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Copyright © 2008 Pearson Education, Inc. Slide 13-2 Chapter 13 Mathematics and Business 13ANetwork Analysis 13BThe Traveling Salesman Problem 13CScheduling Problems

Copyright © 2008 Pearson Education, Inc. Slide 13-3 Unit 13A Network Analysis

Copyright © 2008 Pearson Education, Inc. Slide 13-4 Network Representation 13-A NetworkA collection of points or objects that are interconnected in some way. VertexAn object such as a computer, phone, city, island, etc. which makes up a network. EdgeRepresented by a line or curve to be a connection between two vertices.

Copyright © 2008 Pearson Education, Inc. Slide 13-5 Bridges of Konigsberg 13-A A map of the Pregel River flowing through Königsberg and a network representation of the bridges of Königsberg. The vertices represent the land masses (capital letters) and the edges represent the bridges (lowercase letters).

Network Analysis and the War in Iraq and Afghanistan http://www.npr.org/2010/12/03/131755378/u-s- connects-the-dots-to-catch-roadside-bombers Copyright © 2008 Pearson Education, Inc. Slide 13-6

Copyright © 2008 Pearson Education, Inc. Slide 13-7 An Office Intranet A layout of computers, servers, and cables in a small office intranet. 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-8 An Office Intranet A network diagram overlaid on the office intranet. A B E D C F G H I 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-9 An Office Intranet A network diagram representing the connections in the office intranet. Vertices represent computers (capital letters) and edges represent cables connecting computers (lowercase letters). 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-10 Euler Circuits 13-A An Euler circuit is a path through a network that starts and ends at the same point and traverses every edge exactly once. An Euler circuit exists for a network if each vertex has an even number of edges. In the figure below, networks (a) and (b) have Euler circuits, but (c) and (d) do not.

Copyright © 2008 Pearson Education, Inc. Slide 13-11 Network Analysis 13-A Which network has an Euler circuit? a) b) c) d)

Copyright © 2008 Pearson Education, Inc. Slide 13-12 Network Analysis 13-A Which network has an Euler circuit? a) b) c) d)

Copyright © 2008 Pearson Education, Inc. Slide 13-13 The Burning Bridges Rule for Finding Euler Circuits You may begin your circuit from any vertex in the network. However, 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. 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-14 Applying the Burning Bridges Rule Find an Euler circuit for this network. 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-15 Applying the Burning Bridges Rule 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-16 Network Terminology 13-A Circuit A path within a network that begins and ends at the same vertex without using any edges more than once. Complete network Every vertex is directly 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.

Copyright © 2008 Pearson Education, Inc. Slide 13-17 Network Analysis 13-A What is the order of the network below? a) 4 b) 5 c) 6 d) 8

Copyright © 2008 Pearson Education, Inc. Slide 13-18 Network Analysis 13-A What is the order of the network below? a) 4 b) 5 c) 6 d) 8

Copyright © 2008 Pearson Education, Inc. Slide 13-19 Network Analysis 13-A Which network is a tree? a) b) c) d)

Copyright © 2008 Pearson Education, Inc. Slide 13-20 Network Analysis 13-A Which network is a tree? a) b) c) d)

Copyright © 2008 Pearson Education, Inc. Slide 13-21 Minimum Cost Spanning Networks A map of seven towns (capital letters) and the routes between them along which telephone lines could be strung, along with the network representation. 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-22 Minimum Cost Spanning Networks 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). 13-A

Copyright © 2008 Pearson Education, Inc. Slide 13-23 Kruskal’s 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