Section Hamilton Paths, and Hamilton Circuits. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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1 Section 14.3 Hamilton Paths, and Hamilton Circuits
2 INB Table of Contents Date Topic Page # September 11, 2013 Section 14.3 Examples/Handout 18 September 11, 2013 Section 14.3 Notes
3 What You Will Learn Hamilton Paths Hamilton Circuits Complete Graphs Traveling Salesman Problems
4 Hamilton Paths A Hamilton path is a path that contains each vertex of a graph exactly once
5 Hamilton Paths The graph shown has Hamilton path A, B, C, E, D. The graph also has Hamilton path C, B, A, D, E. Can you find some others?
6 Hamilton Paths The graph shown has Hamilton path A, B, C, F, H, E, G, D. The graph also has Hamilton path G, D, E, H, F, C, B, A. Can you find some others?
7 Hamilton Circuits A Hamilton circuit is a path that begins and ends at the same vertex and passes through all other vertices of the graph exactly one time
8 Hamilton Circuits The graph shown has Hamilton circuit A, B, C, E, D, A. Can you find another?
9 Hamilton Circuits The graph shown has Hamilton circuit A, B, D, G, E, H, F, C, A. A Hamilton circuit starts and ends at the same vertex
10 Complete Graph A complete graph is a graph that has an edge between each pair of its vertices
11 Example 2: Finding Hamilton Circuits Determine a Hamilton circuit for the complete graph shown
12 Number of Unique Hamilton Circuits in a Complete Graph The number of unique Hamilton circuits in a complete graph with n vertices is (n 1)!, where (n 1)! = (n 1)(n 2)(n 3) (3)(2)(1)
13 Example 4: American Idol Travel Keith Urban, Jennifer Lopez, and Harry Connick, Jr., the judges for the television show American Idol, are in Hollywood (H). They need to travel to the following cities to judge contestants auditions: San Antonio (SA), East Rutherford (ER), Birmingham (B), Memphis (M), and Seattle (S). In how many different ways can Keith, Jennifer, and Harry, traveling together, visit each of these cities and return to Hollywood?
14 Example 4: American Idol Travel Solution Represent this problem with the complete graph. 6 vertices represent the 6 locations. Edges represent the one-way flights between these locations
15 Traveling Salesman Problems Complete graphs can represent cities and the process of traveling between these cities. Our goal in a traveling salesman problem is to find the least expensive or shortest way to visit each city once and return home. In terms of graph theory, our goal is to find the Hamilton circuit with the lowest associated cost or distance
16 Traveling Salesman Problems The Hamilton circuit with the lowest associated cost (or shortest distance, etc.) is called the optimal solution. We will discuss two methods, the brute force method and the nearest neighbor method, for determining the optimal solution. A complete, weighted graph is a complete graph with the weights (or numbers) listed on the edges
17 The Brute Force Method of Solving Traveling Salesman Problems To determine the optimal solution: 1. Represent the problem with a complete, weighted graph. 2. List all possible Hamilton circuits for this graph. 3. Determine the cost (or distance) associated with each of these Hamilton circuits. 4. The Hamilton circuit with the lowest cost (or shortest distance) is the optimal solution
18 Example 5: Using the Brute Force Method Julienne Ward is the Southeast District Sales Director for Addison Wesley. On the next slide is a complete, weighted graph, whose numbers represent one-way fares, in dollars, between the cities. We want to use the brute force method to determine the optimal solution for Julienne to visit her regional sales offices
19 Example 5: Using the Brute Force Method She will start in Orlando (O); visit offices in Atlanta (A), Memphis (M), and New Orleans (N); and then return to Orlando
20 Example 5: Using the Brute Force Method Circuit Cost per leg Total Cost
21 Nearest Neighbor Method A method for finding an approximate solution to a traveling salesman problem. Approximate solutions can be used in cases where determining the optimal solution is not reasonable. One method for finding an approximate solution is the nearest neighbor method
22 Nearest Neighbor Method To approximate a optimal solution: 1. Represent the problem with a complete, weighted graph. 2. Identify the starting vertex. 3. Of all the edges attached to the starting vertex, choose the edge that has the smallest weight. This edge is generally either the shortest distance or the lowest cost. Travel along this edge to the second vertex
23 Nearest Neighbor Method con t To approximate a optimal solution: 4. At the second vertex, choose the edge that has the smallest weight that does not lead to a vertex already visited. Travel along this edge to the third vertex. 5. Continue this process, each time moving along the edge with the smallest weight until all vertices are visited. 6. Travel back to the original vertex
24 Example 6: Using the Nearest Neighbor Method In Example 4, we discussed the American Idol judges plan to visit five cities in which auditions would take place and then return to Hollywood. Use the nearest neighbor method to determine an approximate solution for the judges visits. The one-way per person flight prices between cities are given in the table on the next slide
25 Example 6: Using the Nearest Neighbor Method
26 Example 6: Using the Nearest Neighbor Method Solution Here s the complete, weighted graph. Start at H, go to S. From S, go to SA. From SA, go to B. From B, go to ER. From ER, go to M. From M, go to H
27 Example 6: Using the Nearest Neighbor Method Solution H to S, $108 S to SA, $119 SA to B, $274 B to ER, $258 ER to M, $104 M to H, $144 Total cost is $
28 Example 6: Using the Nearest Neighbor Method Solution For comparison, here are 4 randomly chosen Hamilton circuits with costs
Section Hamilton Paths, and Hamilton Circuits. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 14.3 Hamilton Paths, and Hamilton Circuits What You Will Learn Hamilton Paths Hamilton Circuits Complete Graphs Traveling Salesman Problems 14.3-2 Hamilton Paths A Hamilton path is a path that
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