizes the existing network of roads; connects all the buildings together; has the least cost.

Key Term Requirements of NETWORK in the example:

1) SUBGRAPH :

______________ come from the _____________________ graph

2) SPAN :

A graph the must include _______________________________ of the original graph

3) MINIMAL :

________ _____weight of the network must be as _______________________ as possible

EXAMPLES: Based on the original graph, draw different examples of subgraphs and spans. Original

A

BC

D E F

Exp 1: SUBGRAPH of 4 Vertices/ 6 Edges

Exp 2: SPAN of 5 Edges

A

B

CD

E3.1

2.5

2.9

2.7

2.2 2.8

2.6

1.8

2.4

2.1

Exp 3: SUBGRAPH of 5 vertices/ 4 edges

Exp: SPAN of 6 Edges Exp 3: SUBGRAPH of 6 vertices and 5 edges

SPECIFIC TYPE OF NETWORK:TREE:

Which of the following graphs a – h represents a tree: Why or why not?

Observation Questions for Graphs that are TREES: 1) Is there a relationship between edges and vertices?

2) Is there a special term you can use to describe the edges used?

SPANNING TREE:

EXAMPLE: For the given graph (right), find a spanning tree.

A B

CD

E

(a)(b) (c)

(d)

(e)(f)

(g) (h)

PROPERTIES # 1 – 3 OF TREES: PROPERTY 1: o In a tree, there is one and only one PATH joining any two vertices. o If there is one and only one path joining any two vertices of a graph, the

graph must be a tree.

PROPERTY 2: o In a tree, every edge is a BRIDGE. o If every edge of a graph is a bridge, then the graph must be a tree.

PROPERTY 3: o A tree with N VERTICES has N-1 EDGES. o If a NETWORK has N vertices and N-1 edges, then it must be a tree.

Practice Problems: A graph G has no loops or multiple edges. Identify if the graph (1) ALWAYS, (2) NEVER, or (3) SOMETIMES is a tree. 1) G has 4 vertices and 3 edges.

2) G is connected with 4 vertices and 3 edges

3) G has 5 vertices and 5 edges.

4) G has 5 vertices and 4 bridges.

5) G has 10 vertices and 9 edges.

6) G has 10 vertices and 11 edges.

7) G has 12 vertices and 11 bridges.

8) G is connected with 8 vertices and every vertex has degree 7.

HW 7.1: p. 259 # 1, 2, 3 – 7 (odd)

Section 7.2: Spanning TreeSPANNING TREE: a subgraph of a network that connects all the vertices and has no circuits

1A. Is the following graph a tree? Why?

1b. Find a spanning tree in the graph.

2A. Is the following graph a tree? Why?

2b. Find a spanning tree in the graph.

A B C

D

E F G

A B C

D

E F G

PROPERTY 4 of Trees:o If a network (CONNECTED GRAPH) has V vertices and E edges, then E ≥ V - 1.

REDUNDANCY =

o If E = V - 1, then the network is a tree; if M > V - 1, the network has circuits and is not a tree. o If R = 0; o If R > 0;

A

B C

D

E

FG

H

If Redundancy is positive, (R > 0), then many SPANNING TREES exist in network

How do you find spanning trees?. 1) Your goal is to remove R edges to create a tree. (Break Up Circuits)2) DO NOT remove edges that are Bridges.

EXAMPLE #1:1) How many edges do you need to remove?

2) How many possible spanning trees exist in this graph?

EXAMPLE #2: How many edges do you need to remove?

METHOD #1: FIND BY DRAWING

METHOD #2: How might we know when we’ve found all spanning trees?

EXAMPLE #3: How many edges do you need to remove?

REMOVE CD REMOVE CH REMOVE HD

A

B C

D

E

FG

H

A

B C

D

E

FG

H

METHOD #1: FIND BY DRAWING

METHOD 2: How might we know when we’ve reached finding all spanning trees? SPANNING TREE PRACTICE

For each of the below graphs, find the total number of possible spanning trees.

REMOVE CD REMOVE CH REMOVE HD

1) 2) 3)

4)

8)

5)

HW 7.2: p. 260 #11 – 12 (all), 16 – 18 (all)

6) 7)

Section 7.3: Kruskal’s AlgorithmMINIMUM SPANNING TREE (MST):

How many spanning trees exist in each of the given networks?Network #1: Network #2: Network #3:

How can we find the minimum spanning tree without checking all of the possible spanning trees?

Kruskal’s Algorithm for N vertices on a Network.Step #1: Pick the cheapest edge available.

o In case of a tie, randomly pick one edge.

o Mark it or somehow identify you chose it.

REPEAT: Pick the next cheapest link or edge available and mark it.

RULE #1 : Cannot make a Circuit by choice

RULE #2 : Stop once you’ve chosen N-1 Edges

RULE #3 : Vertices are allowed to have any degree > 0

710

11

13

9

8

53

6

14

9

15

2314

8

197

18

2123

36

4532

811

2942 5

4

12

17

14

Example: The following graph represents roads connecting seven towns in a developing jungle region that is planning to lay fiber-optic cable. The cost of laying the fiber optic cable is represented by the weights of graph. Let’s find the minimum spanning tree (MST) using Kruskal’s Algorithm.

Positives of Kruskal’s Algorithm:

EXAMPLE PROBLEMS:A

B

CD

E

2.22.32.1

2.4

2.8 3.5

2.63.1

1.4

1.2

B

C D

E F

A

5

2040

4015

45 5 25

35

45

2010

7.3: pp. 260 – 261 # 19 – 25 (odd)

7.4 Introduction TrianglesGEOMETRY - TRIANGLES REVIEW:

BASIC TRIANGLE RELATIONSHIPS: ANGLES v. SIDE LENGTHS

LONGEST Side of a triangle is opposite (across from) the ____________________________ angle

SHORTEST Side of a triangle is opposite (across from) the _________________________ angle

MIDDLE Side of triangle is opposite (across from) the ________________________________ angle.

Triangles are Not Drawn to Scale!!

RIGHT TRIANGLES:State the Pythagorean Theorem Label the Right Triangle based on your statement.

PRACTICE: Find the value of the missing side x. (Triangles not drawn to scale)

SPEICAL RIGHT TRIANGLES: 30 0 – 60 0 – 90 0 ALL 300 – 600 – 900 triangles are ____________________ and the sides are proportional.

4

8x

6

6x

5

12

x

3

5x

A

B

C10

7 6

Which angle is the largest?

Which angle is the smallest?

Which angle is the middle?

P

Q

R

35º47º

What is measure of angle Q?

Which side is the shortest?

Which side is the longest?

105º

A

B

C

10 14

Which angle is larger A or C? Why?

Write an inequality to describe the length of AC.

D E

F

27º18 18

X Z

Y

33º

2519

What is measure of angle E and F?

Is DE longer or shorter than 18? Why?

What is of angle X or Z larger? Why?

What is the largest total of angle X and Z?

What value must angle Y be larger than?

HYPOTENUSE =

SHORT Leg =

LONG Leg =

To move between sides because sides are always proportional.

Bigger to Smaller = ________________ Smaller to Bigger = _____________

PRACTICE: Use your knowledge of 30 0 – 60 0 – 90 0 to find all side lengths.

600600

50

300

100600

8

300

15

300

40

600

30

300

25 600

10

300

600 900

GENERAL TRIANGLE PRACTICE IDENTIFY missing information as largest, smallest, or middle angles and lengths.

If possible, provide an exact value for missing measurement. If not possible, then provide approximate angle or side values with inequalities.

A

BC

1518

20

A

BC

36

450

200 A

BC530

110 89

A B

C

120

600 800

A

BC350 350

10A

BC

420

6570

A

BC

A B

C

23 23800100

1300

57

41

A ________________

B ________________

C ________________

AC_______________

CB_______________

B ________________

A _______________

C ________________

CB_______________

AC_______________

CB_______________

C ________________

AC_______________

CB_______________

A ________________

BC_______________

C _______________

A ________________

AC_______________

A_______________

CB_______________

B_______________

C ________________

Section 7.4: The Shortest Network Connecting Three Points

Minimum Spanning Tree (MST): The spanning tree of a given network that has the least weight

MST is the ___________________________________________ way to connect a set of vertices based on the

assumption that all ______________________ should belong to the _______________________NETWORK.

Example: Consider the network (right) that describes 3 isolated towns (A, B, C) which are equidistant from each other and located in the heart of the Australian outback. The towns are connected by three unpaved straight roads as described by the edges. What is the minimum spanning tree solution to this network?

Could there be a shorter network than the Minimum Spanning Tree (MST) in a graph?What would we need to do to the graph to check for shorter answer?

SHORTEST NETWORK = the network of all vertices that has the ____________________ total weight

without __________________________ on the edges you can use or create.

Example: Do power cables have to follow existing roads OR highways?

EXAMPLE #1 OF ADDING NEW VERTEX, T, AND EDGE: Vertex T is at the midpoint of the edge BC and creates edge TA.

What is the length of TA?

What is the total weight of the T-Network (AT, BT, CT)?

Is this T-Network shorter than the MST?

500 miles

500 miles

500 miles

A

B C

500 miles

A

B C

500miles

500miles

500 500

A

B C500 miles

EXAMPLE #2 OF ADDING NEW VERTEX, Y, AND EDGES: Put a new vertex, Y, in the exact middle of the triangle and create edges YA, YB, and YC. (Y-NETWORK).

What is the length of YA, YB, and YC?

What is the total weight of the Y-Network (YA, YB, and YC)?

Is this Y-Network shorter than the MST and the T-Network?

What is important about the angles the edges of the Y-network form? _____________

Shortest Network Key Terms:JUNCTION Point : Any point where ________________________________ segments (edges) of the network come together.

NATIVE JUNCTION Point: a junction point located at an ______________________ vertex of the network

INTERIOR JUNCTION Point: a “nonnative” junction point located somewhere __________________ than an __________________________vertex of the network

STEINER POINT: A specific ________________________ junction point consisting of three

line segments (edges) coming together forming equal ___________ angles

7.4 Shortest Network Continued…

TRUE STORY EXAMPLE: In 1989, a consortium of several of the world’s biggest telephone companies (AT&T, MCI, Spring, and British Telephone) completed the Third Trans-Pacific Cable (TPC-3) line, a network of submarine fiber-optic lines linking Japan and Guam to the US (via Hawaii). The approximate straight line distances between the three locations was 1620 Japan to Guam, 3910 Japan to Hawaii, and 3820 Guam to Hawaii (in miles). The solution to these companies problem was to find the SHORTEST NETWORK between the Japan-Guam-Hawaii Triangle.

SOLUTION: _________________________________________ of the Triangle to create a network of 5690 miles.

Is the SHORTEST NETWORK for THREE POINTS always a STEINER POINT? Use the transparency of 120o angles to check if a Steiner Point Exists in each triangle.

SHORTEST NETWORK CONNECTING THREE POINTS:

If the LARGEST angle of the triangle is _____________________________________, the shortest network linking the three vertices consists of the two shortest sides of the triangle. In this situation, the shortest network is the minimum spanning tree.

If LARGEST angle of the triangle is _______________________________________, the shortest network is obtained by finding a STEINER POINTS inside the triangle and joining S to each of the vertices.

INTERIOR JUNCTION RULE for Shortest Networks:In a shortest network the interior junction points are all Steiner Points.

Japan

Guam

Hawaii

YES or NO

YES or NO

YES or NO

YES or NO YES or NO

Consider the triangle ABC with interior junction points of P, Q, and R. We know that triangle ABC has no angle larger than 120o. The table below gives the lengths of all the created edges from each junction point to the vertices of the triangle. In each table, which junction point must be the STEINER POINT?

P Q R

A 54 72 41B 61 94 87C 125 77 104

P Q R

A 380 390 700B 620 680 260C 300 190 550

SUMMARY OF TYPE OF SHORTEST NETWORKAngles Largest Angle ≥ 120 Largest Angle < 120

Shortest Network Minimum Spanning Tree (MST) STEINER POINT (Y – NETWORK)

Weight of

Network

Pick the two smallest sides of the original triangle.

Draw the y-network and find the 3 weight of the edges drawn.

(Normally requires trig)

Special Case: For an equilateral triangle, use the 30-60-90 triangle knowledge.

Example

Apply the Shortest Network for Connecting Three PointsFor each triangle, identify the shortest network as a Minimum Spanning Tree (MST) or a Steiner Point.

Hint: Determine exact or approximate size of all angles in the triangle.

TRIANGLES ARE NOT DRAWN TO SCALE!!

83º

38º 59º

22º

19º

118º

25 25

2537

37

27º

30º

6342

15º

30

35

25º