1. Copyright © 2005 Pearson Education, Inc. Slide 13-1

2. Copyright © 2005 Pearson Education, Inc. Chapter 13

3. Copyright © 2005 Pearson Education, Inc. Slide 13-3 Bridges of Konigsberg 13-A

4. Copyright © 2005 Pearson Education, Inc. Slide 13-4 Network Terminology 13-A network A collection of points or objects that are interconnected in some way. vertex 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. circuitA path within a network that begins and ends at the same vertex without using any edges more than once. Euler circuit A path through a network that starts and ends at the same point and traverses each edge exactly once. complete network Every vertex is connected to every other vertex. treeA 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.

5. Copyright © 2005 Pearson Education, Inc. Slide 13-5 Kruskals Algorithm for Finding Minimum Cost Networks Step 1:Make a list of the edges from the least expensive to the most expensive. Step 2:Begin with the least expensive edge. Highlight it to indicate that it is part of the minimum cost spanning network. Continue to select edges in order of increasing cost until every vertex is connected, either directly or indirectly, to every other vertex. Step 3:If a closed circuit has been created within the spanning network, remove the most expensive edge. The final result is the minimum cost spanning network. 13-A

6. Copyright © 2005 Pearson Education, Inc. Slide 13-6 Hamiltonian Circuits and Five National Parks 13-B

7. Copyright © 2005 Pearson Education, Inc. Slide 13-7 The Nearest Neighbor Method and the Traveling Salesman Problem 13-B Courtesy of Bill Cook, David Applegate and Robert Bixby, Rice University and Vasek Chvatal, Rutgers University. 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.

8. Copyright © 2005 Pearson Education, Inc. Slide 13-8 A House Building Project 13-C

9. Copyright © 2005 Pearson Education, Inc. Slide 13-9 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. 13-C

10. Copyright © 2005 Pearson Education, Inc. Slide 13-10 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. 13-C

11. Copyright © 2005 Pearson Education, Inc. Slide 13-11 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 13-C