Y11 m02 networks
- 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.
- 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.
Editor's Notes
- #4: Six degrees of separation is the idea that all living things and everything else in the world are six or fewer steps away from each other so that a chain of "a friend of a friend" statements can be made to connect any two people in a maximum of six steps.
- #12: An important consideration in defining a path is to remember that you cannot use an edge or a vertex more than once. When listing a path, we can write the vertices that we are passing through in the order we are using them.