Year 12 Maths A Textbook - Chapter 7

Maths A Yr 12 - Ch. 07 Page 355 Friday, September 13, 2002 9:33 AM

7

Networks

syllabus reference
eference
Elective topic
Operations research —
networks and queuing

In this chapter
chapter
7A
7B
7C
7D

Networks, nodes and arcs
Minimal spanning trees
Shortest paths
Network ﬂow

Maths A Yr 12 - Ch. 07 Page 356 Wednesday, September 11, 2002 4:24 PM

356

M a t h s Q u e s t M a t h s A Ye a r 1 2 f o r Q u e e n s l a n d

Introduction to networks
Mathematical models may be computer programs, drawings, a system of equations or a
combination of these. Through models, people attempt to understand real situations. A
postman plans the shortest delivery route or a builder schedules jobs on a large construction project so that the formwork is done as soon as the foundations are completed
and the plasterers do not arrive before the walls have gone up.
Models allow these people to think about and plan tasks before actually doing them.
In particular operations research is the science of planning and executing an operation to make the most economical use of available resources.

Networks are maps that can represent an amazing variety of different things: simplified
maps, relationships between people, sub-tasks in a building project, computer terminals
or the flow of traffic through a city. In each case the network provides a means of
studying real-life situations so that decisions can be made. When drawing a network,
irrelevant information, such as bends in the roads of a map, is ignored.
1. A network is a collection of objects connected to each other in some way.
2. Networks are made up of nodes joined by arcs. If nodes are connected they are
joined by an arc.
3. When the arcs have arrows they are called directed networks and travel is
possible only in the direction of the arrows.
There are many examples where networks can be used to model a situation. The first
worked example uses a network to plan a drive that takes the shortest possible path or
distance.
The network can be drawn and each node labelled. A path is a specific set of arcs
connecting nodes and can be represented by the letters in the nodes, as we will see.


357

Chapter 7 Networks

WORKED Example 1
George and Effie want to drive from Airlie to Gillespie
using the map at right.
a Draw a network which represents the map.
b Given that each road taken must bring them
closer to Gillespie, list the number of ways from
Airlie to Gillespie. How many ways are possible?
c Identify the shortest path from the possible routes
in b.

68 km

Charles

Friday
Moon Mountain

66 km

39 km

48 km
50 km

Davis

Airlie

60 km

46 km

Barnard

THINK
Represent towns with circles,
called nodes, labelled with the
first letter of the town.
Ignore the bends in the roads and
use straight lines to represent
roads connecting the towns.
Check that towns not connected
by roads on the map are not
joined with an arc.

a

Each road taken from Airlie must
go towards Gillespie. Indicate the
direction on each arc with an
arrow.

b

54 km

Ellis

83 km

WRITE/DRAW

a

Lake Kawana
41 km
Gillespie

1

2

3

b

1

66 39
A
46

60
B

C

3

c

Use the network to list the
number of ways from A to G.

Answer the question.

1

Add the lengths of the nodes to
calculate the distances of the 6
routes in part b.

2


G

D
41 54
83

68

E

F
48 50

D
60

F
48 50

66 39
A
46
B

2

68

C

83

G
41 54
E

A–B–D–E–G
A–B–D–F–G
A–B–E–G
A–C–D–E–G
A–C–D–F–G
A–C–F–G
There are 6 ways to go from Airlie to Gillespie.
c A–B–D–E–G (60 + 46 + 41 + 54) 201 km
A–B–D–F–G
204 km
A–B–E–G
197 km
A–C–D–E–G
200 km
A–C–D–F–G
203 km
A–C–F–G
184 km
The shortest path is A–C–F–G: Airlie to
Charles to Friday to Gillespie.

If Effie and George were more concerned with time, rather than distance, they might
have consulted their travel adviser about the times for each of these stages and redrawn
the network with the arcs representing average times for travelling on each connecting
road. This network would help them to find the shortest time.


358


WORKED Example 2
Efﬁe consults the local travel adviser about the travel times
for the stages in the journey planned in worked example 1.
She then redraws the network with the average time
(in minutes) taken to drive between the towns as shown
at right. Which path would take the least time and what is
that time?
THINK

C
29

50
A

68

F
36 38

D

45

36
B

67

31
E

41

G

WRITE

1

List all the possible paths and the times
they will take.

2

The path of least time is ACDEG.

A-B-D-E-G
153 min
A-B-D-F-G
155 min
A-B-E-G
153 min
A-C-D-E-G
151 min
A-C-D-F-G
153 min
A-C-F-G
156 min
The path ACDEG takes 151 min, the least time.

So, Efﬁe and George would plan different routes depending on whether they were interested n shortest distance or shortest time. In addition to distances and times, arcs may
also represent other relationships between nodes. In the following worked example we
look at cost relationships between nodes.

WORKED Example 3
The costs of connecting various locations on a university campus with computer cable are
given in the table below. A blank space indicates no direct connection.
A

B

A

——

4000

B

——

——

C

——

D

——

C

D

E

5000

3000

1500

2200

4500

——

——

2200

1500

——

——

——

2500

Draw a network to represent this situation, showing the cost of connection along each arc.
THINK
1

There are 5 nodes. Draw them as
labelled circles. Because A and C have
3 connections, put them on the outside.

WRITE/DRAW
B

A
D
E

C


359

Chapter 7 Networks

THINK
2

WRITE/DRAW

From the table insert, in a systematic
way, each arc and label each arc with
its cost.

A
3000

4000
2200
5000
D

4500
1500

2500
E
1500

B

2200
C

remember
remember
1. A network is a collection of objects connected to each other in some speciﬁc
way.
2. A network consists of nodes which may be connected by arcs.
3. In a directed network, the arcs will have a direction indicated by arrows.
4. Networks can be used to model situations and calculate shortest paths.

7A


1 Examine the network at right. (All the lengths are in metres.)
a Which is the longest path?
1c
b Which is the shortest path?

A

WORKED

Example

6m
C
4m

5m
B
3m
D
E 9m

2 A traveller plans a journey from Ulawatu to
Example
Yallingup
Bargara (shown on the road map at right).
160 km
1
a Draw a network to represent this situation. 120 km
b Calculate the longest path if no road is
118 km
travelled twice.
Ulawatu
Bargara
c Calculate the shortest path.
45 km
Angourie
d The travelling times between each town are:
100 km
Ulawatu–Yallingup
85 min
109 km
Ulawatu–Black Rock
75 min
Black Rock
Yallingup–Angourie
80 min
Black Rock–Angourie
82 min
Yallingup–Bargara
120 min
Angourie–Bargara
34 min.
i Draw a network of this situation showing
the time taken to travel between towns on
each arc of the network.
ii Calculate the longest time taken to travel
from Ulawatu to Bargara.
WORKED


360


iii Calculate the shortest time taken to travel from Ulawatu to Bargara.
iv Complete the table showing the shortest distance between each of the towns.
Black Rock
Ulawatu

Yallingup

Angourie

Ulawatu

——

120

100

Yallingup

——

——

220

Black Rock

——

——

——

Angourie

——

——

——

Bargara

209

——

v Produce a similar table showing the travelling times between each of the towns
shown on the map.
3 A traveller plans a journey from Renoir to
85
Gauguin. The distances between various nearby
Matisse
2
Pissarro
towns are shown on the map at right.
62
Renoir
a Calculate the shortest path.
46
60
b The travelling times between the following
Van Gogh
58
41
65
towns are:
38
Monet
Renoir–Pissarro
47 min
30
Renoir–Monet
44 min
46
Monet–Cezanne
40 min
75 Gauguin
Pissarro–Cezanne
45 min
Cezanne
Pissarro–Van Gogh
34 min
Pissarro–Matisse
75 min
Pissarro–Monet
25 min
Cezanne–Van Gogh
20 min
Van Gogh–Matisse
38 min
Cezanne–Gauguin
59 min
Matisse–Gauguin
28 min
i Draw a network of this situation showing the time taken to travel between towns
on each arc of the network.
ii Calculate the longest time to travel from Renoir to Gauguin.
iii Calculate the shortest time to travel from Renoir to Gauguin.
c Complete the table below showing the shortest distance between each of the towns.

WORKED

Example

Renoir

Pissarro

Monet

Cezanne

Van
Gogh

Renoir

——

Pissarro

——

——

41

Monet

——

——

——

——

——

——

Van Gogh

——

——

——

——

——

Matisse

——

——

——

——

——

Gauguin

——

Cezanne

Matisse

179
123

——

d Produce a similar table showing the travelling times between each of the towns
shown on the map.


Chapter 7 Networks

WORKED

Example

361

4 The cost of trips on McFlaherty’s Bus service are given in the table below.

3

Port St

Land St

Port St

——

2.40

Land St

——

——

Tork Rd

——

——

Bell St

——

——

Tork Rd

Bell St

Key St
1.80

2.40

1.50

——

1.80

1.50

——

——

2.00

a Draw a network representing this information.
b What is the minimum cost of travelling from Port St to Tork Rd?
c What is the minimum cost of travelling from Bell St to Port St?

The distances, in kilometres, between towns in a region are given in the table below.
Note: Where a blank appears no direct link between the towns exists.
Grantha

Tamwor

Armida

Beech

Kianga

85

104

122

Grantha

——

Tamwor

——

——

43

Armida

——

——

——

100
85

In a big storm the bridge on the Armida to Beech road was washed out. How far is the
journey from Beech to Armida now?
A 163 km
B 128 km
C 189 km
D 154 km

The diagram at right represents a farm complex.
240
Sheds
Workshop
Each site needs to be connected directly or
250
200
indirectly to the transformer so that it can get
200
Garage
250
electrical power. For example, the garage can get
150
250
its power directly from the transformer or indirectly
from the house, if the house is connected. The
Pump
House
350
numbers represent the distance between each
site. How should the connections be arranged so
390
that the minimum length of cabling is used?
350
400
To answer this question in a systematic way
Transformer
we consider the following aspects of networks.
A tree is a series of connections in a network that does not contain a loop.
A spanning tree in a network is a tree that contains each node.

HEET

SkillS

5 multiple choice

7.1


362


To identify a minimal spanning tree, we use the minimal spanning algorithm
which has the following steps.
Step 1 Choose any node at random and connect it to its closest neighbour.
Step 2 Choose an unconnected node which is the closest to any connected node.
Connect this node to the nearest connected node. (If two or more nodes
are nearest; that is have the same value, just select any one.)
Step 3 Repeat step 2 until all the nodes are connected.
The minimal spanning algorithm can be used to determine the least length of cable
needed to connect each building of the farm complex considered above.

WORKED Example 4

240

Sheds

Find the minimal spanning tree to determine the
minimum amount of cable needed to connect all the
buildings in this farm complex to the transformer.
Distances between locations are shown in this plan
and are in metres.

Workshop

250
200
Garage

250
150

200
250

Pump

House

350
390
350

400

Transformer

THINK
1
2
3

4
5
6

7

Draw a network with nodes using the ﬁrst
letter of each building.
Use dotted lines for the arcs and label each
arc with distances between the nodes.
Start with the transformer and ﬁnd the
shortest arc. The unconnected node closest to
T is P, so join T to P with an arc.
Find the unconnected node closest to P or T.
It is G. Connect P and G with an arc.
Find the unconnected node closest to P, T, or
G. It is W. Connect G and W with an arc.
Find the unconnected node closest to P, T, G
or W. It is H. Connect W and H.
The sheds, S, are still not connected. Find the
node closest to P, T, G, W or H which is
closest to the unconnected node S. Connect
W and S with an arc.

WRITE/DRAW
240

S

W

200

200

250
250

250

G

150

H
350

P
390

400

350
T

240

S

W
200
200

150

G
H

P
350
T

8
9

Add up the lengths in the minimal spanning
tree.

350 + 150 + 200 + 200 + 240 = 1140
The minimal length of cable to connect
the buildings is 1140 m.


Chapter 7 Networks

363

For the minimal spanning tree in the previous worked example it does not matter which
node was used as the starting point. The same spanning tree would have resulted. However, suppose that the distance between the sheds and the pump had been 240 — the
same distance from the sheds to the workshop. Then we could have chosen the final
arc as either SW or SP but not both. However, the total length of the minimal spanning
tree would have been the same.

History of mathematics
JOHN FORBES NASH (1928–)

When the movie A Beautiful Mind won an
Oscar for best film in 2002, John Nash was in
the audience. The movie, based on a book by
the same name, is his story.
John Nash was born in Bluefield, West
Virginia in the United States. His schoolteachers did not recognise his brilliance and
they focussed on his lack of social skills.
His mother was a schoolteacher who
encouraged his love of books and
experiments. One of his chemistry
experiments with explosives caused the death
of a school friend. He enjoyed Compton’s
Pictured Encyclopedia, and the book, Men of
Mathematics by E T Bell, first excited him
about mathematics. He succeeded in proving
difficult mathematical problems such as
Fermat’s Theorem for himself.
He entered Carnegie Technical College in
Pittsburgh to follow his father’s footsteps in
engineering. He moved to chemistry to avoid
the rigidity of mechanical drawing. Then,

encouraged by the mathematics faculty, he
moved from chemistry to major in
mathematics, realising that it was possible to
make a good career in America as a
mathematician.
He excelled in mathematics and graduated
with an MS as well as a BS because of his
advanced mathematical knowledge. On
graduation from Carnegie, where an elective
course in international economics influenced
his mathematical ideas, he was offered
fellowships at both Harvard and Princeton.
In 1948, he chose Princeton where he was
closer to his family in Bluefield. He avoided
lectures and studied on his own, and was full
of mathematical ideas. His interest in game
theory grew and he developed the mathematics of equilibrium strategies to predict
behaviour. In two papers Equilibrium Points
in n-person Games and Non-cooperative
Games, Nash proved the existence of a strategic equilibrium for non-cooperative games,
the Nash equilibrium, and suggested
approaching the study of cooperative games
by their reduction to non-cooperative form. In
his two papers on bargaining theory, he
proved the existence of the Nash bargaining
solution and provided the first execution of
the Nash program.
He was awarded the Nobel Prize in
Economic Science in 1994, for this work on
game theory 45 years earlier.
In the movie, A Beautiful Mind, we see a
version of how his ideas were stimulated by
thinking about non-predictable strategies in a
bar scene. In another scene we see him


364


mapping the interactions between pigeons
and saying that he is developing an algorithm
to predict their behaviour. An algorithm is a
step-by-step procedure for a particular mathematical problem and is the idea that lies at
the heart of all the computer programming
and the code which drives digital computers.
After obtaining his degree in 1950 he
worked as an instructor at Princeton but
moved to the mathematics faculty of
Massachusetts Institute of Technology (MIT)
where he met his wife, Alicia, a physics
graduate. In 1958 he was described as the
most promising mathematician in the world.
He became mentally disturbed in 1959 when
Alicia was pregnant.
Nash attributes his recovery from mental
illness to a determined effort to think
rationally, aided by light mathematical work.
He rejected his delusions and in his
acceptance speech for the Nobel Prize in

1994 said, ‘I am still making the effort and it
is conceivable that with the gap period of
about 25 years of partially deluded thinking
providing a sort of vacation, my situation
may be atypical. Thus I have hopes of being
able to achieve something of value through
my current studies or with any new ideas that
come in the future.’
In 1999 John Nash was also awarded the
Leroy P Steele Prize by the American
Mathematical Society for contributions to
research.
Questions
1. Which book first stimulated John Nash’s
interest in mathematics?
2. Which two prizes did John Nash receive?
3. What is an algorithm?
Research
1. Find out about game theory.
2. What opportunities are there to study
mathematics after finishing school?

WORKED Example 5
The cost, in dollars, of connecting 7 offices with a computer network is given in the table.

A
——
——
——
——
——

A
B
C
D
E

B
45
——
——
——
——

C
70
150
——
——
——

D
100
50
100
——
——

E
65
90
85
40
——

F
140
95
50
55
70

Use the minimal spanning algorithm to calculate the minimum cost of connecting the offices.
THINK
1
2

Draw a network to represent the
information given in the table.
Select any starting point, say C.

WRITE/DRAW

A
140

45
70
B

100

65
F

95
90

55
70

150
50

E

50
100

C
85

D

40


Chapter 7 Networks

THINK
3

365

WRITE/DRAW

Identify the shortest arc connected to C.
This is arc CF.

A
140

45
70
B

100

65
F

95
90

55
70

150
50

E

50
100

C

D

40

85

4

Identify the shortest arc connected to C
or F to an unconnected node. This is arc
FD.

A
140

45
70
B

100

65
F

95
90

55
70

150
50

E

50
100

C

D

40

85

5

Continue, using the minimal spanning
algorithm to get the ﬁgure opposite.

A
140

45
70
B

100

65
F

95
90

55
70

150
50

E

50
100

C

D

40

85

6

Use the minimal spanning tree to
answer the question.

The minimum cost of linking the ofﬁces is
$45 + $50 + $50 + $55 + $40 = $240.


366


remember
remember
1. A spanning tree connects all nodes in the network and does not contain any
loops.
2. A minimal spanning tree is the smallest spanning tree.
3. To ﬁnd the minimal spanning tree use the minimal spanning tree algorithm.
Step 2 Choose an unconnected node which is the closest to any connected
node. Connect this node to the nearest connected node.
Step 3 Repeat Step 2 until all nodes are connected.

7B
WORKED

Example

4


1 Find the minimal spanning tree for each of the following networks.
a
b
B
B
4

8

4

A

D

5

A

5

9

6

7

4

C

c

C

12

A

D

d

20

A

D
40

21

30
18

C
17
B

D

20
40

30
15

15

C

E

2 The rail authority plans to connect the country
centres shown with a rail network (distances are
in kilometres). What is the minimum length of
track required to achieve this? Use a minimal
spanning tree algorithm as follows.
a Begin at Pallas and connect it to its nearest
neighbour. Which town is this?
b Which unconnected town is closest to Pallas
or to the town selected in a?
c Connect this town to the existing link in the
shortest way possible.
d Continue by connecting the closest
unconnected nodes to any connected ones,
one at a time, until all nodes are connected.

30

B

E

15

Yule
65
View

42
Zenith
50

80
70

Pallas

52
88
82

65

79

Xavier

50

55

Walga

88
67
Rockdale
55
52
Urchin
62
Sturt


367

Chapter 7 Networks

3 The paths between the various cages at the
Monkeys
Nolonger Park Zoo are dirt and when it rains
70
55
65
they become muddy. The figure at right shows
Crocodiles
65 Lions
50
all paths, with distances in metres. Management
has decided to put in concrete paths.
Kiosk
a What total length of path would be required if
30
60
60
50
each dotted line was to become a concrete path?
b Use the minimal spanning tree algorithm to
find the minimum length of concrete path
Entrance
Birds
80
that is required so that patrons could
see each exhibit and visit the kiosk without walking on a dirt path.
c Repeat the minimal spanning tree algorithm using a different starting point and
show that it does not matter where you start.
4 Use the minimal spanning tree algorithm to find the minimal spanning tree for the
following networks.
30
a
b
54
54
C
E
B
E
A
23
18

20

45

D
18

B

31

23
C

40

A
C

D

20

d

B
30

30

23

A

F

24

20

50

6
B
7

23
6

C

E

D

9

4

8

8

G

40

F
6

E

5 Find the minimal spanning tree for each of the following networks.
a
b
c A
D
12
14
D
18
17
15
A

F
20

22

G
E

15

I
5

C
15

B

18
22

6 A number of small, private mines have opened
up in Waller Flats and the local shire council
wants to link them by bitumen roads as
shown in the figure at right. What is the
minimum length of road that is needed?
(Assume the only connections that can be
made are those marked on the map of
Waller Flats at right.)

F

E

G

D

5

5
J

12

5

5

8

8

H
5

12

C

F

5

8

8

22
17

10

12
13

10

18

15

B

E

12
14

C
B

A

10

18

F

48

7

7

50

F

D

48

45

60

55

A

c

45

8

8

G

K

5

Mine 1
Mine 4
5 km
15 km
Mine 5

Mine 2
6 km
14 km

Mine 6

12 km
10 km Mine 3
7 km
15 km

11 km
Mine 7


368


7 In question 6 the dotted lines connecting the mines represent dirt roads. If an
inspector wants to visit all the mines and is willing to travel on dirt roads, what is the
shortest distance he or she needs to travel to visit each of them, starting from Mine 1?
WORKED

Example

5

8 A gas pipeline is to be connected between 5 towns so that each town has at least one
connection to the system. The gas pipeline costs $25 000 per kilometre. The distance
(in km) between the towns is given in this table.
A

B

C

D

E

A

——

16

23

10

43

B

——

——

32

17

19

C

——

——

——

35

43

D

——

——

——

——

38

a Find the length of the network connecting these towns in the shortest way.
b What is the cost of this connection?
9 An ofﬁce computer system requires the linking of 8 terminals. Each terminal has to
have at least one connection with the system. The cost (in dollars) of connecting each
terminal with another is given in the table.
A

B

C

D

E

F

G

H

A

——

35

50

75

50

100

65

105

B

——

——

100

40

65

70

90

105

C

——

——

——

70

60

40

55

15

D

——

——

——

——

30

40

105

100

E

——

——

——

——

——

55

40

30

F

——

——

——

——

——

——

25

50

G

——

——

——

——

——

——

——

75

a What is the smallest possible cost for linking the computer terminals if each terminal has at least one connection with the system?
b If each terminal is connected to every other terminal, what is the cost of the
linking?
Use the network at right to answer questions 10 and 11.
The dimensions are in km.
10 multiple choice
Which of the following arcs are not in the spanning tree?
A AB
B AC
C BC
D BG

G

A

B

12
27

C

15 18
20 D 25 E

11 multiple choice
What is the length of the minimal spanning tree?
A 120 km
B 105 km
C 98 km

F
40

18
22

18

22

D 103 km


369

Chapter 7 Networks

Shortest paths
Given a network representing the distance between towns, consider the question, ‘How
far is it from town A to town X?’.
In earlier sections we have approached such a question using a trial and error
method. However, when networks become more complex, a systematic method is
required. The method used is called the shortest path algorithm.

Shortest path algorithm
To ﬁnd the shortest path between A and X in a network, follow these steps.
Step 1 For all nodes that are one step away from A, write the shortest distance from A
inside the circle representing the closest node.
Step 2 For all nodes which are two steps away from A, write the shortest distance
from A inside the circle representing the closest node two steps away.
Step 3 Continue in this way until X is reached.
Step 4 The shortest path can be identiﬁed by starting at X and moving back to the
node from which the minimum value at X was obtained, then continuing this
process until A is reached. This will be explored in the next worked example.

WORKED Example 6

A
2
3E
3
5I
2
3
M
3

Find the shortest path from A to P in the network at right.
The units are in minutes and represent time taken.
Note: We have placed the labels outside the nodes so that the
times can be placed inside the circles.

THINK
1

2

3

4

5
6

Beginning at A write inside the nodes at
B and E the shortest time taken to get to
them.
Then write in the shortest time for all
nodes which are two steps away from
A. That is, C = 4, F = 5 and I = 8.
Continue in this way until P is reached.
For example, at node J, the time from I
would be 10, so the shorter time, 9,
from F is put in the node.
Now back-track from P moving from
node to node along the arcs which
produced the minimum values. Check
to see if this is the shortest path.
This is the shortest path. Put arrows on
this path.
Write the answer.

WRITE/DRAW
A 2
3
E
3 3
5
I
8
2
3
M
11 3

2

B 2

3

4

C 2

3

5

F
3

4
9
1

7

10

2

D

6

H
3

12

9
5

K
11 2
3

N

C
2
3G
3
5K
2
3
O
3

3
G

5
J 2

B
2
3F
3
4J
2
1
N
2

13

L

1
O
3

14

P

The shortest path from A to P is
A–B–F–J–K–L–P and is 14 minutes long.

D
3H
5L
1

P


370


remember
remember
To ﬁnd the shortest path from A to X in a network:
1. For all nodes one step away from A, write the shortest distance.
2. For all nodes two steps away from A, write the shortest distance.
3. Continue until X is reached.
The shortest path is located by starting at X and working backwards to A.

7C
WORKED

Example

6

Shortest paths

1 Find the length of the shortest path from A to B in each of the following networks.
a

b

7
10

5

5

4

3

8

A

20
B

4

10

20
13

10

A

16
12

12
25

15

25

25

2

B

14

c

d A

23
23

30

24

A

16

45

2
2

3
B

4

4
6

4
7

27

34

3

34

18

4

5
4

12

5

30

8

10
4
10
7

20

6

7

6

5
7
6
B

e A

5

6

6

7

5

25
50

18

3
4

4

45
35

7

5
4

A
35

5

5
5

f

4

5

14
19

16
23

15

3

12
29

25
40

25
15

16

26
30

16
32

22
24
B

5
B


371

Chapter 7 Networks

64

2 From the map at right, where the units are km,
answer the following questions.
a What is the shortest distance from Fourier to
Rolle?
b What is the shortest distance from Fourier to
Stokes?
c What is the shortest distance from Fourier to
Stokes travelling through Reynolds?

Aiken

Stokes

32
24

60
25 Feynman
56
95
44
Reynolds Rolle
Hardy 36
45
32
34 Gauss
Ahmes
51
45
27
Fourier
26
Lebesgue

3 For each of the following networks, ﬁnd the shortest path from A to B.
a
b
25
10
7
10
7
A

25

15

35
45

25

B

28

25

10

5
6

50

31

20
40

12

15

d

15
5

30

7

20

13

14

8

8

7

8

A

12
7

9

12

35

25

B

8

37

15 30

15
26

12

13

8

7
6

8

10
12
11
10
10

7
8
8

6

8

6
6

26

6

8

12

11

6

A

f

12

7
8

B

35

8
6

7

40

8
14

e

34

26
8

6

8

15

25

13
11

23

12
15

16
A

14

15

9

7

50

30

8

10

B

30

35

8

c

35

50

35

32

10

25

40

A

15

5

40

6

11
14

13

6

13

7

A

9

12
8

10

6

11

12

12

8

B
8

12

5

7

11

8

7

8

15

14

B

13
11

13

12

14

9


372


4 This table shows the travelling times in minutes between towns which are connected
directly to each other. Note: The line indicates that towns are not connected directly to
each other.
Addisba

Eric

0

50

20

25

—

50

0

25

30

30

Callop

20

25

0

—

60

Dilger
Work

Dilger

Bundong

7.1

Callop

Addisba

ET
SHE

Bundong

25

30

—

0

70

Eric

—

30

60

70

0

a Draw a network to show the connection of the towns by these roads.
b Find the shortest travelling time between Addisba and Eric.

1
This network at right represents the potential cost of a covered
walkway between various locations on a campus.
1 How many nodes are there in this network?

3600
A
6400

2 How many arcs are there in this network?

B 4000 3000 E
8000
D

C

2000
6000

4000
6400

F

3 Which node/s have more than 4 arcs meeting?
The cost of the walkway is to be kept to a minimum but it should be possible to go from
any location to any other via a covered walkway.
4 Find the minimal spanning tree.
5 What arcs are not included in the minimal spanning tree?
6 What is the minimum cost of such an arrangement of walkways?
7 If one is to travel from D to F under cover, what path should be taken?
It is found that there was an error in the estimate for the walkway connecting A to C. The
correct value should be $3600.
8 Find the new minimal spanning tree
9 What is the new minimum cost for a suitable arrangement of walkways?
10 If one were to travel from B to C under cover, what path should be taken?


Chapter 7 Networks

373

Network flow
An application of networks used to analyse flow of traffic or water is network flow.
These usually involve directed networks where arrows show the direction of flow. An
example is described below.
A driver starts for work in the city at 7.30 am each morning. He lives in an outer
suburb and as he travels from his driveway through a few streets in his local neighbourhood, there is not much traffic on the roads. As he joins the road that connects his
suburb to the next suburb, he notices an increase in the volume of the traffic. As this
two-lane road joins the four-lane freeway into the city, the flow of traffic becomes
immense. Cars are following bumper to bumper, with drivers changing lanes to drive in
the fastest lane. The costs involved, financial and otherwise, for those who participate
in the morning rush are significant.
It is in everyone’s best interest that the traffic flow smoothly and that traffic jams
be avoided at all costs. Engineers use mathematical models of network flow to ensure
smooth flow of traffic.

Flow capacities and
maximum flow
The network’s starting node(s) is
called the source. This is where all
flows commence. The flow goes
through the network to the end node(s)
which is called the sink.
The flow capacity (capacity) of an
arc is the amount of flow that an arc
can allow through if it is not connected
to any other arcs.
The inflow of a node is the total of
the flows of all arcs leading into the
node.
The outflow of a node is the minimum
value obtained when one compares the
inflow to the sum of the capacities of all
the arcs leaving the node.
Consider the following figures.
D
A

B

C

Source

F
Sink

E

All flow commences at A. It is
therefore the source. All flow
converges on F indicating it is
the sink.

100

B

30
20
10

B has an inflow of 100. The flow
capacity of the arcs leaving B is
30 + 20 + 10 = 60. The outflow
is the minimum of 100 and 60,
which is 60.

30
100

B 20
80

B still has an inflow of 100 but
now the capacity of the arcs
leaving B is (80 + 20 + 30) = 130.
The outflow from B is now 100.

The flow capacity of the network is the total flow possible through the entire
network.


374


WORKED Example 7
Consider the information presented in the following table.

From

Quantity
(litres per minute)

Demand (E)

1000

—

200

To

200

Rockybank Reservoir

(R)

Marginal Dam

Marginal Dam

(M)

Freerange

Marginal Dam

(M)

Waterlogged (W)

200

200

Marginal Dam

(M)

Dervishville (D)

300

300

(F)

a Convert the information to a network diagram, clearly indicating the direction and
quantity of the flow.
b Determine the flow capacity of the network.
c Determine whether the flow through the network is sufficient to meet the demand of all
the towns.
THINK

WRITE

a Construct and label the required number
of nodes. The nodes are labelled with
the names of the source of the flow and
the corresponding quantities are
recorded on the arcs.

a

F
200

R

1000

M

200

200

W

300

200

E

300

D

b

1

Examine the flow into and out of the
Marginal Dam node. Record the
smaller of the two at the node. This
is the maximum flow through this
point in the network.

b

F
200

R

1000

M

200

W

300

D

Even though it is possible for the
reservoir to send 1000 L/min (in theory),
the maximum flow that the dam can pass
on is 700 L/min (the minimum of the
inflow and the sum of the capacities of
the arcs leaving the dam).
2

In this case the maximum flow
through Marginal Dam is also the
maximum flow of the entire network.

Maximum flow is 700 L/min.


Chapter 7 Networks

THINK

WRITE

c

c

375

1

Determine that the maximum flow
through Marginal Dam meets the
total flow demanded by the towns.

F
200
200

M

200

W

200

300

E

300

D

2

If the requirements of step 1 are able
to be met, then determine that the
flow into each town is equal to the
flow demanded by them.

Flow through Marginal Dam = 700 L/min
Flow demanded = 200 + 300 + 200
= 700 L/min
By inspection of the table, all town
inflows equal town demands (capacity of
arcs leaving the town nodes).

Consider what would happen to the system if
Rockybank Reservoir continually discharged 1000
L/min into Marginal Dam while its output remained
at 700 L/min.
Such flow networks enable future planning.
Future demand may change, the population may
grow or a new industry that requires more water
may come to one of the towns. The next worked
example will examine such a case.
Excess flow capacity is the surplus of the
capacity of an arc less the flow into the arc.

es

in
ion v

t i gat

n inv
io

es

The seven bridges of Königsberg
On the River Pregel in the European town of Königsberg, there were 7 bridges
arranged as below.

Island

Land

People wondered if it was possible to cross all 7 bridges without crossing any
bridge more than once.
Can you see if it can be done?

t i gat


376


WORKED Example

8

A new dairy factory, Creamydale (C), is to be set up on the outskirts of Dervishville. The
factory will require 250 L/min of water.
a Determine whether the original flow to Dervishville is sufficient.
b If the answer to part a is no, is there sufficient flow capacity into Marginal Dam to allow
for a new pipeline to be constructed directly to the factory to meet their demand?
c Determine the maximum flow through the network if the new pipeline was constructed.
THINK
a 1 Add the demand of the new
factory to Dervishville’s
original flow requirements. If
this value exceeds the flow into
Dervishville then the new
demand cannot be met.

WRITE
a

F
200

R

1000

M

200

200

W

300

200

E

300 + 250

D
2

The new requirements exceed
the flow.

The present network is not
capable of meeting the new
demands.

M

E
550

300

D

b

1

Reconstruct the network
including a new arc for the
factory after Marginal Dam.

b

F
200

R

1000

M

200

200

W

E

300

300
250

200

D

250

C
2

Repeat step 1 from worked
example 7 to find the outflow
of node M.

Marginal Dam inflow = 1000
Marginal Dam outflow
= 200 + 200 + 300 + 250 R
= 950

F
200
1000

M

200

W

300
250

D
C

3

Determine if the flow is
sufficient for a new pipeline to
be constructed.

c This answer can be gained from
part b step 2 above.

There is excess flow capacity of 300 into Marginal
Dam which is greater than the 250 demanded by
the new factory. The existing flow capacity to
Marginal Dam is sufficient.
c The maximum flow through the new network
is 950 L/min.


377

Chapter 7 Networks

C

D-

extension

extension — Paths and circuits: Eulerian and Hamiltonian
remember
remember

C

7D
WORKED

Example

7a

Network flow

1 Convert the following flow tables into network diagrams, clearly indicating the direction and quantity of the flow.
a

To

Flow capacity

A
A
B
C
D
c

From

B
C
C
D
E

100
200
50
250
300

From

To

Flow capacity

M
M
N
N
Q
O
R

N
Q
O
R
R
E
E

20
20
15
5
10
12
12

b

From

To

Flow capacity

R
S
T
T
U
d

AC
ER T

D-

1. In a network flow diagram, the arcs have quantities that indicate rates of flow;
for example, litres per minute, cars per second, people per hour and so on.
2. The starting node(s) from which all flows commence is called the source.
3. The flow goes through the network to the end node(s) which is called the sink.
4. The flow capacity (or capacity) of an arc is the amount of flow that an arc
would allow if it were not connected to any other arcs.
5. The flow capacity of the network is the total flow possible through the network.
6. The inflow of a node is the total of the flows of all arcs leading into the node.
7. The outflow of a node is the minimum of either the inflow or the sum of the
capacities of all the arcs leaving the node.
8. Excess flow capacity of an arc equals the flow capacity of an arc minus the
flow into the arc.

S
T
U
E
E

250
200
100
100
50

From

To

Flow capacity

D
D
G
G
F
F
J
H

F
G
H
J
H
J
E
E

8
8
5
3
2
6
8
8

RO

IVE

INT

For more information on paths and circuits, click here when using the CD-ROM.

M

extension — Minimum cut–Maximum flow

M

extension

AC
ER T

IVE

INT

The maximum flow through most simple networks can be determined using these
methods, but more complex networks require different methods to be used.

RO


378


2 For node B in the network at right, state:
a the inflow at B

B
23

b the arc capacities flowing out of B
c

16

A

the outflow from B.

D
27

34
C

3 Repeat question 2 for the network at right.

A

B
2
C
2
D

4
5
3

WORKED

Example

3
4

E

6

4 For each of the networks in question 1, determine:
i the flow capacity

7b, c

ii whether the flow through the network is sufficient to meet the demand.
5 Convert the following flow diagrams to tables as in question 1.
a
b
B
B
4
5

A

3

c
4
5

A

3

3
4

2
C
2
D

A

E

3

6

B
2
D

4
5

d

3
E

4
5

A

6
12
7

8

7
C

2
C
2
D
B

2
6
D
4 F
7
C

3
4

E

6

3
E
8

6 Calculate the capacity of each of the networks in question 5.
WORKED

Example

8

7 i Introduce new arcs, from the information which follows, to each of the network
diagrams produced in question 1
ii calculate the new network flow capacities.
a

From

To

Flow capacity

A
A
B
C
D
B

B
C
C
D
E
E

100
200
50
250
300
100

b

From

To

Flow capacity

R
S
T
T
U
S

S
T
U
E
E
T

250
200
100
100
50
100


379

Chapter 7 Networks

c

From

To

Flow capacity

M
M
N
N
Q
O
R
N

N
Q
O
R
R
E
E
E

d

From

20
20
15
5
10
12
12
5

To

Flow capacity

D
D
G
G
F
F
J
H
D

F
G
H
J
H
J
E
E
E

8
8
5
3
2
6
8
8
10
ET
SHE

In question 7c the outflow from N is:
A 5
B 20

Work

8 multiple choice
C 15

D 25

2
Questions 1 to 4 refer to the network at right.
The network represents the distance between towns
in kilometres.

B

20
A

1 What is the shortest path from A to F?

20 15
C

40
32

30

20
D

E
10
F

32

2 Give the length of the shortest path from A to F.
3 Give the shortest path from B to F.
4 What is the length of the shortest path from B to F?
5 In the network at right, what is the inflow at the node?

20
40

6 In the same network as question 5, what is the outflow
at the node?
7 What is the excess flow capacity of
arc BC in the network at right?

A

20

10

B

30

C

Questions 8 to 10 also refer to the first network above. This network shows the capacity
of irrigation pipes in kilolitres per hour.
8 What is the inflow at C?
9 What is the outflow at C?
10 What is the maximum
flow in the network?

7.2

Maths A Yr 12 - Ch. 07 Page 380 Friday, September 13, 2002 10:36 AM

380


summary
• A network consists of a number of nodes connected by arcs.
• When the arcs have arrows the network is called a directed network and travel is
possible only in the direction of the arrows.

Minimal spanning tree
• A tree is a series of connections in a network that does not contain a loop.
• A spanning tree in a network is a tree that contains each node of the network.
• A minimal spanning tree is the arrangement of arcs in which every node is connected
to at least one other node in such a way as to minimise the total length of these arcs.
• To find the minimal spanning tree use the minimal spanning tree algorithm:
Step 2 Choose an unconnected node which is the closest to any connected node.
Connect this node to the nearest connected node.
Step 3 Repeat Step 2 until all the nodes are connected.
• A path is a series of nodes connected by arcs.
• The shortest path is the shortest distance from a given starting point to a given end point.

Shortest path
• The shortest path is the shortest distance from a given starting point to a given end point.
• To find the shortest path between A and X:
1. For all nodes that are one step away from A, write the shortest distance from A
inside the circle representing the node.
2. For all nodes which are two steps away from A, write the shortest distance from
A inside the circle representing the node.
3. Continue in this way until X is reached.
4. The shortest path can be identified by starting at X and moving back to the node
from which the minimum value at X was obtained, then continuing this process
until A is reached.

Network flow
• A network can be used to represent the network flow of quantities such as water,
traffic or telephone calls.
• Arcs indicate rates of flow. The inflow at a node is the sum of the capacities of the
arcs leading into the node. The outflow at a node is the minimum of either the
inflow or the sum of the capacities of the arcs leaving the node.
• In a network flow diagram, the arcs have quantities that indicate rates of flow, for
example, litres per minute, cars per second people per hour and so on.
• The starting node(s) is called the source, from which all flows commence.
• The flow goes through the network to the end node(s) which is called the sink.
• The flow capacity (or capacity) of an arc is the amount of flow that an arc would
allow if it were not connected to any other arcs.
• The flow capacity of the network is the total flow possible through the entire network.
• The inflow of a node is the total of the flows of all arcs leading into the node.
• The outflow of a node is the minimum of either the inflow or the sum of the
capacities of all the arcs leaving the node.
• Excess flow capacity equals the flow capacity of an arc minus the flow into the arc.


381

Chapter 7 Networks

CHAPTER
review
10

B

1 For the network at right, write down:
a the number of nodes
b the number of arcs.

E

20
10
A

10

C
10

G
10

20
D

7A

20

5

F

10

2 The following table represents the cost, in tens of thousands of dollars, of resurfacing roads
connecting various locations in a district. Draw a network representing this situation.
A

B

C

D

A

——

5

E

11

B

——

——

4

C

——

——

——

8

D

——

——

——

——

E

——

——

——

——

7A

12
7

——

3 Describe an algorithm used to identify the minimal spanning tree.
4 Give the minimal spanning tree for the network in question 1.
5 Determine the minimal spanning tree for the ﬁgure
at right.

30

B
15

15

20

E

H

15

40

30

7B
7B
7B

30

I 15
15
F
K
15
35
35 25
30 30
25

A

25

C

30

30
D

35

G

25

J

6 It is planned to join the towns shown on the map
Caerleon
16 km
at right by a rail link. Use a minimal spanning
15 km
Freshwater
Brownsville
algorithm to ﬁnd the shortest length of track
15 km
needed to connect each town by rail.
18 km
Miriam
34 km
Amesbury

19 km
Manto
18 km

61 km

41 km
Gaine

7B


382


7C
7C
7D

7 Identify the shortest path from A to G in question 1. What is the length of this path?

7D

10 If the arcs in question 5 represent capacity for flow, calculate each of the following:
a inflow at C
b outflow at C
c maximum flow from A to K.

7D

11 From the table at right
produce a network flow
diagram.

8 Identify the shortest path from A to K in question 5. What is the length of this path?
9 If the arcs in the network in question 1 represent capacity for flow, calculate the following:
a inflow at C
b outflow at C
c the maximum flow.

13

C

6

C

10

D

4

D

3

C

E

14

D

F

10

E

F

15

From

To

Flow quantity

A

B

13

A

C

6

A

G

16

B

C

10

B

D

4

B

G

2

C

D

3

C

E

14

D

F

10

E

F

15

G
CHAPTER

B

C

7

A

B

test
yourself

Flow quantity

B

12 Draw the network flow
diagram for the table at
right.

To

A

7D

From

D

3

G

H

10

H

F

13

Year 12 Maths A Textbook - Chapter 7

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