This document provides information about networks and algorithms used to solve network problems. It discusses what a network is, including nodes, edges, arcs, weights, paths, circuits, trees, and connectivity. Different network problems are introduced, such as finding the shortest path, minimum spanning tree, and number of paths between nodes. Algorithms for solving these problems are described, including Dijkstra's algorithm, Prim's algorithm, and Kruskal's algorithm. The document provides examples and exercises for students to work through. It outlines the requirements for a network problem assignment, where students will formulate a real-world network problem to explore.

2.  A Network Problem (maximum8 pages)
You will be required to devise a network problem of your own,
and investigate possible changes to this problem.

4.  But according to a study from Facebook, the world is actually
smaller than we think. The social network has found there are
on average just three and a half people between any one
person and another.

5. Networks are used to show connections or relations between
things of interest. They can be used to find optimum solutions
to practical problem. These may be the solutions which cost the
least, take the least time, or require the smallest amount of
material.
A network is also called a graph.

6.  The points on a network are known
as nodes or vertices.
 The connection or relationships are
represented by lines.

7.  If we are allowed to move in either direction along the lines,
the graph is an undirected graph and the lines are called
edges.
Node or Vertex
Edge

8.  If we are only allowed to move in one direction along the lines,
the graph is a graph or directed graph and the lines are called
arcs.
Node or Vertex
Arc

9.  An edge or arc may sometimes be assigned a number called
its weight. This may represent the cost, time, or distance
required to travel along the line.
 The lengths of the arcs and edges are not drawn in proportion
to their weight.
3
1
5 2
8
7
3

10.  A connected network is a network in which it is possible to
travel from every node to every other node by following
edges. The network is otherwise said to be disconnected.

11.  A path is a connected sequence of edges or arcs showing a route
that start at one node and ends at another.
 When listing a path, an edge or a vertex cannot be used more than
once
Path A B D C

12.  A circuit is a path that starts and ends at the same node.
 A circuit can use nodes or vertices more than once but edges
cannot repeat.
Circuit A B D C A

13.  If a connected network graph has no cycles or circuits, no
loops, and no multiple edges then it is a tree.

14.  Every connected graph will have at least one subgraph that is
a tree. If this tree subgraph connects all the vertices than we
call it a spanning tree.

17.  15A.1
1, 2, 4, 7
 15A.2
2, 4

18.  Draw a network diagram of friendships if:
◦ A has friends B, D and F;
◦ B has friends A, C and E;
◦ C has friends D, E and F.

20.  Use a network to model the room access of this house.
 Does the network contain any circuit? Describe any circuits in
the context of the house plan

21.  To find an optimum solution, it could be a good idea to
identify how many options we have first.

22.  In ‘number of paths’ problems we are interested in finding the
total number of paths which go from one node to another
node without backtracking. Each connection on the graph
allows traffic in one direction only.
 When listing a path, an edge or a vertex cannot be used more
than once.

23.  How many paths are there from A to B?

24.  Step 1: From A, there are two possible nodes we can go to.
There is one possible path to each of them so we write 1 near
each

25.  Step 2 : We then look at the next step in the journey. At each
successive node we add the numbers from the previous nodes
which lead to that point.

26.  Step 3: Repeat Step 2 until we finish at B.

27.  How if we must avoid or pass through the node in the middle?

28.  If we must avoid a given node, immediately place a 0 at that
node.
 If we must pass through a particular node, place a 0 at nodes
that do not allow passage through it.

30.  15B.2 1, 2 ,3

31.  How many paths are there from A to B?

32.  Step 1: From A, there are two possible nodes we can go to.
There is one possible path to each of them so we write 1 near
each

33.  Step 2 : We then look at the next step in the journey. At each
successive node we add the numbers from the previous nodes
which lead to that point.

34.  Step 3: Repeat Step 2 until we finish at B.

35.  If we must avoid a given node, immediately place a 0 at that
node.
 If we must pass through a particular node, place a 0 at nodes
that do not allow passage through it.

37.  A shortest path problem is an optimal distance problem where
we must find the shortest path between two nodes on a
network.
 A ‘shortest path’ problem can not only be used to find the
shortest distance between nodes. A ‘shortest path’ may also
be used to find the path that costs minimum time and money.

39.  try to find the shortest path from each node to the starting
node;
 discard previously found ‘shortest’ paths when a shorter path
is found;
 obtained the actual shortest path from start to finish.

41.  Step 1: Assign a value of 0 to the starting vertex. Draw a box
around the vertex label and the 0 to show the label is permanent.
 Step 2: Consider all unboxed vertices connected to the vertex you
have just boxed. Label them with the minimum weight from the
starting vertex via the set of boxed vertices.
 Step 3: Choose the least of all of the unboxed labels on the whole
graph, and make it permanent by boxing it.
 Step 4: Repeat steps 2 and 3 until the destination vertex has been
boxed.
 Step 5: Back-track through the set of boxed vertices to find the
shortest path through the graph.

42.  Find the shortest path from A to O.

43. If we are forced to pass through a given node on the way to the
destination, we need to find the minimum length from starting
node to this given node and then add to it the minimum length
from this node to the finishing node.

44.  Find the shortest path from A to O that passes through M.

45.  15C.2 1, 2

46.  Find the length of the shortest path between towns A and
C.

47.  Trial and Error Method
 Step1: Label the starting node 0.
 Step2: Find an unlabelled node that is only accessible from
nodes that are labelled. Label it with the maximum length
possible.
 Step 3: Repeat Step 2 until you reach the finishing node.
 Step 4: Back-track through the nodes to establish the longest
path.

48. Start
Finish
4
7 4 4
5
5
2
8
1
3
9
6

49.  Exercise 15D
1, 2

50.  A school has decided to
add four new drinking
fountains for its students.
The water is to come from
an existing purification
system at point A; the
points B, C, D and E mark
the locations of the new
fountains.

51.  The problem of finding the most efficient way to connect a set
of nodes is called a shortest connection problem or minimum
spanning tree problem.
 The solution to the shortest connection problem will not
contain any circuits.

52.  In this algorithm we begin with any vertex. We add vertices
one at a time according to which is nearest.
 For this reason Prim’s algorithm is quite effective on scale
diagrams where edges are not already drawn in.
 It also has the advantage that because we add a new vertex
with each step, we are certain that no circuits are formed.

54.  Step 1: Start with any vertex.
 Step 2: Join this vertex to the nearest vertex.
 Step 3: Join on the vertex which is nearest to either of those
already connected.
 Step 4: Repeat until all vertices are connected.
 Step 5: Add the lengths of all edges included in the minimum
length spanning tree.

55.  In Kruskal’s algorithm, we keep choosing the shortest edge in
the network and examining each edge chosen to make sure
no circuit is formed.
 For this reason Prim’s algorithm is quite effective on an
existed networks where edges are already determined.
 We need to check if a circuit is formed when choosing an edge.

57.  Step 1: Start with the shortest edge. If there are several,
choose one at random.
 Step 2: Choose the shortest edge remaining that does not
complete a circuit. If there is more than one possible choice,
pick one at random.
 Repeat Step until all nodes are connect.

58.  15E.1 5
 15E.2 5

59.  A Network Problem (maximum8 pages)
You will be required to devise a network problem from the local
environment and investigate possible changes to this problem.

60.  Part 1: Formulate the problem to be solved.
 Part 2: Solve the basic problem.
 Part 3: Investigate the effects of possible changes.
 Part 4: Conclusion

61.  Formulate a problem in the ‘local environment’ means
somewhere you are familiar with – it could be in your home or
school grounds, local council area or it could include the
whole state or country.

62.  Construct the network diagram from the information you have
collected and solve the problem(s) you have posed.

63.  Devise one or more changes to conditions in the initial
problem, and make a prediction about the possible effect
these changes would have on the original solution. These
changes to conditions could include:
◦ Restrictions on the original conditions
◦ Using a different algorithm to find the solution
◦ Possible upgrades to improve the solution to the original problem.

65.  Analyse and compare your results from Parts 2 and 3 above,
including the reasonableness of your prediction. Your
discussion should include a consideration of the effects of
simplifying assumptions and the limitations on the practicality
or reliability of your solution.

66.  an outline of the problem to be explored
 the method used to find a solution
 the application of the mathematics, including
 generation or collection of relevant data and/or information, with a
summary of the process of collection
 mathematical calculations and results, using appropriate representations
 discussion and interpretation of results, including consideration of the
reasonableness and limitations of the results
 the results and conclusions in the context of the problem
 a bibliography and appendices, as appropriate.

67. Formulate the problem and submit the outline of the problem
to be explored by Next Tuesday.

68.  Example 1: The students at your school have requested that drinking fountains
fed from a rainwater tank be placed at convenient places around the grounds.
Using a map of the school, decide where the fountains would be located and
how they can be connected to the tank in the most cost efficient way.
 Example 2: A charity walk is planned for your council area. Part of the proposed
route has all the walkers passing through a local park that has a network of
paths of different widths. Analyse the maximum possible flow rate of people
through the park and report to the council on the implications of your findings.
 Example 3: You are planning a road trip between two major cities in Australia
(or towns in your state). There are several routes that can be taken without
backtracking. Determine the ‘best’ route to take using a variety of different
‘costs’ (such as distance, time, road conditions, number of tourist attractions
available, etc) along each of the arcs of the road network.

69.  Example 2: A charity walk is planned for your council area.
Part of the proposed route has all the walkers passing through
a local park that has a network of paths of different widths.
Analyse the maximum possible flow rate of people through
the park and report to the council on the implications of your
findings.

71.  Some directed networks have a clear beginning (called
the source) and a clear end (called the sink).
 Arcs of such networks could also be considered as having a
capacity to indicate the maximum possible rate of flow.
 Each vertex has an amount of inflow (total weight of all arcs
arriving at the vertex) and an outflow (total weight of all arcs
leaving the vertex).
 The maximum flow through the vertex is the smallest of these
values.

72.  Step 1: Choose one possible route and find the arc with the
smallest capacity. Record this number. Subtract this number
from the capacities of every pipe along the route. This is the
new capacity for each arc along that path.
 Step 2: Choose another route and repeat Step 1 once again
recording the smallest capacity.
 Step 3: Choose another route and repeat Step 1, etc. until all
possible routes have been exhausted.
 Step 4: Add the smallest capacities of all routes. This is the
maximum carrying capacity of the network.

73. Source Sink

74.  15F 1, 3, 5

75.  Part 1: Formulate the problem to be solved.
 Part 2: Solve the basic problem.
 Part 3: Investigate the effects of possible changes.
 Part 4: Conclusion

76.  an outline of the problem to be explored
 the method used to find a solution
 the application of the mathematics, including
 generation or collection of relevant data and/or information, with a
summary of the process of collection
 mathematical calculations and results, using appropriate representations
 discussion and interpretation of results, including consideration of the
reasonableness and limitations of the results
 the results and conclusions in the context of the problem
 a bibliography and appendices, as appropriate.

77.  Formulate a problem in the ‘local environment’ means
somewhere you are familiar with – it could be in your home or
school grounds, local council area or it could include the
whole state or country.

78.  State what your ‘local environment’ problem is.
 Describe the motivation for the investigation and provide
relevant background information.
 Describe the goals of the investigation.