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### nossi ch 6

1. 1. A Mathematical View of Our World 1 st ed. Parks, Musser, Trimpe, Maurer, and Maurer
2. 2. Chapter 6 Routes and Networks
3. 3. Section 6.1 Routing Problems <ul><li>Goals </li></ul><ul><ul><li>Study graphs, vertices, and edges </li></ul></ul><ul><ul><li>Study paths and circuits </li></ul></ul><ul><ul><li>Study connected graphs </li></ul></ul><ul><ul><ul><li>Use Euler’s Theorem </li></ul></ul></ul><ul><ul><ul><li>Use Fleury’s Algorithm </li></ul></ul></ul>
4. 4. 6.1 Initial Problem <ul><li>Can the people of Konigsberg, pictured here, take a walk crossing each bridge exactly one time? </li></ul><ul><ul><li>The solution will be given at the end of the section. </li></ul></ul>
5. 5. Graphs <ul><li>A graph is a collection of one or more points, called vertices , and the paths connecting them, called edges . </li></ul><ul><ul><li>The singular of vertices is vertex . </li></ul></ul>
6. 6. Graphs <ul><li>The vertices are drawn as dots. </li></ul><ul><li>The edges are drawn as straight or curved lines. </li></ul><ul><ul><li>Each edge has two ends with a vertex at each end, or else it is a loop connecting a vertex to itself. </li></ul></ul>
7. 7. Equivalent Graphs <ul><li>Two graphs are considered the same if they have the same number of vertices connected in the same way, even if they look different. </li></ul>
8. 8. Example 1 <ul><li>The three graphs shown are equivalent. </li></ul>A is connected to B, B to C, and C to D
9. 9. Vertices and Edges <ul><li>Two edges may appear to cross, but they do not actually intersect unless there is a vertex. </li></ul><ul><li>Example: Graph (a) shows two edges crossing where there is no vertex. Think of a road that crosses an interstate but there is not an exit. </li></ul>
10. 10. Example 2 <ul><li>Graphs can be drawn to represent real-life connections, such as roads between cities. </li></ul><ul><li>Example: Draw a graph to represent highway connections between Boise, Olympia, Portland, Salem, Seattle, and Spokane. </li></ul>
11. 11. Example 2 <ul><li>The vertices represent the cities, and the edges represent the roads. </li></ul><ul><li>4 of the 5 cities are along the same interstate. </li></ul>
12. 12. Example 3 <ul><li>List the pairs of adjacent vertices in the graph. </li></ul><ul><li>Two vertices are called adjacent vertices if there is an edge connecting them. </li></ul>
13. 13. Example 3, cont’d <ul><li>Solution: The pairs of adjacent vertices are: </li></ul><ul><ul><li>A and B, A and C, A and D, A and E </li></ul></ul><ul><ul><li>B and D </li></ul></ul><ul><ul><li>C and E </li></ul></ul><ul><ul><li>E and E, since there is a loop at E </li></ul></ul>
14. 14. Example 4 <ul><li>Draw two different graphs with the following properties : </li></ul><ul><ul><li>Vertices: A, B, C, and D </li></ul></ul><ul><ul><li>Adjacent vertices are: </li></ul></ul><ul><ul><ul><li>A and C, B and C, B and D, C and D. </li></ul></ul></ul>
15. 15. Example 4, cont’d <ul><li>Solution: Two possible graphs are shown below. </li></ul><ul><ul><li>Vertices: A, B, C, and D </li></ul></ul><ul><ul><li>Adjacent vertices are: </li></ul></ul><ul><ul><ul><li>A and C, B and C, B and D, C and D. </li></ul></ul></ul>
16. 16. Degree of a Vertex <ul><li>The degree of a vertex is the total number of edges at the vertex. </li></ul><ul><ul><li>Find the degree of a vertex by counting the number of edges leaving or arriving at the vertex. </li></ul></ul><ul><ul><li>A loop contributes 2 to the degree of a vertex. </li></ul></ul>
17. 17. Example 5 <ul><li>Find the degree of each vertex in the graph. </li></ul>
18. 18. Example 5 <ul><li>Find the degree of each vertex in the graph. </li></ul><ul><li>Solution: </li></ul><ul><ul><li>A has degree 5, </li></ul></ul><ul><ul><li>B has degree 3, </li></ul></ul><ul><ul><li>C has degree 2, </li></ul></ul><ul><ul><li>D has degree 2, </li></ul></ul><ul><ul><li>E has degree 4. </li></ul></ul>
19. 19. Example 5 <ul><ul><li>A has degree 5, </li></ul></ul><ul><ul><li>B has degree 3, </li></ul></ul><ul><ul><li>C has degree 2, </li></ul></ul><ul><ul><li>D has degree 2, </li></ul></ul><ul><ul><li>E has degree 4. </li></ul></ul><ul><ul><li>How many edges are there? _____ </li></ul></ul><ul><ul><li>How many total degrees are there? </li></ul></ul>
20. 20. Theorem <ul><li>If d is the sum of all the degrees of the vertices in a graph and e is the number of edges in the graph, then </li></ul><ul><li>d = 2 e . </li></ul><ul><ul><ul><li>Note: This theorem is true because each edge adds 2 to the degree total. </li></ul></ul></ul>
21. 21. Example 6 <ul><li>Verify that the theorem is true for the following graph. </li></ul><ul><li>Solution: The sum of the degrees is 16. The number of edges is 8. Check that 16 = 2(8). </li></ul>
22. 22. Paths <ul><li>A path is a route in a graph. </li></ul><ul><ul><li>Each vertex used is adjacent to the next vertex used. </li></ul></ul><ul><ul><li>Each edge in the graph is used at most one time. </li></ul></ul><ul><li>A path is written as a list of vertices. </li></ul>
23. 23. Example 7 <ul><li>Find three different paths in the graph. </li></ul><ul><li>Solution: </li></ul><ul><ul><li>One possible path is A,B,D,A,E . </li></ul></ul><ul><ul><li>A second path is B,A,B,D. </li></ul></ul><ul><ul><li>A third path is C,E,E,A </li></ul></ul><ul><ul><ul><li>There are many other paths. </li></ul></ul></ul>
24. 24. Circuits <ul><li>A circuit is a path that begins and ends at the same vertex. </li></ul><ul><li>Example: One circuit in the graph is A,B,D,A. </li></ul>
25. 25. Euler Paths <ul><li>A path that uses every edge of a graph exactly once is called an Euler path . (Euler is pronounced “oiler.”) </li></ul><ul><li>Example: The edges are numbered in the order used. </li></ul>
26. 26. Euler Circuits <ul><li>A circuit that uses every edge of a graph exactly once is called an Euler circuit . </li></ul><ul><li>Example: The edges are numbered in the order used. </li></ul>
27. 27. Connected Graphs <ul><li>A graph is said to be connected if, for every pair of vertices, there is a path that contains them. </li></ul><ul><li>If a graph is not connected, it is said to be disconnected . </li></ul><ul><ul><li>An edge is called a bridge if its removal from the graph would change the graph from connected to disconnected. </li></ul></ul>
28. 28. Example 8 <ul><li>Explain whether or not the graphs are connected. </li></ul><ul><ul><li>Solution: The graph is connected because there is a path, using one edge or more, between every pair of vertices. </li></ul></ul>
29. 29. Example 8, cont’d <ul><ul><li>Solution 2: The graph is disconnected because, for example, there is no path between vertex A and vertex G. </li></ul></ul>
30. 30. Example 9 <ul><li>Identify any bridges in the (connected) graph. </li></ul><ul><li>Solution: There are three bridges: the edge between D and E, the edge between D and F, and the edge between F and G. </li></ul>
31. 31. Example 10 <ul><li>Find all components of the graph. </li></ul><ul><li>There are three components: the piece containing A, E, F, and G; the piece containing B and C; and the piece containing D. </li></ul>
32. 32. Degree Theorem <ul><li>Any graph must have an even number of vertices with odd degree. </li></ul><ul><ul><li>We saw earlier that the sum of all the degrees in a graph is always an even number, so: </li></ul></ul><ul><ul><ul><li>Vertices with even degrees do not have to come in pairs. </li></ul></ul></ul><ul><ul><ul><li>Vertices with odd degrees come in pairs. </li></ul></ul></ul>
33. 33. Euler’s Theorem <ul><li>For a connected graph: </li></ul><ul><ul><li>If the graph has zero vertices of odd degree, then it has at least one Euler circuit. </li></ul></ul><ul><ul><li>If a graph has an Euler circuit, then it has no vertices of odd degree. </li></ul></ul>
34. 34. Euler’s Theorem cont’d <ul><li>For a connected graph: </li></ul><ul><ul><li>If the graph has exactly two vertices of odd degree, then </li></ul></ul><ul><ul><ul><li>It does not have an Euler circuit . </li></ul></ul></ul><ul><ul><ul><li>It has at least one Euler path . </li></ul></ul></ul><ul><ul><ul><li>Any Euler path starts at one of the two vertices of odd degree and ends at the other. </li></ul></ul></ul>
35. 35. Euler’s Theorem cont’d <ul><li>For a connected graph: </li></ul><ul><ul><li>If the graph has more than two vertices of odd degree, then </li></ul></ul><ul><ul><ul><li>It does not have an Euler circuit. </li></ul></ul></ul><ul><ul><ul><li>It does not have an Euler path. </li></ul></ul></ul>
36. 36. Example 11 <ul><li>Does the graph have an Euler circuit, Euler path, or neither? </li></ul>2 F 2 E 4 D 2 C 4 B 4 A Degree Vertex
37. 37. Example 11 <ul><ul><li>Solution: Every vertex is of degree 2 or degree 4, so by Euler’s theorem the graph has an Euler circuit. </li></ul></ul><ul><li>Does the graph have an Euler circuit, Euler path, or neither? </li></ul>
38. 38. Example 11, cont’d <ul><ul><li>Solution: There are two vertices of odd degree, E and G, so by Euler’s theorem the graph has only an Euler path. </li></ul></ul>
39. 39. Fleury’s Algorithm <ul><li>If Euler’s Theorem indicates that there is an Euler circuit, Fleury’s Algorithm can be used to find the circuit. </li></ul><ul><li>The step-by-step procedure is as follows: </li></ul><ul><ul><li>Copy all the vertices, but not the edges, of the original graph. </li></ul></ul><ul><ul><li>Select any vertex on the original graph and mark it as the present position. </li></ul></ul>
40. 40. Fleury’s Algorithm, cont’d <ul><ul><li>Choose one edge at the marked vertex and transfer it to the new graph. </li></ul></ul><ul><ul><ul><li>Do not choose a bridge, unless it is the only edge available. </li></ul></ul></ul><ul><ul><ul><li>Number the edges in the order they are transferred. </li></ul></ul></ul>
41. 41. Fleury’s Algorithm, cont’d <ul><ul><li>Use the vertex on the other end of the edge as the new present position. Repeat step 3. </li></ul></ul><ul><ul><li>When you transfer the last edge you are done. </li></ul></ul>
42. 42. Example 12 <ul><li>Use Fleury’s Algorithm to find an Euler circuit for the graph. (See ex.6.10 pgs 350-352 in text.) </li></ul><ul><li>Possible Final Solution: </li></ul>
43. 43. 6.1 Initial Problem Solution <ul><li>The idea of Euler circuits can be used to solve the question of whether the people of Konigsberg can take a walk crossing each bridge exactly one time. </li></ul>
44. 44. Initial Problem Solution, cont’d <ul><li>The vertices represent the land and the edges the bridges. The described walk would be an Euler path. </li></ul><ul><li>We use Euler’s Theorem to show that there is no such path for their walk because there are four vertices of odd degree. </li></ul>
45. 45. Section 6.1 Assignment <ul><li>Pg 353 (3, 11a,b, 17-19,25,27,39) </li></ul>
46. 46. Section 6.2 Network Problems <ul><li>Goals </li></ul><ul><ul><li>Study weighted graphs </li></ul></ul><ul><ul><li>Study spanning trees </li></ul></ul><ul><ul><ul><li>Study minimal spanning trees </li></ul></ul></ul><ul><ul><ul><li>Use Kruskal’s algorithm </li></ul></ul></ul>
47. 47. 6.2 Initial Problem <ul><li>How can the walkways be constructed so that: </li></ul><ul><ul><li>Students can go from any building on campus to any other building without needing to walk outside. </li></ul></ul><ul><ul><li>The total length of the walkways is minimized. </li></ul></ul><ul><ul><li>The solution will be given at the end of the section. </li></ul></ul>
48. 48. Weighted Graphs <ul><li>A graph in which each edge has a number associated with it is called a weighted graph . The number corresponding to an edge is called the weight of the edge. </li></ul>
49. 49. Example 1 <ul><li>Suppose Ed has: </li></ul><ul><ul><li>A 35-minute drive from his home to his workplace </li></ul></ul><ul><ul><li>A 15-minute drive from his workplace to his health club </li></ul></ul><ul><ul><li>A 25 minute drive from his health club to his home </li></ul></ul><ul><li>Draw a weighted graph to represent this situation. </li></ul>
50. 50. Example 1, cont’d <ul><li>Solution: </li></ul><ul><ul><li>The vertices of the graph will represent Ed’s home ( H ), his workplace ( W ), and his health club ( C ). </li></ul></ul><ul><ul><li>The weight of each edge will represent the driving time between the two places. </li></ul></ul><ul><ul><li>For simplicity, we can draw all the edges as straight lines and with the same length. </li></ul></ul>
51. 51. Example 1, cont’d <ul><li>Solution, cont’d: One possible weighted graph is shown below. </li></ul>
52. 52. Redundant Edges <ul><li>Redundant edges are edges that can be removed while still leaving a path between the two vertices. </li></ul><ul><li>Example: If edge CH is removed from this graph, Ed can still get from home to his club by driving to work and then to the club. </li></ul><ul><ul><li>The edge CH is a redundant edge. </li></ul></ul>
53. 53. Subgraphs <ul><li>A subgraph of a given graph is a set of vertices and edges chosen from among those of the original graph. </li></ul><ul><li>Example: Removing edge CH creates the subgraph shown at right. </li></ul>
54. 54. Subgraphs <ul><li>Example: Another subgraph could be created by removing vertex C and its two incident edges. </li></ul>
55. 55. Example 2 <ul><li>A bus line goes between the cities shown in the table. </li></ul><ul><ul><li>Draw a weighted graph to represent the situation. </li></ul></ul><ul><ul><li>Find two different routes from Selma to Anniston. </li></ul></ul>
56. 56. Example 2 <ul><li>Solution: A weighted graph is shown below. </li></ul>
57. 57. Example 2 <ul><li>Solution: One possible route from Selma to Anniston is SMGA , with a total weight (length) of 49 + 127 + 27 = 203 miles. </li></ul>
58. 58. Example 2 <ul><li>Solution, cont’d: Another possible route from Selma to Anniston is SBA , with a total weight (length) of 160 + 62 = 222 miles. </li></ul>
59. 59. Trees <ul><li>One way to recognize that a graph contains redundant edges is to find a circuit in the graph. </li></ul><ul><ul><li>Recall that a circuit is a path that begins at a vertex and returns to that vertex without using any edges twice. </li></ul></ul><ul><li>A connected graph that has no circuits is called a tree . </li></ul>
60. 60. Trees, cont’d <ul><li>The following figure gives examples of graphs that are trees and a graph that is not a tree. </li></ul>
61. 61. Example 3 <ul><li>This graph is not a tree because it contains several circuits. </li></ul><ul><li>Find a subgraph that is a tree. </li></ul>
62. 62. Example 3 <ul><li>Solution: Circuits are highlighted and edges are removed until the remaining subgraph has no circuits left. </li></ul>
63. 63. Spanning Trees <ul><li>A spanning tree is a subgraph that: </li></ul><ul><ul><li>Contains all the original vertices of the graph. </li></ul></ul><ul><ul><li>Is connected. </li></ul></ul><ul><ul><li>Contains no circuits. </li></ul></ul>
64. 64. Spanning Trees <ul><li>Every connected graph has at least one spanning tree. </li></ul><ul><li>An example of a spanning tree is shown below. </li></ul>
65. 65. Minimal Spanning Trees <ul><li>A spanning tree with the smallest possible total weight is called a minimal spanning tree . </li></ul><ul><ul><li>Since a connected, weighted graph can have multiple spanning trees, there may be one or more than one of those spanning trees with the smallest total weight. </li></ul></ul>
66. 66. Minimal Spanning Trees <ul><li>For the connected, weighted graph shown below: </li></ul><ul><ul><li>Subgraph (b) is not a minimal spanning tree. </li></ul></ul><ul><ul><li>Subgraph (c) is a minimal spanning tree. </li></ul></ul>
67. 67. Kruskal’s Algorithm <ul><li>A method known as Kruskal’s algorithm can be used to find a minimal spanning tree of a connected, weighted graph. </li></ul><ul><li>Kruskal’s algorithm begins with all the vertices of a graph and adds edges one by one, using the idea of acceptable and unacceptable edges. </li></ul>
68. 68. Kruskal’s Algorithm <ul><li>Acceptable edges are </li></ul><ul><ul><li>Edges that do not share a vertex with any edges already chosen. </li></ul></ul>
69. 69. Kruskal’s Algorithm <ul><li>Acceptable edges are </li></ul><ul><ul><li>Edges that connect two components of the subgraph. </li></ul></ul>
70. 70. Kruskal’s Algorithm <ul><li>Acceptable edges are </li></ul><ul><ul><li>Edges that connect a component of the subgraph to a new vertex. </li></ul></ul>
71. 71. Kruskal’s Algorithm <ul><li>Unacceptable edges are </li></ul><ul><ul><li>Edges that add to a component of the subgraph, but do not add a new vertex. </li></ul></ul>
72. 72. Kruskal’s Algorithm <ul><li>Consider only the vertices of the graph. </li></ul><ul><li>Add the edge with the smallest weight. </li></ul><ul><li>Add the acceptable edge with the smallest weight. </li></ul><ul><li>Determine whether all vertices are connected by a path. </li></ul><ul><ul><li>If so, you have a minimal spanning tree. </li></ul></ul><ul><ul><li>If not, repeat Step 3. </li></ul></ul>
73. 73. Example 4 <ul><li>Construct a minimal spanning tree for the weighted graph. </li></ul>
74. 74. Example 4 <ul><li>Solution: </li></ul><ul><ul><li>Start with the 4 vertices. </li></ul></ul><ul><ul><li>Add the edge with the smallest weight. </li></ul></ul>
75. 75. Example 4 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>All unused edges are acceptable, so we simply choose the edge with the smallest weight. </li></ul></ul><ul><ul><ul><li>Select the edge with weight 2.6. </li></ul></ul></ul>
76. 76. Example 4 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>The acceptable edges at this step were the ones with weights 4.8 and 3.7. </li></ul></ul><ul><ul><ul><li>Select the edge with weight 3.7. </li></ul></ul></ul>
77. 77. Example 4 <ul><li>Solution, cont’d: </li></ul><ul><ul><li>The subgraph is now connected, so this is a minimal spanning tree. </li></ul></ul><ul><ul><ul><li>The weight of the tree is 1.5 + 2.6 + 3.7 = 7.8. </li></ul></ul></ul>
78. 78. 6.2 Initial Problem Solution <ul><li>How can a college build walkways of the minimum length possible so that students can go from any building on campus to any other building without needing to walk outside? </li></ul>
79. 79. Initial Problem Solution <ul><li>Create a weighted graph to represent the buildings and existing sidewalks on the campus. </li></ul><ul><ul><li>The weight of each edge is the distance between the two buildings. </li></ul></ul>
80. 80. Initial Problem Solution <ul><li>Use Kruskal’s algorithm to find a minimal spanning tree. </li></ul><ul><ul><li>The edges included in the minimal spanning tree represent the sidewalks that should be converted into covered walkways. </li></ul></ul>
81. 81. Initial Problem Solution <ul><li>Under this plan, students can get from any building to any other building without going outside, although they may have to walk farther inside. </li></ul><ul><li>The length of the walkways was minimized to a total of 210 + 220 + 120 + 200 = 750 feet. </li></ul>
82. 82. Section 6.2 Assignment <ul><li>Pg 370 (5 a,b and list 2 different routes from St Louis (StL) to Cleveland and the total distance of each route, 9,11,27,29) </li></ul>
83. 83. Section 6.3 Traveling-Salesperson Problem <ul><li>Goals </li></ul><ul><ul><li>Study Hamiltonian circuits </li></ul></ul><ul><ul><li>Study the Traveling-Salesperson Problem </li></ul></ul><ul><ul><ul><li>Use the Brute-Force Algorithm </li></ul></ul></ul><ul><ul><ul><li>Use the Nearest-Neighbor Algorithm </li></ul></ul></ul><ul><ul><ul><li>Use the Cheapest-Link Algorithm </li></ul></ul></ul>
84. 84. 6.3 Initial Problem <ul><li>An author needs to find a route that will minimize the total distance she must drive between the cities. </li></ul><ul><ul><li>The solution will be given at the end of the section. </li></ul></ul>
85. 85. Hamiltonian Circuits <ul><li>A path that visits each vertex in a graph exactly once is called a Hamiltonian path . </li></ul><ul><li>If a Hamiltonian path begins and ends at the same vertex , it is called a Hamiltonian circuit . </li></ul>
86. 86. Example 1 <ul><li>The highlighted edges in the graph represent a Hamiltonian path. </li></ul><ul><ul><li>The path can be written AGFEDBC or CBDEFGA. </li></ul></ul>
87. 87. Example 2 <ul><li>The highlighted edges in the graph represent a Hamiltonian circuit. </li></ul><ul><ul><li>The circuit, starting at A , can be written AGFEDCBA </li></ul></ul>
88. 88. Hamiltonian Circuits, cont’d <ul><li>Note: A Hamiltonian path or circuit must pass through every vertex of the graph, but it does not have to use every edge. </li></ul><ul><li>A Hamiltonian path can be used, for example, to model a delivery truck stopping at every house in a neighborhood but not using every street. </li></ul>
89. 89. Example 3 <ul><li>Find a Hamiltonian path in each graph or explain why you cannot. </li></ul>
90. 90. Example 3 <ul><li>Solution: The path ABCD is a Hamiltonian path. </li></ul><ul><ul><li>Adding an edge to the path forms the Hamiltonian circuit ABCDA . </li></ul></ul><ul><ul><li>Other Hamiltonian paths and circuits are possible. </li></ul></ul>
91. 91. Example 3, cont’d <ul><li>Solution: The path ABCD is a Hamiltonian path. </li></ul><ul><ul><li>There is no Hamiltonian circuit. </li></ul></ul>
92. 92. Example 3, cont’d <ul><li>There is no Hamiltonian path. </li></ul><ul><ul><li>Any Hamiltonian path must start at either A or F . </li></ul></ul><ul><ul><li>Any path that starts at A will leave out either C , D , or F. </li></ul></ul><ul><ul><li>Any path that starts at F will leave out either C , D , or A. </li></ul></ul>
93. 93. Complete Graphs <ul><li>A complete graph is a graph in which every pair of vertices is connected by exactly one edge. </li></ul>
94. 94. Example 4 <ul><li>For each graph, determine whether or not it is complete. </li></ul>
95. 95. Example 4, cont’d <ul><li>Solution: Check each pair of vertices. </li></ul><ul><ul><li>There is exactly one edge between each pair of vertices, so the graph is complete. </li></ul></ul>
96. 96. Example 4, cont’d <ul><li>Solution: Check each pair of vertices. </li></ul><ul><ul><li>There is no edge from A to C , for example. The graph is not complete. </li></ul></ul>
97. 97. Example 4, cont’d <ul><li>Solution: Check each pair of vertices. </li></ul><ul><ul><li>There are two edges from A to F , for example. The graph is not complete. </li></ul></ul>
98. 98. Complete Graphs <ul><li>In general, a graph may or may not have a Hamiltonian path or circuit. </li></ul><ul><li>A complete graph, however, always has a Hamiltonian circuit. </li></ul><ul><ul><li>Any list of all the vertices in the graph will be a Hamiltonian path. </li></ul></ul><ul><ul><li>Adding the starting vertex to the end of the list describing the path will create a Hamiltonian circuit. </li></ul></ul>
99. 99. Example 6 <ul><li>List all the Hamiltonian paths and all the Hamiltonian circuits in the graph. </li></ul>
100. 100. Example 6, cont’d <ul><li>Solution: </li></ul>
101. 101. Example 6, cont’d <ul><li>Note that the complete graph in the previous example had 3 vertices and 6 different Hamiltonian circuits. </li></ul><ul><li>This is the case because each circuit can begin at one of the 3 vertices, travel to one of 2 vertices second, and travel to the 1 remaining vertex last. </li></ul><ul><ul><li>There are 3(2)(1) = 6 Hamiltonian circuits. </li></ul></ul>
102. 102. Theorem on Complete Graphs <ul><li>The number of Hamiltonian paths in a complete graph with n vertices is </li></ul><ul><li>n ! = n ( n – 1)( n – 2)( n – 3)…(1). </li></ul><ul><ul><li>The symbol n ! is read “n factorial”. </li></ul></ul>
103. 103. Question: <ul><li>How many possible Hamiltonian circuits exist in a complete graph with 9 vertices? </li></ul><ul><li>a. 81 </li></ul><ul><li>b. 40,320 </li></ul><ul><li>c. 362,880 </li></ul><ul><li>d. 3,265,920 </li></ul>
104. 104. Question: <ul><li>How many possible Hamiltonian circuits exist in a complete graph with 9 vertices? </li></ul><ul><li>9*8*7*6*5*4*3*2*1 = 362,880 </li></ul><ul><li>a. 81 </li></ul><ul><li>b. 40,320 </li></ul><ul><li>c. 362,880 correct answer </li></ul><ul><li>d. 3,265,920 </li></ul>
105. 105. Traveling-Salesperson Problem <ul><li>A salesperson who must visit various cities and then return home wants to minimize the cost (measured in distance, time, or money) of his or her trip. </li></ul><ul><li>Because Hamiltonian circuits can be applied to find this type of least-cost trip, the problem of finding a least-cost Hamiltonian circuit is called the Traveling-Salesperson Problem . </li></ul>
106. 106. Complete Weighted Graphs <ul><li>The cost of a path in a weighted graph is the sum of the weights assigned to the edges in the path. </li></ul><ul><li>When costs are assigned to each edge in a complete graph, the graph is called a complete weighted graph . </li></ul>
107. 107. Question: <ul><li>What is the cost of the path ABD in the graph below? </li></ul><ul><li>a. 26 </li></ul><ul><li>b. 14 </li></ul><ul><li>c. 24 </li></ul><ul><li>d. 41 </li></ul>
108. 108. Question: <ul><li>What is the cost of the path ABD in the graph below? </li></ul><ul><li>a. 26 10+16=26 </li></ul><ul><li>b. 14 </li></ul><ul><li>c. 24 </li></ul><ul><li>d. 41 </li></ul>
109. 109. Example 7 <ul><li>List all possible Hamiltonian paths, and their costs, in the complete weighted graph. </li></ul><ul><li>Find the path of least cost. </li></ul>
110. 110. Example 7, cont’d <ul><li>Solution: The graph has 4 vertices, so there are 4! = 24 different Hamiltonian paths. </li></ul><ul><li>These paths and their costs are listed on the next slide. </li></ul>
111. 111. Example 7, cont’d
112. 112. Example 7, cont’d <ul><li>Solution, cont’d: There are 2 Hamiltonian paths with the lowest cost of 37. </li></ul><ul><ul><li>CDAB </li></ul></ul><ul><ul><li>BADC </li></ul></ul>
113. 113. Example 7, cont’d <ul><li>List all possible Hamiltonian circuits that start and end at vertex A. </li></ul><ul><li>Find the circuit at A of least cost. </li></ul>
114. 114. Example 7, cont’d <ul><li>Solution: </li></ul>
115. 115. Example 7, cont’d <ul><li>Solution, cont’d: There are 2 least-cost Hamiltonian circuits with a cost of 74. </li></ul><ul><ul><li>ABCDEA </li></ul></ul><ul><ul><li>AEDCBA </li></ul></ul><ul><ul><ul><li>These 2 circuits are mirror images of each other. </li></ul></ul></ul>
116. 116. Traveling-Salesperson, cont’d <ul><li>The method used to solve the previous example is called the Brute-Force Algorithm . </li></ul><ul><ul><li>List all possible Hamiltonian circuits. </li></ul></ul><ul><ul><li>Calculate the cost of each circuit. </li></ul></ul><ul><ul><li>Find the 2 mirror image circuits of least cost. </li></ul></ul><ul><li>This method is very time-consuming. </li></ul>
117. 117. Traveling-Salesperson, cont’d <ul><li>Instead of the Brute-Force Algorithm, faster approximation algorithms are typically used. </li></ul><ul><li>An approximation algorithm finds a Hamiltonian circuit that is either the least-cost circuit or is one that is not much more costly than the least-cost circuit. </li></ul><ul><ul><li>The Nearest-Neighbor Algorithm </li></ul></ul><ul><ul><li>The Cheapest-Link Algorithm </li></ul></ul><ul><ul><li>(We will not study these other than the names.) </li></ul></ul>
118. 118. 6.3 Initial Problem Solution <ul><li>An author needs to find a route that will minimize the total distance she must drive between the cities shown below. </li></ul>
119. 119. Initial Problem Solution, cont’d <ul><li>Imagine a graph in which the vertices represent the cities and the weighted edges represent the distances. </li></ul><ul><li>This is a Traveling-Salesperson problem, and there are 9! = 362,880 possible Hamiltonian circuits. </li></ul><ul><ul><li>Of these, 8! = 40,320 circuits start at New Orleans. </li></ul></ul>
120. 120. Initial Problem Solution, cont’d <ul><li>With such a large number of circuits, it is foolish to try to solve this by hand using the Brute-Force Algorithm. </li></ul><ul><li>The Cheapest-Link Algorithm will be used to find a low-cost Hamiltonian circuit representing a low-cost trip for the author. </li></ul><ul><ul><li>Rather than drawing the large graph, view the costs (distances) in the table. </li></ul></ul>
121. 121. Initial Problem Solution, cont’d <ul><li>Using the algorithm, add acceptable edges one at a time. </li></ul><ul><ul><li>The first 7 edges added are shown below. </li></ul></ul>
122. 122. Initial Problem Solution, cont’d <ul><li>The cheapest acceptable edge at this point is the route between Charlotte and Orlando. </li></ul><ul><li>After that, the route from New Orleans to Orlando must be added to complete the circuit. </li></ul>
123. 123. Initial Problem Solution, cont’d <ul><li>The approximate least-cost Hamiltonian circuit is shown below. </li></ul><ul><li>The total cost of the circuit is 2462, so the author will drive 2462 miles if she takes this route on her trip. </li></ul>
124. 124. A fun website with Traveling Salesman Problem (TSP) Artwork: http://www.oberlin.edu/math/faculty/bosch/tspart-page.html
125. 125. Section 6.3 Assignment <ul><li>Pg 396 (1,5,7,17,19,25, 27) For problems 25 and 27 you may look at the charts in the back of the book rather than making one yourself. Then, write a sentence explaining your choice of an answer and how the chart helps you make the choice you did.) </li></ul>
126. 126. Chapter 6 Assignments <ul><li>Pg 353 (3, 11a,b, 17-19,25,27,39) </li></ul><ul><li>Pg 370 (5 a,b and list 2 different routes from St Louis (StL) to Cleveland and the total distance of each route, 9,11,27,29) </li></ul><ul><li>Pg 396 (1,5,7,17,19,25, 27) For problems 25 and 27 you may look at the charts in the back of the book rather than making one yourself. Then, write a sentence explaining your choice of an answer and how the chart helps you make the choice you did.) </li></ul>