Finite Math A Chapter 6 Notes Hamilton Circuits

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Chapter 6: The Mathematics of Touring (Hamilton Circuits) and Hamilton Paths 6.1 Traveling Salesman Problems/ 6.2 Hamilton Paths and Circuits A traveling salesman has clients in 5 different cities.

1 Chapter 6: The Mathematics of Touring (Hamilton Circuits) and Hamilton Paths 6.1 Traveling Salesman Problems/ 6.2 Hamilton Paths and Circuits A traveling salesman has clients in 5 different cities. He knows the cost to travel between each pair of cities. He would like to start at his hometown, travel to each of the five cities, and return back home. What is the cheapest way to make this happen? These types of problems are similar to the type of problems we solved in chapter 5. However, we are no longer concerned with traveling EDGES; we are traveling to places (VERTICES). Euler Circuit: Covers each edge of a graph exactly once. Starts and ends at the same vertex. Euler Path: Covers each edge of a graph exactly once. Starts and ends at a different vertex. Examples: 1. Find a Hamilton Path that starts at G and ends at E. 2. Find a Hamilton Circuit for the graph at the right that starts and ends at A. 3. Is there another way to write the same set of edges using the same starting vertex? Writing a circuit backwards is called a. These are considered to be different circuits. 4. Is there another way to write the same set of edges starting at a different vertex? The vertex you start at is called the. Rewriting the same order of edges starting at a different vertex is considered to be the same circuit. 1

A B A, B, D, C, A & A, C, D, B, A are called mirror images not considered equal D C A, B, D,C,A All considered the same circuit, B, D, C, A, B each has a different reference point.

2 A B A, B, D, C, A & A, C, D, B, A are called mirror images not considered equal D C A, B, D,C,A All considered the same circuit, B, D, C, A, B each has a different reference point. C, A, B, D, C D, C, A, B, D Example: Trace a Hamilton Circuit in the graph: Name it: Name it s mirror image: Name it from two different reference points: For consistency, we will try to always name circuits from the same reference point. If the problem is not specific, name it alphabetically. The circuit or it s mirror image are both considered to be correct answers. Complete Graphs A complete graph is a graph in which every vertex is adjacent to every other vertex in the graph. We use: K N to represent The complete graph with N vertices. Draw: K 3 K 4 K 5 K 6 2

3 In a Complete Graph with N vertices: Each vertex has degree: N 1 Total number of edges: N( N 1) 2 Total number of Hamilton Circuits: N 1! (including Mirror-Images) Example: How many Hamilton Circuits are there in: 1. K 3 2. K 4 3. K 5 As the number of edges increases, the number of Hamilton Circuits GREATLY increases. If we are looking for the best Hamilton Circuit for our traveling salesman, we need to have a plan! 4. K 6 5. K 10 Example: How many edges are there in 1. K K 150 Example: How many vertices (what is N) if 3. K N has 5040 Hamilton Circuits? 5. K N has 120 Hamilton Circuits? 4. K N has 990 edges? 6. K N has 136 edges? 3

4 Traveling Salesman Problems Shown is a graph of five cities (A, B, C, D, E) that our salesman, Willy Loman must travel. Notice that this is a complete graph with 5 vertices. The weight of an edge is the cost to travel that edge. What is the weight of edge AB = EB = Find a Hamilton Path that starts at A and ends at D and give its weight: Find a Hamilton Circuit that starts with edge CD and give its weight: Find a Hamilton Circuit that ends with edge EA and give its weight REAL EXAMPLES OF TRAVELING SALESMAN PROBLEMS Read: pg Read: pg 183 School buses. Circuit boards. Errands around town. Bees. Table 6-2, 6-3 SUPERHERO Example: Find a Hamilton Circuit and give its weight. 4

6.3: Optimal Hamilton Circuits the Brute Force Method The only way to

It is very time consuming and is considered to be optimal but

5 6.3: Optimal Hamilton Circuits the Brute Force Method The only way to find an OPTIMAL Hamilton circuit is to actually find ALL POSSIBLE circuits Check the cost one by one. This is called the Brute Force Method. It is very time consuming and is considered to be optimal but inefficient. Willy s graph has 5 vertices. How many circuits would he have to check? Which Hamilton Circuit gives the OPTIMAL tour through the graph? Weight: 5

6 Example 1: Find the OPTIMAL Hamilton Circuit for the following A 10 B graph. Use the Brute Force Method. Write your answer using A as your reference point D 7 C A Example 2: Find the optimal Hamilton Circuit and its weight D B 2 6 C Example 3: Find the optimal Hamilton Circuit and its weight. IND RSW LAX MSP IND *** RSW 253 *** LAX *** 248 MSP *** 6

6.4: The Nearest-Neighbor Algorithm and Repetitive Nearest Neighbor Algorithm Brute Force Algorithm takes too long for N>4. We need a faster way to come up with a good answer.

7 6.4: The Nearest-Neighbor Algorithm and Repetitive Nearest Neighbor Algorithm Brute Force Algorithm takes too long for N>4. We need a faster way to come up with a good answer. The Nearest Neighbor Algorithm 1. START: Start at the designated starting vertex. If there is no designated starting vertex, pick any vertex. 2. First Step: From the starting vertex, go to its nearest neighbor (the vertex for which the corresponding edge has the smallest weight). 3. Middle Steps:From each vertex, go to its nearest neighbor among the vertices that haven t been visited yet. Keep doing this until all the vertices have been visited. 4. Last Step: From the last vertex, return to the starting vertex. The tour we get is called the nearest-neighbor tour. Use the nearest neighbor algorithm, starting at vertex A, to find a Hamilton Circuit and its weight. Notice that our answer is NOT an optimal answer, but the process to find it is relatively quick. Willy Loman has to decide if the time he saved not doing Brute Force is worth the money he loses from the difference. We say that the nearest-neighbor method is and. What if Willy Loman s territory gets expanded to 10 cities? How many Hamilton Circuits are in this graph? How many edges are in this graph? 7

A much more practical approach is to organize the data in a table. Use the nearest neighbor algorithm to find the Hamilton Circuit starting at vertex A.

8 A much more practical approach is to organize the data in a table. Use the nearest neighbor algorithm to find the Hamilton Circuit starting at vertex A. Example: a) Use the nearest neighbor method to find the Hamilton Circuit starting with vertex A. b) Use the nearest neighbor method to find the Hamilton Circuit starting with vertex B. A 1 B D 4 C Example: The chart below represents the price of a bus ticket between the following cities. You would like to start and end at your hometown of Z. Find a Hamilton Circuit and its weight. V W X Y Z V ** $35 $10 $20 $45 W $35 ** $42 $37 $60 X $10 $42 ** $30 $50 Y $20 $37 $30 ** $77 Z $45 $60 $50 $77 ** 8

The distances are given in the chart (yes, I know I used this example earlier for TSPs hang on, I m making a point) It is logical that the MORE Hamilton Circuits that we check, the better the chance

9 Example: Use the nearest neighbor algorithm to find a Hamilton Circuit from vertex D. Write your final answer starting at the reference point A. Example: A band is planning a concert tour through 10 cities including their hometown A. The distances are given in the chart (yes, I know I used this example earlier for TSPs hang on, I m making a point) It is logical that the MORE Hamilton Circuits that we check, the better the chance of finding a circuit that is closer to the optimal answer. The Nearest Neighbor Method is dependent on the vertex you start at. If you start at a different vertex, you will most likely get a different answer. (See table above) Remember you can rewrite any circuit so that it starts with a different reference point. With 10 vertices, there are possible circuits including mirror images. The table shows the nearest-neighbor tour found from each vertex. So we are only looking at tours. And really we only found distinct values. We could rewrite any of these circuits starting at a different reference point A! 9

10 The Repetitive Nearest Neighbor Algorithm 1. Let X be any vertex. Find the nearest-neighbor tour with X as the starting vertex and calculate the cost of this tour. 2. Repeat the process with each of the other vertices of the graph as the starting vertex. 3. Choose the tour with the least cost. If necessary, rewrite the tour so that it starts at the designated starting vertex. The tour we get is called the repetitive nearest-neighbor tour. Example: Use the Repetitive Nearest Neighbor Method to find the Hamilton Circuit for Willy Loman s sales district. 10

11 Examples: Use the Repetitive Nearest Neighbor Method to find a Hamilton Circuit and its weight. 1. Home City A A 1 B D 4 C 2. Home city B 3. Home City Z V W X Y Z V ** $35 $10 $20 $45 W $35 ** $42 $37 $60 X $10 $42 ** $30 $50 Y $20 $37 $30 ** $77 Z $45 $60 $50 $77 ** 11

6.5 Cheapest Link Method So far we ve been considering the creation of a Hamilton Circuit in order from a specific starting point. What if we ignored that?

12 6.5 Cheapest Link Method So far we ve been considering the creation of a Hamilton Circuit in order from a specific starting point. What if we ignored that? Let s look at the graph as a set of potential LINKS between cities. As long as we are very careful they are all connected at the end, it doesn t matter if at some point in the process they are disconnected. BE CAREFUL: Don t close your circuit until the very end! You only need to go IN and OUT of each vertex, so you never need three edges from any one vertex. If you have N vertices, you will have N edges in your final circuit. The Cheapest Link Algorithm 1. Step 1: Pick the cheapest link available. (If there is more than one, randomly pick on among the cheapest links) Highlight the link in red (or any other color). 2. Step 2: Pick the next cheapest link and highlight it. 3. Steps 3 to N-1: Continue picking and highlighting the cheapest link available that (a) DOES NOT close a partial circuit (b) DOES NOT create three edges meeting at the same vertex. 4. Step N: Connect the two vertices that close the highlighted circuit. Once you have the Hamilton circuit, write it from the designated starting vertex. Either clockwise or counter-clockwise travel give you a tour we call the cheapestlink tour. Example: Use the cheapest link algorithm to find a Hamilton Circuit. 12

13 Examples: Use the Cheapest Link Method to find a Hamilton Circuit. 1. Home City A A 1 B D 4 C 2. Home city D 3. Home City Z V W X Y Z V ** $35 $10 $20 $45 W $35 ** $42 $37 $60 X $10 $42 ** $30 $50 Y $20 $37 $30 ** $77 Z $45 $60 $50 $77 ** 13

14 Relative Error The Brute Force Method is but. The Nearest Neighbor, Repetitive Nearest Neighbor, and Cheapest Link Methods are but only. So far, no one has been able to produce an optimal, efficient algorithm for solving TSP s. Also, no one has proved that an optimal, efficient algorithm is impossible. We can determine how accurate our answers are using something called relative error. What were the values from Willy Loman s Hamilton Circuits using: Brute Force: (optimal) Nearest Neighbor: Relative Error = Repetitive Nearest Neighbor Relative Error = Cheapest Link Relative Error = 14

15 Putting it all together: 1. In order to find an optimal Hamilton Circuit for the graph at the right you would need to check how many Hamilton Circuits to use the BRUTE FORCE METHOD? 2. Find a Hamilton Circuit and its weight starting at C using the NEAREST-NEIGHBOR METHOD. 3. Find a Hamilton Circuit and its weight starting at C using the REPETITIVE NEAREST-NEIGHBOR METHOD. 4. Use the CHEAPEST LINK ALGORITHM to find a Hamilton Circuit and its weight that starts at C. (Picture repeated for your highlighting convenience) 15

1. The Highway Inspector s Problem

MATH 100 Survey of Mathematics Fall 2009 1. The Highway Inspector s Problem The Königsberg Bridges over the river Pregel C c d e A g D a B b Figure 1. Bridges f Is there a path that crosses every bridge