Math for Liberal Arts MAT 110: Chapter 13 Notes

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in some way. Vertex: A point to represent an object such as a computer, phone, city, island, etc. which makes up a network.

1 Math for Liberal Arts MAT 110: Chapter 13 Notes Graph Theory David J. Gisch Networks and Euler Circuits Network Representation Network: A collection of points or objects that are interconnected in some way. Vertex: A point to represent an object such as a computer, phone, city, island, etc. which makes up a network. Network Representation Example 13.1: Create a network representation of the following situation. Edge: Represented by a line or curve to be a connection between two vertices. 1

3: Draw a diagram where the vertices represent the countries and the edges represent shared borders.

2 Network Representation Example 13.2: Create a network representation of the following situation. Country Borders Example 13.3: Draw a diagram where the vertices represent the countries and the edges represent shared borders. Circuit Circuit: A path that starts and stops at the same point is a circuit. Euler Circuits An Euler circuit is a path through a network that starts and ends at the same point and traverses (travels) every dge exactly once. 2

Order & Degree The degree of a vertex is the number of distinct edges connecting to that vertex. The order of a network is the number of vertices in that network.

3 Order & Degree The degree of a vertex is the number of distinct edges 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). Euler Circuit Even Order Example 13.3: State whether each of the following has an Euler circuit. Finding the Euler Circuit If an Euler circuit exists: 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. Euler Circuit Example 13.4: Find an Euler circuit for the bridges of Konigsberg. 3

4 Euler Circuit Example 13.5: Find an Euler circuit for the computer network. Euler Circuit Example 13.6: Find an Euler circuit for the country border network. Euler Circuit Example 13.7: Find an Euler circuit for the following network. Euler Circuit Example 13.8: Find an Euler circuit for the following network. 4

5 Hamilton Circuit A Hamiltonian 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 Complete network is a network where every vertex is directly connected to every other vertex. The number of Hamilton Circuits in a complete network with order is 1! 2 Factorial 5! ! ! Most calculators have factorial button. Look under the PRB button for it. 5

Complete Network Traveling Salesman A Hamilton circuit is very practical.

each city once. The twelve Hamiltonian circuits for a complete network of order 5. 5 and therefore there are! Hamilton circuits.

The route on the left has us traveling 664 miles, while the route on the right has us traveling 499 miles.

6 Complete Network Traveling Salesman 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! Hamilton circuits. 12possible Travel Salesman We want to visit each park once on our route. Below are two ways to do that. The route on the left has us traveling 664 miles, while the route on the right has us traveling 499 miles. Is the one on the right the shortest router or is there another? Finding the Shortest Hamilton Circuit Unfortunately the only way to guarantee you can find the shortest Hamilton circuit is to check the length of each possible route. PROBLEM! If the graph only has 15 vertices then it has 15 1! 44 2 Possible Hamilton circuits. 6

Nearest Neighbor Method To find the quickest approximation we can use the nearest neighbor method.

The Nearest Neighbor Method and the Traveling Salesman Problem The near-optimal solution to finding the shortest path among 13,509 cities with populations over 500.

Courtesy of Bill Cook, David Applegate and Robert Bixby, Rice University and Vasek Chvatal, Rutgers University. Hamilton Circuit Example 13.

7 Nearest Neighbor Method To find the quickest approximation we can use the 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. The Nearest Neighbor Method and the Traveling Salesman Problem The near-optimal solution to finding the shortest path among 13,509 cities with populations over 500. 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 Example 13.9: Use the nearest neighbor method to find a good approximation for the shortest route. Hamilton Circuit Example 13.10: The following table show thee distances 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

8 Tree Tree is a network in which all of the vertices are connected and no circuits appear. Spanning Tree and Optimization Hamilton Circuit Example 13.11: Which if the following are trees? Why or why not? Spanning Tree A spanning tree is a tree within any network. Original Network 8

Minimum Spanning tree If we have a network with weighted edges, the spanning tree with the least weight or total value is called the minimum spanning tree.

9 Minimum Spanning tree If we have a network with weighted edges, the spanning tree with the least weight or total value is called the minimum spanning tree. Minimum Cost 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 Spanning 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

Complete network Every vertex is directly connected to every other vertex.

10 Find the Minimum Cost Spanning Tree Find the Minimum Cost 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 connected and no circuits appear. Order The number of vertices in a network. Degree of vertex The number of edges connected to the vertex. Scheduling Problems 10

11 A House Building Project Each edge is labeled with the number of months needed to complete the task. In some phases of the project, only one task can be undertaken at a time. During other phases, two or more tasks can be carried out concurrently. Limiting Tasks and Critical Path When two (or more) tasks can occur at the same time between two stages of the project, the task that requires the most time is called the limiting task. The critical path through the network is the path that includes all the limiting tasks. The length of the critical path is the completion time for the project. Finding Earliest Start and Finish Times The earliest start time (EST) of a task leaving a particular vertex is the largest of the earliest finish times of the tasks entering that vertex. The earliest finish time (EFT) of a task is the earliest start time of that task plus the time required for the task. That is, EFT = EST + time for task. Finding Latest Start and Finish Times The latest finish time (LFT) of a task entering a particular vertex is the smallest of the latest start times of the tasks leaving that vertex. The latest start time (LST) of a task is the latest finish time of that task minus the time required for the task. That is, LST = LFT time for task 11

12 Examples What is the limiting task between C and F? Examples What is the limiting task between F and H? Examples What is the critical path and therefore the completion time? Examples What is earliest start time (EST) for ordering appliances? What is earliest start time (EST) for trim work? 12

13 Examples What is earliest finish time (EFT) for ordering appliances? What is earliest finish time (EFT) for construction? Slack Time Slack time is the longest you can delay a task without disrupting the earliest finish time. What is slack time for construction? What is slack time for ordering appliances? Examples What is latest start time (LST) for ordering appliances? What is latest start time (LST) for construction? Homework Check out page 778 #21-28 We will not hand this in. You will have problems very similar to these on your final. 13

Chapter 14 Section 3 - Slide 1

AND Chapter 14 Section 3 - Slide 1 Chapter 14 Graph Theory Chapter 14 Section 3 - Slide WHAT YOU WILL LEARN Graphs, paths and circuits The Königsberg bridge problem Euler paths and Euler circuits Hamilton