Mathematics Standard 2 Year 12
Networks Topic Guidance
Mathematics Standard 2 Year 12 Networks Topic GuidanceTopic focus ....................... 3
Terminology .......................................................................................................................... 3
Use of technology ................................................................................................................ 3
Background information ...................................................................................................... 3
General comments ............................................................................................................... 4
Future study.......................................................................................................................... 4
Subtopics .............................................................................................................................. 5
MS-N2: Network Concepts ..................................................................................................... 5
Subtopic focus ................................................................................................................. 5
N1.1 Networks ........................................................................................................................ 5
Considerations and teaching strategies .......................................................................... 5
Suggested applications and exemplar questions ............................................................ 6
N2.2 Shortest Paths ............................................................................................................... 7
Considerations and teaching strategies .......................................................................... 7
Suggested applications and exemplar questions .......................................................... 10
MS-N3: Critical Path Analysis............................................................................................... 12
Subtopic focus ............................................................................................................... 12
Considerations and teaching strategies ........................................................................ 12
Suggested applications and exemplar questions .......................................................... 17
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Topic focus
Networks involve the graphical representation and modelling of situations as an approach to
decision-making processes.
Knowledge of networks enables development of a logical sequence of tasks or a clear
understanding of connections between people or items.
The study of networks is important in developing students’ ability to interpret a set of
connections or sequence of tasks as a concise diagram in order to solve related problems.
Terminology
activity chart
arc
backward scanning
critical path ⚑
critical path analysis
degree of a vertex
directed network ⚑
earliest starting time (EST)
edge
float time
flow capacity ⚑
forward scanning
interdependence
Königsberg bridge problem ⚑
node
path
Prim’s algorithm ⚑
Kruskal’s algorithm ⚑
latest starting time (LST)
map
maximum-flow minimum-cut
minimum spanning tree ⚑
network
network diagram
network flow
shortest path ⚑
sink
source
spanning tree ⚑
tree ⚑
vertex
vertices
weighted edge
Use of technology
The use of appropriate software could be used in order to construct network diagrams and
minimum spanning trees to represent and analyse networks.
Background information
Very few areas of Mathematics have their history as clearly recorded as Graph Theory and
consequently Networks. The foundation of Graph Theory can be traced officially to 1735 when
Euler sought to prove the Königsberg bridge problem. The problem related to a puzzle the
locals pondered: Can a person walk all seven bridges and never cross any given bridge twice?
It was Euler’s translation of the map of the seven bridges of Königsberg into an abstract
mathematical representation, a graph, using points and lines to represent quantities and
connections to prove the problem that represented the foundation of Graph Theory.
In 1736 Euler published Solutio problematis as geometriam situs pertinentis, which translates
to ‘the solution of a problem relating to the theory of position’ and describes the Königsberg
bridge problem and related proofs. This history is represented in texts related to Graph Theory
or Network theory. Interestingly the term ‘graph’ in this context did not appear until James
Sylvester in the nineteenth century used it in relation to diagrams of molecules.
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The use of networks is becoming increasingly prolific, particularly in representing and
analysing data in areas such as Technological networks, Biological networks, Information
networks and Social and Economic networks.
The Critical Path Methods and analysis has a rich and vibrant history. The critical path is the
pathway through a set of activities that optimises the sequence of tasks or scheduled activities
in an activity network. The origin of tools like the Critical Path method can be traced back to
ancient civilisations. For example archaeologists have found Egyptian records related to the
construction of the Great Pyramid of Giza, which show how workers were grouped into four
construction teams (one for each face of the pyramid) and how organisation occurred to
ensure the correct stones were located, cut, transported and set in place to form the pyramid.
Formalisation of techniques such as those used by the ancient Egyptians and the use of
networks progressed in the 1940s and 1950s, made possible by the geniuses of Bletchley
Park and the emergence of computers and computer-powered algorithms for better transport
and logistics. It became an important way for people to manage tasks and projects and to
create efficiencies, minimise costs and time, and protect against potential issues and hold-ups
to a project. It is commonly used in areas such as software development, research projects,
construction, product development and engineering.
General comments
Network theory is sometimes described as graph theory where the word ‘graph’ has a different
meaning in this context than that in other mathematical contexts; for instance, it is not the
graph of a function or a data graph.
Other algorithms that are used to determine minimum spanning trees could be explored, for
example Solin’s (Borůvka’s) minimum spanning tree algorithm.
Applications should be tailored to be as relevant as possible to the students, using community
issues, for example local distribution networks, or current personal considerations, for example
an activity table of the tasks involved in the production of the school play.
Analysing current social networks using technology is a relevant application of Network theory
and has become a vibrant field of research referred to as Social Network Analysis. Similar
applications of Network theory are found in many other areas, including linguistics,
criminology, medicine and law.
Future study
The study of networks can be continued at a tertiary level in various vocational contexts, for
example, to examine the efficiency of a delivery service or to minimise costs of connecting
services.
Network theory is becoming increasingly relevant in further study in areas such as Project
Management, Data Analysis and scientific research.
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Subtopics
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MS-N2: Network Concepts
MS-N3: Critical Path Analysis
MS-N2: Network Concepts
Subtopic focus
The principal focus of this subtopic is to identify and use network terminology and to solve
problems involving networks.
Students develop their awareness of the applicability of networks throughout their lives, for
example social networks and their ability to use associated techniques to optimise practical
problems.
N1.1 Networks
Considerations and teaching strategies
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A network is a collection of objects connected to each other in some way. Networks
are made up of vertices joined by edges.
Different terminology exists to describe aspects of networks, including ‘node’ instead of
‘vertex’ and ‘arc’ instead of ‘edge’.
A directed network is a graph in which each edge has a direction, as with a one-way
street. In a directed network the edges have arrows to indicate the direction you may
travel along it.
Networks may or may not have weighted edges, depending on the context.
The weight of an edge may be distances or times, but it could also be something
entirely different, such as depreciation costs or petrol charges.
Edge weights can for instance represent lengths, times, weights, costs, or something
entirely different. An example might be a map of a railway system, where each station
is represented by a vertex and where each edge represents the train lines between two
neighbouring stations. Appropriate edge weights would include travel times between
stations and cost of journeys between stations.
The terminology should be discussed in detail and students encouraged to use the
appropriate terms. Students could be encouraged to write their own glossary for the
topic of Networks.
Students should be given the opportunity to construct network diagrams from real-life
situations, for example, the Königsberg bridge problem, a simple rail network or an
electricity cable network.
Practical experiments could be conducted and represented by a network diagram,
such as every person at a party shaking hands with every other person at that party
(which can lead to discovery of the Handshake Lemma which states that an even
number of people must have shaken an odd number of people’s hands). Another
example that could be considered is networks involving personal connections, such as
film actors who are connected (the popular notion “six degrees of separation from
Kevin Bacon”).
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It is important to have a way of representing networks which does not rely on a
diagram. In particular, computers cannot work with diagrams. A table can be
constructed that shows all the information about the weights in a network, such as the
one below:
𝐴
𝐵
A

B2
C 4

D  

𝐶
𝐷
2 4 

 3 
3  1

 1  
A
B
C
D
A
-
2
4
-
B
2
-
3
-
C
4
3
-
1
D
-
-
1
-
A directed network can also be represented in a table. For example:
From:
To:
A
B
C
D
A
-
2
4
-
B
-
-
3
-
C
5
-
-
1
D
-
-
1
-
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When a network diagram is constructed from a table, reconstructions may differ in
appearance. For example the network diagram below may be constructed from the
table above.
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The construction of network diagrams where multiple edges emerge from each vertex
should be explored.
Suggested applications and exemplar questions
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Current social networks using technology are a relevant application of Network theory.
Model the following house plan as a network, showing the doors (doorways) as edges
and the rooms as vertices.
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Given a road atlas or access to an online map, students note the distances and routes
between certain cities in NSW, and draw a network diagram that represents the map.
Investigate network representations of songlines or kinship within the Aboriginal or
Torres Strait Islander cultures.
Draw a network diagram to illustrate the following table:
A
B
A
-
B
250
C
200 400
C
250 200
-
D
-
E
500
-
F
300
70
G
-
-
D
-
400 200
-
200 300
E
F
500 300
-
300 400
G
-
70
-
-
300
-
-
-
350
400
-
-
-
500
-
-
-
-
-
-
-
300 350 500
N2.2 Shortest Paths
Considerations and teaching strategies
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A tree is a connected network, or part of a network that does not contain a cycle. An
example of a tree network is an actual tree, where the vertices are where the trunk and
the branches meet and branch off and the edges are the branch and trunk segments
between vertices. There are no cycles here since branches don’t re-join the trunk or
other branches, at least not usually.
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A spanning tree in a network is a tree that contains each vertex.
For a connected graph with 𝑛 vertices, each spanning tree has precisely 𝑛 − 1 edges.
Students should be encouraged to consider and explain why this is true.
A minimal spanning tree in a network is the spanning tree with the total length of its
edges at a minimum. Every vertex is connected to every other vertex by the network,
but the total length of the reduced network is a minimum.
Students could consider how to minimise the cost of a computer wireless network at
their school connecting the main hubs of the school in order to maximise connectivity,
for example considering the following diagram:
Options for spanning trees include:
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and
Students can then investigate which tree would be a minimal spanning tree for the
network.
Prim’s algorithm is a quick method of finding a minimum spanning tree. The sequence
of steps is as follows to find a minimum spanning tree, 𝑇:
 Select any vertex to be the first vertex of 𝑇.
 Consider the edges which connect vertices in 𝑇 to vertices outside 𝑇.
Pick the one with the minimum weight. Add this edge and the extra vertex to 𝑇.
(If there are two or more edges of minimum weight, choose any one of them).
 Repeat the step above until 𝑇 contains every vertex of the graph.
Students should notice that when they apply Prim’s algorithm, they choose the edge
which is immediately ‘best’ without considering the long-term consequences of their
choice.
Kruskal’s algorithm uses the edge weights directly rather than considering the
connections between vertices. The sequence of steps to find a minimum spanning tree
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for a connected graph with 𝑛 vertices is as follows:
 Choose the edge of least weight.
 Choose from those edges remaining, an edge of least weight which does not
form a cycle with already chosen edges. (If there are several such edges,
choose any one of them).
 Repeat the step above until 𝑛 − 1 edges have been chosen (until all vertices
are connected by chosen edges).
An example of applying Kruskal’s algorithm to a network diagram is as follows:
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There are 6 vertices, so we need 5 edges.
Edge 1: Choose edge BC (least weight)
Edge 2: Choose edge DE (next least weight)
Edge 3: Choose edge AF (could also have chosen CD)
Edge 4: Choose edge CD
Edge 5: Reject edge CE as it creates a loop with CDE
Edge 5: Choose edge CF
5 edges chosen so the minimal spanning tree is complete:
Students can begin investigating the shortest path by considering a network
representing distances between towns and asking how far it is from town 𝐴 to town 𝐽.
The shortest path can also be thought of as a path of minimum weight.
To find the shortest path from 𝐴 to 𝐽 in a network follow this sequence of steps:
 Redraw the network diagram, with circles at each vertex except for 𝐴.
 For all vertices one step away from 𝐴, write down the shortest distance inside a
circle representing the closest vertex.
 For all vertices two steps away from 𝐴, write down the shortest distance from 𝐴
inside each circle representing a vertex.
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Continue this way until 𝐽 is reached.
The shortest path can then be identified by starting at 𝐽 and moving back to the
vertex from which the minimum value at 𝐽 was obtained, then continuing this
until 𝐴 is reached.
An example of finding the shortest path is as follows:
The shortest path is 𝐴𝐵𝐸𝐹𝐽 and has weight 8.
Students should recognise that there are often several shortest paths (of equal length)
between two given vertices.
Suggested applications and exemplar questions
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Students could investigate problems relevant to their local area, such as water supply
to an industrial estate, the local rail network or the minimum length of bitumen road
needed so enough vehicles can visit a number of destinations without getting bogged
in bad weather.
Students could investigate problems involving personal or social networks; for
example, if some friends want to be sure that they keep each other updated about
urgent matters, then they could agree on a network of communication between friends,
possibly including online groups. If they wanted to make sure that everyone was
reliably updated on news and that there wasn’t wasted energy spent on two friends
telling the same friend the news, then the network would have no cycles and would be
a spanning tree. Here, the weights are just 1, and the vertices are each of the friends
and each of the social networks. The edges indicate lines of communication.
(a) Find all the possible spanning trees for the following network:
(b) Which of the spanning trees has minimum weight?
(c) Use Prim’s algorithm, starting with vertex 𝐴, to find the minimum spanning tree.
(d) Show that Kruskal’s algorithm produces the same minimum spanning tree as
Prim’s algorithm.
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Find a minimum spanning tree for the following network:
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Find the length of the shortest path from 𝑎 to 𝑒 in the above network.
This table shows the travelling times in minutes between towns which are connected
directly to each other. Note: The dash in a box indicates that towns are not connected
directly to each other:
A
A
B
D
E
0
50 20 25
-
B 50
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25 30 30
C 20 25
0
-
60
D 25 30
-
0
70
30 60 70
0
E
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0
C
-
(a) Draw a network diagram showing the information on this table.
(b) Find the shortest travelling time between A and E.
As a practical experiment, model a network diagram such as the one above by using
pieces of string tied to small key rings. The shortest paths between two vertices could
then be found by pulling the two key rings tightly apart. The shortest path is shown by
the tight strings. Students can then discuss the limitations of this type of physical model
and compare results with that of using theoretical processes.
The distinction between minimum spanning trees and shortest paths could be
explored. For example, the mayor of a small outback town would like to upgrade the
dirt tracks in the town to paved roads so that every home is connected to every other
home by a paved road. The mayor has two choices: minimise the overall kilometres
required (ie the cost) to find a minimum spanning tree or choose the roads to create a
collection of shortest paths from the mayor’s house to every other home.
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MS-N3: Critical Path Analysis
Subtopic focus
The principal focus of this subtopic is to use critical path analysis in the optimisation of real-life
problems.
Students develop awareness that critical path analysis is a useful tool in project planning,
management and logistics.
Considerations and teaching strategies
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Students could use technology to draw the network diagram for a project, including the
labelling of earliest starting times and latest starting times on the vertices. Online
project management programs such as TeamGantt could be explored.
Activity tables can be constructed when considering a number of activities that are
involved in the completion of a task or project. Some activities can be performed at the
same time and in other cases they must be done one after the other sequentially. For
example, when making spaghetti bolognese, you can cook the spaghetti while the
bolognese is bubbling, but you cannot make the bolognese without adding tomatoes.
Students should be given the opportunity to explore networks that sequence from a
beginning to an end point and involve activities/pathways that can occur
simultaneously. Networks where all activities must occur in order for the project to be
completed should be considered.
When drawing a directed network diagram of an activity table, use a rectangular box
for Start and End vertices. Activities can be represented by arrows, and must start and
end at a vertex. Connect any activity that does not have any prerequisites to the start
box. Activities with the same prerequisites begin at the same vertex. For example:
Draw a directed network diagram to show the activities involved in putting on a rock
eisteddfod production. The activity table is given:
Code
Activity
Time
(weeks)
Prerequisites
A
Select theme and music
2
None
B
Hire a director
1
A
C
Obtain copyright for music
2
A
D
Hold auditions for dancers
1
B, C
E
Confirm dancers
2
D
F
Rehearse
12
E
G
Book venue
1
B
H
Build and paint sets
8
G
I
Obtain props
8
B
J
Make costumes
8
E
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K
Set up lights and sound system
2
H
L
Hold main dress rehearsal
1
F, I, J, K
M
Advertise rock eisteddfod
3
E, G
N
Stage rock eisteddfod
1
L, M
Note that a dummy activity is needed, with no weighting, to show that the dancers must
be confirmed before the rock eisteddfod can be advertised.Once the diagram is drawn,
the weightings of how many weeks each activity takes can also be put onto each edge
in the diagram if needed.
An alternative method to the above, is for the vertices to represent the activities. To
schedule the production, each activity (vertex) can only be commenced once all
prerequisite activities (vertices) have been completed. Note the times (weights) are
attached to the vertices not the edges in this instance. This method would produce a
diagram as follows:
Sometimes a ‘dummy activity’ is needed to draw the most effective diagram in order to
show that a possible pathway is to move from one activity to another. This activity
would have a time value of zero and no letter assigned to it. It is shown on a network
diagram as a directed edge with weight zero.
Determination of critical activities whose completion are essential to whether a project
can be carried out in the least possible time should be explored.
The critical path through a project network is the longest path from the start to the
finish. Each activity on the critical path is called a critical step. Any delay to a critical
step means a delay of the whole project. The critical path gives the minimum time for
completion of a project.
An example of examining critical paths is as follows:
The following diagram is a project network for preparing and serving a curry for a
family of four. The tasks are: A = Chop vegetables, B = Mix spices, C = Cook curry, D
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= Cook papadums, E = Place papadums and chutney on table, F = Cook rice, G = Put
curry and rice on plates, H = Put plates on table.
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The critical path through this network is FGH, giving a critical time of 23 minutes.
A five-minute delay in cooking the poppadoms has no effect on the critical time, as it
does not affect the critical path. A five-minute delay in cooking the rice increases the
critical time by five minutes, as it is a critical step, that is, it is on the critical path. A fiveminute delay in mixing the spices changes the critical path to ABCGH, and increases
the critical time to 26 minutes.
The same method as used for finding the shortest path in N2.2 can be used, but
finding the longest path instead. This method is sometimes called dynamic
programming, but is more commonly known as a forward scan.
Forward scanning consists of following these steps:
 Redraw the activity chart, with empty circles at each vertex.
 Work along each path from the start, writing the total time of the path at each
vertex.
 When two or more paths join, select the highest total as the path to follow.
 Continue until you reach the finish.
For example, the following diagram shows an activity chart having been forward
scanned. The critical path is ADKHLN, of length 25.
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Forward scanning finds the earliest time each activity can be started after the project
has begun. For example, the circle at the beginning of the arrow for activity E has a 3
in it, therefore the earliest start time for activity E is 3 (because you have to have
completed both A and B before beginning E). Similarly, the earliest start time for
activity M is 13. These activities could take place later in the project, but this process
identifies the earliest possible time they can begin, known as the Earliest Starting Time
(EST).
Backward scanning consists of following these steps:
 Redraw the activity chart, with empty circles at each vertex.
 Note the critical time of the project at the Finish point.
 Work along each path from the finish, subtracting the total time of the path at
each vertex.
 When two or more paths join going backwards, select the lowest total as the
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path to follow.
Continue until you reach the start.
For example, the following diagram shows the activity chart from above having been
backward scanned. The critical path is still ADKHLN, of length 25.
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Backward scanning finds the latest time each activity can be started after the project
has begun in order to complete the project within the critical time. For example, the
circle at the beginning of the arrow for activity M includes the number 18, therefore the
latest start time for activity M is 18 (because after completing M, which takes time 2,
you also have to complete N, which takes time 5, before you can complete the project).
Similarly, the latest start time for activity E is 11. These activities could take place
earlier in the project, but this process identified the latest possible time they can begin,
known as the Latest Starting Time (LST).
The float time of an activity is the maximum delay possible in starting that activity that
does not affect the completion time of the project.
As any alteration to an activity on the critical path will affect the critical time, any
activities on the critical path have a float time of zero (no delay is possible).
Float times for non-critical activities (activities not on the critical path) can be calculated
by performing both a forward and backward scan and then comparing the results. For
example, on the diagrams above, activity F has an EST of 4 and a LST of 6 and
therefore has a float time of 2 (in practical terms this means that activity J must begin
between 4 and 6 time units after the project begins for the critical time to be achieved).
It is useful to enlarge the vertices and divide each into two halves. The forward scan
results can be recorded in one half, and the backward scan results can be recorded in
the other half, as illustrated in the following diagram:
There are many applications of problems involving flows through networks, for
example the movement of traffic through an airport, the flow of water in a system of
pipes and the transfer of data across a computer intranet.
In a flow network diagram, the weights on the edges are often referred to as capacities
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(rates of flow). The start of the network is called the source (often labeled 𝑠) and the
end of the network is called the sink (often labeled 𝑡).
Flows in a network are constrained by the capacity of both edges and vertices.
The flow capacity (or capacity) of an edge is the amount of flow that would be possible
if the edge were not connected to any other edges.
The flow capacity of the network is the total flow possible through the network.
The inflow of a vertex is the total of the capacities of all edges leading into the vertex.
The outflow of a vertex is the minimum of either the inflow or the sum of the capacities
of all the edges leaving the vertex. For example, if a vertex has an inflow of 1000, and
then three edges leading out of it with capacities 200, 300 and 400, then the outflow is
900. In practical terms, although 1000 is flowing into the vertex, it is only capable of
allowing 900 to flow through it, therefore the outflow is 900. In another example, if the
inflow was 600 and the three edges leading out of it still had capacities 200, 300 and
400, the outflow would be 600 as there is not more than 600 flowing into the vertex.
Excess flow capacity of an edge is the flow capacity of the edge minus the flow into the
edge.
Consider the following network representation of cars through a series of junctions,
where the capacities are expressed in terms of maximum number of cars per hour
along each edge (road):
By considering the inflow and outflow of each vertex in turn, it is discovered that there
is a flow capacity of the network of 650 cars per hour. In this solution, as illustrated in
the following diagram, edges such as sD and CB are said to be saturated (at maximum
capacity).
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When considering the ‘maximum-flow minimum-cut’ theorem, a cut is a continuous line
or curve that separates the start (or source) from the end (or sink) of a network but
does not pass through any vertices. Any cut can be thought of as a collection of
potential bottlenecks restricting the flow of the network.
For example, in the above example of cars travelling through a road system, we can
place a cut across the network as illustrated by the dashed line in the following
diagram:
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The maximum flow across this cut is found by considering the flow at each point the
cut intersects with an edge, so in this example it is 300+150+200=650. This is the
same as the flow capacity of the network, so therefore this flow cannot be improved
upon unless the capacity of one of the edges sD, AD or AB is increased.
The ‘maximum-flow minimum-cut’ theorem states that the flow through a network
cannot exceed the value of any cut, and the maximum flow equals the value of the
minimum cut. In other words, if you can find, by inspection, a flow and cut which have
the same value, then the maximum-flow minimum-cut theorem tells you that you have
obtained the maximum flow (flow capacity) for the network.
Suggested applications and exemplar questions
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Students could determine the minimum time in which a local project can be completed
and the optimal use of resources required to achieve this minimum completion time.
The applications of network flow problems could include movement of air traffic
through a small airport, the flow of water through a system of house pipes, foot traffic
through a school precinct or the transfer of information in a local computer network.
Start and end points should involve genuine activities that students can relate to, for
example, renovating a bathroom; getting ready to go out; obtaining a driver's license.
Work out an activity table for the project of organising a barbeque and trophy
presentation for the end-of-season of a local soccer club.
Draw a directed network for the task of cooking a roast dinner.
Code
Activity
Time (mins)
Prerequisites
A
Cook the meat
120
None
B
Prepare the potatoes
10
None
C
Roast the potatoes
60
B
D
Set the table
20
None
E
Prepare the vegetables
10
None
F
Cook the vegetables
15
E
G
Carve the meat
10
A
H
Make the gravy
5
None
I
Put the gravy jug on the table
0.5
H
J
Serve at the table
5
C, D, F, G, I
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The following table gives details of a set of six tasks which have to be completed to
finish a project. The ‘immediate predecessors’ are those tasks which must be
completed before a task may be started:
Task
Duration
(days)
Immediate
predecessors
A
5
-
B
2
A
C
1
A
D
2
B, C
E
4
B, C
F
1
D, E
(a) Draw an activity network.
(b) Perform a forward scan and a backward scan to find the critical path, the critical
time and the critical activities.
(c) Identify the float time of activity D.
(d) If a time lag of one day is needed, due to resources, between activities B and E
describe the effect it will have on the critical path.
The following diagram illustrates the amount of water that flows from a main dam at s
to three dams at A, B and C before then supplying a town with water at t. The edge
capacities are given in thousands of litres of water per minute.
(a) Find the maximum flow through this network and show that it is the same as the cut
of minimum value.
(b) What is the outflow of C?
(c) What is the excess flow capacity of Ct?
(d) If the capacity of only one edge could be increased, which one should it be?
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