MEI Further Mathematics Support Programme

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1 Further Mathematics Support Programme

the Further Mathematics Support Programme www.furthermaths.org.

2 the Further Mathematics Support Programme Modelling and problem solving with Networks Sharon Tripconey Let Maths take you Further Nov Feb 2010

3 Network problems and algorithms Topic AQA Edexcel MEI OCR A Graphs Graphs D1 D1 D1 D1 Networks Prim D1 D1 D1 D1 Kruskal D1 D1 D1 D1 Dijkstra D1 D1 D1 D1 Floyd s algorithm D2 TSP D1 D2 D2 D1 Route inspection D1 D1 D2 D1 Network Flows D2 D2 D2 Critical Path Analysis Activity networks D2 node D1 arc D1 arc D2 arc Optimisation Matchings D1 D1 D2 Decision analysis D2

4 Graph theory Graph theory was until recently considered to be just recreational but is now regarded as a subject in its own right. It has widespread applications in all areas of mathematics and science. Many problems can be modelled as graphs (circuit diagrams, molecules in chemistry) or weighted graphs, called networks (distances networks, cost networks, decision trees) Graph theory is also widely used in sociology as a way, for example, to measure an individual s prestige or through the use of social network analysis software.

5 Leads to network analysis Graph theory forms the basis of all the network algorithms

6 Some graph theory (maths as a foreign language) A walk moves along an edge from one vertex to another and can visit a vertex or an edge more than once. A trail moves along an edge from one vertex to another and can visit a vertex more than once but cannot traverse an edge more than once. A path moves along an edge from one vertex to another but cannot visit a vertex more than once.

7 Some other terms used with graphs A graph is connected if there is a path between every pair of vertices A simple graph is one with no loops and no multiple edges between any two vertices

8 Some other terms used with graphs A tree is a simple connected graph with no cycles or loops A digraph (directed graph) is one in which at least one edge has a direction associated with it.

9 Some other terms used with graphs A complete graph is a simple graph in which every pair of vertices is connected A planar graph is a which can be drawn without any edges crossing

10 Teaching tips Give students a vocabulary list to learn Students often seem to think that the graph theory is merely background to the network algorithms and is not a topic in its own right, however there is always a graph theory question on the D1 paper. But do make sure that students realise that many of the network algorithms use graph theory (e.g. Eulerian Trails in Route Inspection, Hamiltonian cycles in TSP). The exam questions can be on any aspect of the topic and are difficult to predict so make sure that students do a wide variety of questions and are prepared for the unexpected.

11 Networks Be able to model and solve problems using networks Key Points: Understand notation and terminology. Minimum spanning trees Kruskal s and Prim s algorithms Shortest path Dijkstra s algorithm Touring Route Inspection and Travelling Salesperson

12 A problem A cable TV company based in Plymouth wants to link all the towns on the map. To keep costs to a minimum they want to use as little cable as possible. What strategy should they use to solve the problem?

13 The Modelling Cycle Accept solution Real life Problem Yes No Review Make simplifying assumptions Compare the solution with reality is it realistic? Interpret the solution in terms of the original problem Define variables and decide on the mathematical techniques to be used Solve the mathematical problem

14 A problem A cable TV company based in Plymouth wants to link all the towns on the map. To keep costs to a minimum they want to use as little cable as possible. What strategy should they use to solve the problem? Model the map as a network

15 A problem A cable TV company based in Plymouth wants to link all the towns on the map. To keep costs to a minimum they want to use as little cable as possible. What strategy should they use to solve the problem? Spanning tree of minimum length Minimum connector

16 Minimum connector algorithms Kruskal s algorithm 1. Select the shortest edge in a network 2. Select the next shortest edge which does not create a cycle 3. Repeat step 2 until all vertices have been connected Prim s algorithm 1. Select any vertex 2. Select the shortest edge connected to that vertex 3. Select the shortest edge connected to any vertex already connected 4. Repeat step 3 until all vertices have been connected

17 A cable company want to connect five villages to their network which currently extends to the market town of Avonford. What is the minimum length of cable needed? Brinleigh 5 Cornwell Avonford 7 Fingley 8 Donster Edan

18 We model the situation as a network, then the problem is to find the minimum connector for the network B 5 C A 7 F 8 D E

19 A 3 4 B 7 5 F 5 Minimum spanning tree C D Kruskal s ED 2 AB 3 CD 4 AE 4 EF 5 Prim s AB 3 AE 4 ED 2 CD 4 EF 5 E Total weight of tree: 18

20 Prim s can also be done on a table A B C D E F Brinleigh Cornwell A B C D Avenford 3 4 Fingley E F Donster The spanning tree is shown 2 4 in the diagram Length Edan = 18Km

21 Some points to note Both algorithms will always give solutions with the same total weight. They will usually select edges in a different order Occasionally they will use different edges this may happen when you have to choose between edges with the same length. In this case there is more than one minimum connector for the network.

22 Teaching tips: Minimum connector Before you have introduced any algorithms define the problem, and ask students to come up with their own algorithm (usually about 50/50 Prim/Kruskal) For Kruskal, it may be helpful to list all the edges in order of length before starting, depending on the size of the network. Make sure that students LIST the order in which they add the edges to the solution

23 Teaching tips: Minimum connector Prim s (matrix form): Introduce this with a simple example, using both network and matrix methods simultaneously to show what is going on. The textbook approach of physically deleting rows can t be replicated by students, there isn t time. Suggest they use coloured highlighter for deleting rows and for adding columns to the solution while first learning this. However, they must be able to do solutions confidently in pencil in the exams. Draw the solution as you go along this can help prevent putting in cycles by mistake. This can also prompt students to look in all relevant columns, not just the one they just added.

24 Networks Be able to model and solve problems using networks Shortest Path Dijkstra s algorithm Googlemaps and Google Earth are brilliant tools for networks

25 Dijkstra s Algorithm This algorithm finds the shortest path from the start vertex to every other vertex in the network. Showing working correctly is vital to getting the marks in these questions B 4 F A C D E G

26 Dijkstra s Algorithm Order in which vertices are labelled. Distance from A to vertex B Working 4 F 1 0 A D Label vertex A 1 as it is the first vertex labelled 3 C E 2 G

27 Teaching tips- shortest path Explain that permanent labels are unbeatable. This includes making the 0 at the starting vertex permanent, since 0 is the shortest possible distance from the start to itself. Stress the importance of working values, examiners will be looking for the correct values in here. When updating temporary values, if there is already a temporary value at a vertex, don t write a bigger one there.

28 Traversable graphs Which of these graphs can be drawn without taking your pen off the paper or repeating any edges?

29 Traversable graphs Yes start and finish in different places Semi-Eulerian Yes- start and finish in the same place Eulerian No What is significant about the results? Can you explain why?

30 Touring: The underlying Graph Theory An Eulerian Trail is a route that travels along every edge once only and returns to the starting vertex A Hamiltonian Cycle is a closed path which visits every vertex (once and only once excluding the start / finish vertex!).

31 Touring algorithms Route Inspection often called Chinese postman after the Chinese mathematician, Mei Ko Kwan, who developed the algorithm in 1962 This algorithm is about trying to find an Eulerian trail in a network. Of course, you usually can t! Travelling salesperson this algorithm is about trying to find a Hamiltonian cycle in a network.

32 Route inspection problems 1. Identify the odd vertices in the network 2. Consider all the routes joining pairs of odd vertices and select the one with the least weight. 3. Find the sum of the weights on all the edges 4. Shortest distance is the sum of the weights plus the extra that must be travelled 5. Find a tour which repeats the edges found in step 2.

33 Teaching tips: Route inspection Make sure that students consider all possible pairings and are systematic where looking for pairs of odd vertices in RI. Modelling skills are very important when tackling all types of network problems. Students must be able to interpret the solution in the context of the original problem. Students may need to work from networks, tables or combinations of both, so make sure they are prepared for any eventuality.

34 Teaching tips: Route Inspection Make sure that students consider all possible pairings and are systematic where looking for pairs of odd vertices in RI. Modelling skills are very important when tackling all types of network problem. Students must be able to interpret the solution in the context of the original problem. Students may need to work from networks, tables or combinations of both, so make sure they are prepared for any eventuality.

35 Flight plan Find the route that visits every city (at least once) and uses the least air miles.

36 Terminology A walk in a network is a finite sequence of edges such that the end vertex of one edge is the start vertex of the next. A tour is a walk that visits every vertex in the network, returning to the starting vertex. The travelling salesman problem is trying to find a walk that gives a minimum tour i.e. find a tour of minimum weight

37 Solution to the TSP problem There is no known algorithm that solves this problem So instead of doing an exhaustive search we can make use of heuristic algorithms where we can have a good answer but probably not the optimal solution.

38 Solution to the TSP problem Lower bound < optimal solution Upper bound We just need to find a largest lower bound that we can We just need to find the smallest upper bound that we can

39 In each network, find a cycle of minimum length, starting and finishing at A. Is it possible to find a cycle that visits each vertex once, and only once? A cycle that that visits each vertex once, and only once including all vertices of a network is a Hamiltonian cycle

40 Two types of problem: Classical and practical In the classical problem you must visit each vertex only once before returning to the start In the practical problem you must visit each vertex at least once before returning to the start Key idea: If you convert a network into a complete network of least distances, the classical problem and the practical problem are the same. All complete graphs do have Hamiltonian cycles

41 Nearest neighbour algorithm for upper bounds 1. Choose any vertex as the starting point. 2. From the vertices not already selected find the nearest vertex to the last one. 3. Repeat step 2 until all vertices have been selected. 4. Join the last vertex to the first vertex. 5. Repeat for different starting vertices. 6. Choose the least upper bound.

42 An algorithm for lower bounds 1. Choose a vertex and delete it and all its edges from the network. 2. Find a minimum connector for the remaining network using Kruskal s or Prim s. 3. Add in the weights of the two least weight deleted edges. 4. Repeat deleting a different vertex. 5. Choose the greatest lower bound.

43 Teaching tips: Travelling salesperson TSP uses the MST algorithms so make sure the students are secure in this first. It is important to distinguish between TSP and RI and which to use for what type of problem. When teaching TSP, make sure students understand the distinction between the classical problem, which uses a complete network, and the practical problem, which almost certainly doesn t. When finding a lower bound for TSP, we use the method where a vertex is deleted. The resulting lower bound DOES NOT give a tour, except in exceptional circumstances. Students don t understand why it is the GREATEST lower bound and LEAST upper bound.

44 Network Problems Activity

45 Exam advice Most marks (70%+) are for clearly demonstrating use of the algorithms, very few of the marks in the exam go on answers. Examiners are trying to give as many marks as possible Students often show signs of running out of time There are extension marks on every paper usually at the end of long questions Poor setting out and illegible handwriting makes it very hard to give marks; clear working is absolutely essential.

46 A satellite navigation system Sainsburys Home Delivery Snowplough clearing all the roads in a town A courier with one urgent delivery to make A milk tanker from a dairy collecting milk from farms Ambulance travelling to an emergency Cable TV company linking towns using as little cable as possible Council re painting the lines in the middle of the roads Road builder joining a few villages as economically as possible Courier with several deliveries to make Highways Authority inspecting roads for fallen trees after a storm Pedestrian precinct being created to connect places of interest in a town centre Water pipelines being laid to connect pumping stations as economically as possible A person planning the route to their holiday house A parking official patrolling all the streets in an area A family on a shopping trip with several shops to visit before they can return to their car

47 Classification Activity This activity is for students who have learnt the topics of Minimum Spanning Trees, Route Inspection Problem, Travelling Salesperson Problem and Dijkstra s Algorithm, to practise matching problems to techniques. Resources needed: sugar paper, scissors, glue, set of cards This is an activity for students working in small groups. Each group is given a set of cards to cut up and a piece of sugar paper which they divide into four as shown Minimum Spanning Trees Dijkstra s Algorithm Route Inspection Travelling Salesperson Students then discuss in their groups and stick the cards in the most appropriate section. They are then to think up one example of their own for each topic which they write on the blank cards and stick in the appropriate section. Groups can then be paired up to compare and discuss their results, or can feedback to the whole class. Posters can be displayed in the classroom

Module 2: NETWORKS AND DECISION MATHEMATICS

Further Mathematics 2017 Module 2: NETWORKS AND DECISION MATHEMATICS Chapter 9 Undirected Graphs and Networks Key knowledge the conventions, terminology, properties and types of graphs; edge, face, loop,