The Olympic Torch Tour

7324 /www/nrich/html/content/id/7324/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/6957">Warm-up</a></li> <li><a href="http://nrich.maths.org/7372">Try this next</a></li> <li><a href="http://nrich.maths.org/2369">Think higher</a></li> <li><a href="http://nrich.maths.org/2585">Read: mathematics</a></li> <li><a href="http://www.sciencenews.org/view/generic/id/72472/title/Cells_take_on_traveling_salesman_problem">Read: science</a></li> <li><a href="http://plus.maths.org/content/all-tied">Explore further</a></li> </ul> <div> </div> If you are a teacher, click <a href="/7324/note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on... In May 2012, the Olympic torch will arrive at Lands End for a 70 day tour of the UK, ending in London. The plan is to cover towns and villages so that 95% of Britons will be within 10 miles of the torch relay. Imagine a mini-Olympic torch tour running between 4 cities in the UK, with the following constraints: <ul> <li>The torch starts and finishes in London</li> <li>The torch should pass each city once and only once</li> <li>The following table lists the distance between each city (in miles as measured by Google Maps)</li> </ul> <table border="1"> <tbody> <tr> <td> </td> <td>London</td> <td>Cambridge</td> <td>Bath</td> <td>Coventry</td> </tr> <tr> <td>London</td> <td style="text-align: center;">0</td> <td style="text-align: center;">50</td> <td style="text-align: center;">96</td> <td style="text-align: center;">86</td> </tr> <tr> <td>Cambridge</td> <td style="text-align: center;">50</td> <td style="text-align: center;">0</td> <td style="text-align: center;">120</td> <td style="text-align: center;">70</td> </tr> <tr> <td>Bath</td> <td style="text-align: center;">96</td> <td style="text-align: center;">120</td> <td style="text-align: center;">0</td> <td style="text-align: center;">80</td> </tr> <tr> <td>Coventry</td> <td style="text-align: center;">86</td> <td style="text-align: center;">70</td> <td style="text-align: center;">80</td> <td style="text-align: center;">0</td> </tr> </tbody> </table> <ul> <li>What is the shortest route?</li> <li>How can you be sure it is the shortest?</li> <li>How many different routes are there?</li> </ul> <div> </div> <div>Let's now try a slightly longer tour of 5 cities. We'll add Oxford to the list: </div> <table border="1"> <tbody> <tr> <td> </td> <td>London</td> <td>Cambridge</td> <td>Bath</td> <td>Coventry</td> <td>Oxford</td> </tr> <tr> <td>London</td> <td style="text-align: center;">0</td> <td style="text-align: center;">50</td> <td style="text-align: center;">96</td> <td style="text-align: center;">86</td> <td style="text-align: center;">60</td> </tr> <tr> <td>Cambridge</td> <td style="text-align: center;">50</td> <td style="text-align: center;">0</td> <td style="text-align: center;">120</td> <td style="text-align: center;">70</td> <td style="text-align: center;">65</td> </tr> <tr> <td>Bath</td> <td style="text-align: center;">96</td> <td style="text-align: center;">120</td> <td style="text-align: center;">0</td> <td style="text-align: center;">80</td> <td style="text-align: center;">54</td> </tr> <tr> <td>Coventry</td> <td style="text-align: center;">86</td> <td style="text-align: center;">70</td> <td style="text-align: center;">80</td> <td style="text-align: center;">0</td> <td style="text-align: center;">46</td> </tr> <tr> <td>Oxford</td> <td style="text-align: center;">60</td> <td style="text-align: center;">65</td> <td style="text-align: center;">54</td> <td style="text-align: center;">46</td> <td style="text-align: center;">0</td> </tr> </tbody> </table> <ul> <li>What is the shortest route now?</li> <li>How many different possible routes did you need to consider?</li> </ul> Is there an efficient way to work out the number of different possible routes when there are 10 cities? 15 cities?... Suppose a computer could calculate one million routes per second. How long would it take to find the optimal route for 10 cities? 15 cities? 20 cities? <div class="framework">NOTES AND BACKGROUND The type of question we have explored above is a famous problem in computation complexity theory known as the <a href="http://en.wikipedia.org/wiki/Travelling_salesman_problem">Travelling Salesman Problem</a>. Perhaps a better question for the torch tour is not to find the shortest or longest route, but to find the maximum number of cities the torch can visit whose route length is at most, say 2000 miles, or visit as many populated towns as possible. These are variants of the original problem known as the 'Orienteering Problem' and the 'Prize Collecting Travelling Salesman Problem'. It is an active area of research among mathematicians and has a wide range of applications.</div> </mdoxml> <mdoxml version="1.0">Niharika was the only one to send us a solution this time round: The possible routes are: 1. London-Cambridge-Bath-Coventry-London 2. London-Cambridge-Coventry-Bath-London 3. London-Bath-Cambridge-Coventry-London 4. London-Bath-Coventry-Cambridge-London 5. London-Coventry-Bath-Cambridge-London 6. London-Coventry-Cambridge-Bath-London. I worked out the distances for each route, and they came to: 1. 336 miles 2. 296 miles 3. 372 miles 4. 296 miles 5. 336 miles 6. 372 miles The shortest routes are routes 2 and 4. Now imagine that each city along our route is a box. We have five boxes to line up. The first and last box must be 'London', so there is only 1 way to fill those boxes. The second box can be filled in one of 3 ways. Once we've used that city up, the third box can be filled in 2 ways. Finally when we come to the fourth box there is only 1 way to fill it in. So there are 1*3*2*1*1 = 6 routes. I guessed that, when extending to 5 cities, we should probably start with the shortest route for 4 cities and then add the extra city in. In this case, we should start with routes 2 and 4, and add Oxford in in all the different places, and see which distance is smallest. In each case, least distance is covered if we add in Oxford between Coventry and Bath. In this case there are 6 boxes and 1*4*3*2*1*1 = 24 routes. When there are n cities, there are n+1 boxes, and so there are $1\times (n-1)\times (n-2)\times \dots \times 2 \times 1\times 1$ routes - just think of how many ways there are to fill in each box. But if you know the shortest routes for the case of n-1 cities, my guess is that you should be able to add the n-th city to those. Great - thanks, Niharika!</mdoxml> <mdoxml version="1.0"><h3>Why do this problem?</h3> As well as encouraging students to work systematically, this problem introduces ideas from Decision Maths and Computer Science such as the Travelling Salesperson problem and the efficiency of algorithms. <h3>Possible approach</h3> This problem may be of interest to students who have worked on <a href="/7822">The Olympic Flame: Are You in the 95%?</a> Hand out <a href="/content/id/7324/Olympic%20torch.pdf">this worksheet</a> and encourage students to work on their own or in pairs on the first part of the problem with four cities. Bring the class together to discuss the sort of strategies they used to find a short route, and to discuss the checking that was necessary to be sure that their route was the shortest. Once students are happy with the idea that a 'brute force' algorithm that checks every possibility is the ONLY way to be certain of finding the shortest route, set them the second challenge to find the shortest route when there are five cities. "You need to be able to convince everyone that the route you find is the shortest. Think about how you could record your work to make sure you don't miss any possible routes." While students are working, circulate and notice anyone who is using a particularly useful representation such as a systematic way of listing possible routes, or a tree-diagram approach. Bring the class together and invite those students to share their representations. "Why does it take longer to check all the routes for five cities than four?" "Is there a quick way to work out how many different possible routes there will be, for six cities?" "How will the number of possible routes continue to grow?" Finally, as students begin to get some appreciation of the way the number of routes grows as the number of cities increases, challenge them to work out the time taken to implement the brute force algorithm by computer. <h3>Key questions</h3> How can you represent the different routes in a way that makes sure you don't miss any? If I add in an extra city, how does that affect the number of different possible routes? <h3>Possible extension</h3> There are rich opportunities for students to go and research topics in Decision Mathematics and Computer Science and to learn about the challenges involved in seeking solutions to problems using computers. <h3>Possible support</h3> <a href="/7822">The Olympic Flame: Are You in the 95%?</a> provides a much simpler context to explore the Olympic torch tour.</mdoxml> <mdoxml version="1.0">You could try to list all the possible routes and calculate the distance the torch would need to travel for each.</mdoxml> <mdoxml version="1.0">1)The shortest route is London -> Cambridge -> Coventry -> Bath -> London with a distance of 296 miles. 2)If you look at an atlas, this route puts the cities in a sensible loop. 3)The only way to be certain this route is optimal is to consider all the other routes. However, it seems likely this is the best route, because it uses most of the smaller numbers from the table. 4)There are 3*2*1=6 routes in total. Starting from London: <ul> <li>3 places to choose to visit first</li> <li>2 places to choose to visit next</li> <li>the third place is fixed - you have to visit the one you haven't visited yet</li> </ul> 5)The longest route is London -> Bath -> Cambridge -> Coventry -> London with a distance of 372 miles. After adding in Oxford to the route: The shortest route is now 314 miles: London -> Cambridge -> Coventry -> Bath -> Oxford -> London The longest route is now 413 miles: London->Bath->Cambridge->Oxford->Coventry-> London For the case, with 5 cities including London, we need to check 4*3*2*1=24 possible routes. In general, for n cities, we need to check (n-1)! routes. (Can you explain why?). Here's a table showing how long this would take: <table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td style="text-align: center;">Number of cities</td> <td style="text-align: center;">(n-1)!</td> <td style="text-align: center;">Time take to search all routes</td> </tr> <tr> <td>4</td> <td>6</td> <td>6*10^-6 s</td> </tr> <tr> <td>5</td> <td>24</td> <td>2.4*10^-5 s</td> </tr> <tr> <td>10</td> <td>362880</td> <td>0.36s</td> </tr> <tr> <td>15</td> <td>1.31*10^12</td> <td>1.31*10^6 s = 15.2 days</td> </tr> <tr> <td>20</td> <td>2.43*10^18</td> <td>2.43*10^12 s = 77100 years</td> </tr> <tr> <td>100</td> <td>9.33*10^157</td> <td>9.33*10^151 s = 1.08*10^147 years!</td> </tr> </tbody> </table> N.B. The age of the universe is 433.6 x 1015s! (We say this algorithm has order of n! time complexity.) The greedy algorithm takes time proportional to n^2, where n is the number of cities in the route. Departing London, it needs to find the smallest of (n-1) numbers, which takes (n-2) comparisons. From this city, it needs to find the smallest of (n-2) numbers, which takes (n-3) comparisons. This continues, until the destination before London is fixed, as all the other cities have been visited. In total, this takes (n-2)+(n-3)+... + 1 = (n-1)*(n-2)/2 = O(n^2) operations as claimed. If n=100, this would take about 0.01s, assuming we can still do 10^6 operations a second. The olympic route can be seen <a href="http://www.bbc.co.uk/news/uk-13391986">here</a>. Was your guess correct? Investigation from the end of the problem - new resource? Suppose that you don't have the time for the brute force approach to complete. You could try a 'greedy algorithm' which starts at London and then simply travels to the next nearest remaining city and so on. How quickly would the greedy algorithm take the computer for 100 cities? Investigation: Implement the greedy algorithm 'by eye' for the major population centres shown on the map below. What order would you place the cities in by using this method? Does it agree with your friends' routes? Did any questions or observations arise in you mind when you implemented the greedy algorithm? Now try to determine a better route using cunning or any other method at your disposal. You could check to find the actual length of the routes to see if you improved on the order provided by the greedy algorithm. You might wish to compare with the actual torch route when it is revealed to see if your ordering for these places matches the actual ordering used on the torch relay. <mdo:image src="map.png"></mdo:image></mdoxml> 2 4 0 0 0 1 0 The Olympic Torch Tour Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route? Decision Mathematics and Combinatorics Combinations Decision Mathematics and Combinatorics Networks/Graph Theory Decision Mathematics and Combinatorics Travelling salesperson problem Applications sport Applications Technology