4/15/2013 Math for Liberal Arts MAT 110: Chapter 13 Notes Networks and Euler Circuits Graph Theory David J. Gisch Network Representation Network Representation • Network: A collection of points or objects that are Example 13.1: Create a network representation of the interconnected in some way. following situation. • Vertex: A point to represent an object such as a computer, phone, city, island, etc. which makes up a network. • Edge: Represented by a line or curve to be a connection between two vertices. 1
4/15/2013 Network Representation Country Borders Example 13.2: Create a network representation of the Example 13.3: Draw a diagram where the vertices following situation. represent the countries and the edges represent shared borders. Circuit Euler Circuits • Circuit: A path that starts and stops at the same point is • An Euler circuit is a path through a network that starts a circuit. and ends at the same point and traverses (travels) every � dge exactly once. 2
4/15/2013 Order & Degree Euler Circuit – Even Order Example 13.3: State whether each of the following has an • The degree of a vertex is the number of distinct edges Euler circuit. connecting to that vertex. • The order of a network is the number of vertices in that network. • An Euler circuit exists for a network if each vertex has an even number of edges (i.e. the degree of each vertex is even). Finding the Euler Circuit Euler Circuit Example 13.4: Find an Euler circuit for the bridges of If an Euler circuit exists: Konigsberg. • Begin your circuit from any vertex in the network. • As you choose edges to follow, never use an edge that is the only connection to a part of the network that you have not already visited. ▫ This is known as the “burning bridge” rule. 3
4/15/2013 Euler Circuit Euler Circuit Example 13.5: Find an Euler circuit for the computer Example 13.6: Find an Euler circuit for the country border network. network. Euler Circuit Euler Circuit Example 13.7: Find an Euler circuit for the following Example 13.8: Find an Euler circuit for the following network. network. 4
4/15/2013 Hamilton Circuit A Ham iltonian circuit is a path that passes through every vertex of a network exactly once and returns to the starting vertex. The paths indicated by arrows in (a) and (b) are Hamiltonian circuits, while (c) has no Hamiltonian circuits. Complete Graphs and Hamilton Circuits Complete network Factorial • Com plete network is a network where every vertex is • 5! � 5 � 4 � 3 � 2 � 1 directly connected to every other vertex. • 6! � 6 � 5 � 4 � 3 � 2 � 1 • The number of Hamilton Circuits in a complete network • 20! � 20 � 19 � 18 � ⋯ � 2 � 1 with order � is � � 1 ! 2 • Most calculators have factorial button. ▫ Look under the PRB button for it. 5
4/15/2013 Complete Network Traveling S alesman • A Hamilton circuit is very practical. • The solution to a traveling salesman problem is the shortest path (smallest total of the lengths) that starts and ends in the same place and visits each city once. The twelve Hamiltonian circuits for a complete network of order 5. ��� ! �� � � 5 and therefore there are � � � 12 possible � Hamilton circuits. Travel S alesman Finding the S hortest Hamilton Circuit • We want to visit each park once on our route. Below are • Unfortunately the only way to guarantee you can find the two ways to do that. shortest Hamilton circuit is to check the length of each possible route. • The route on the left has us traveling 664 miles, while the route on the right has us traveling 499 miles. • PROBLEM! • Is the one on the right the shortest router or is there ▫ If the graph only has 15 vertices then it has 15 � 1 ! another? � 44 ������� 2 Possible Hamilton circuits. 6
4/15/2013 The Nearest Neighbor Method and the Traveling Nearest Neighbor Method S alesman Problem The near-optimal solution to finding the shortest • To find the quickest approximation we can use the path among 13,509 cities with populations over 500. nearest neighbor method. • Beginning at any vertex, travel to the nearest vertex that has not yet been visited. Continue this process of visiting “nearest neighbors” until the circuit is complete. • This method does not guarantee you the shortest route but generally gets you a path close to the shortest amount. Courtesy of Bill Cook, David Applegate and Robert Bixby, Rice University and Vasek Chvatal, Rutgers University. Hamilton Circuit Hamilton Circuit Example 13.9: Use the nearest neighbor method to find a Example 13.10: The following table show thee distances good approximation for the shortest route. between pairs of towns in a rural county. Create a network showing the five towns and distances between them and find the shortest route by checking each possibility. 7
4/15/2013 Tree • Tree is a network in which all of the vertices are connected and no circuits appear. Spanning Tree and Optimization Hamilton Circuit S panning Tree Example 13.11: Which if the following are trees? Why or • A spanning tree is a tree within any network. why not? Original Network 8
4/15/2013 Minimum S panning tree Minimum Cost S panning Tree • If we have a network with weighted edges, the spanning tree with the least “weight” or total value is called the m inim um spanning tree . This is very similar to the nearest neighbor idea with Hamilton circuits but with spanning trees we do not want a circuit. A map of seven towns (capital letters) and the routes between them along which telephone lines could be strung, along with the network representation. Minimum Cost S panning Tree Kruskal’s Algorithm Two spanning networks. The total cost of each spanning network is the sum of the individual costs on its edges. The total cost for spanning network (a) is much higher than the total cost for spanning network (b). 9
4/15/2013 Find the Minimum Cost Find the Minimum Cost S panning Tree Network Terminology Review Circuit A path within a network that begins and ends at the same vertex. Complete network Every vertex is directly connected to every other vertex. Tree A network in which all of the vertices are Scheduling Problems connected and no circuits appear. Order The number of vertices in a network. Degree of vertex The number of edges connected to the vertex. 10
4/15/2013 A House Building Proj ect Limiting Tasks and Critical Path • Each edge is labeled with the number of months needed • When two (or more) tasks can occur at the same time to complete the task. between two stages of the project, the task that requires the most time is called the lim iting ta sk . • In some phases of the project, only one task can be undertaken at a time. • The critica l p a th through the network is the path that • During other phases, two or more tasks can be carried includes all the limiting tasks. The length of the critical out concurrently. path is the completion time for the project. Finding Earliest S tart and Finish Times Finding Latest S tart and Finish Times • The ea rliest sta rt tim e (EST) of a task leaving a • The la test finish tim e (LFT) of a task entering a particular vertex is the largest of the earliest finish times particular vertex is the sm allest of the latest start times of the tasks entering that vertex. of the tasks leaving that vertex. • The ea rliest finish tim e (EFT) of a task is the earliest • The la test sta rt tim e (LST) of a task is the latest finish start time of that task plus the time required for the task. time of that task m inus the time required for the task. That is, That is, LST = LFT time for task EFT = EST + time for task. 11
4/15/2013 Examples Examples • What is the limiting task between C and F? • What is the limiting task between F and H? Examples Examples • What is the critical path and therefore the completion • What is earliest start time (EST) for ordering appliances? time? • What is earliest start time (EST) for trim work? 12
4/15/2013 Examples S lack Time • What is earliest finish time (EFT) for ordering • Sla ck tim e is the longest you can delay a task without appliances? disrupting the earliest finish time. • What is slack time for construction? • What is earliest finish time (EFT) for construction? • What is slack time for ordering appliances? Examples “ Homework” • What is latest start time (LST) for ordering appliances? • Check out page 778 #21-28 • What is latest start time (LST) for construction? • We will not hand this in. You will have problems very similar to these on your final. 13
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