 Graphs  Paths  Circuits  Euler

Traveling Salesman Problems

 You are invited to go on a lecture circuit to colleges across the country.  The company said they will cover for a part of your travel expense and you will have to cover the rest.  The cities you are able to visit are…

 Seattle  Portland  Spokane  Boise  Salt Lake City  Reno  Sacramento  San Francisco  San Jose  Las Vegas  Los Angeles  San Diego  Phoenix  Charleston  Tampa  Orlando  Ft. Lauderdale  Raleigh  Richmond  Washington DC  Philadelphia  New York  Providence  Boston  Denver  Albuquerque  San Antonio  Austin  Houston  Dallas  Oklahoma City  Kansas City  Little Rock  New Orleans  Jackson  Memphis  St. Louis  Pensacola  Atlanta  Nashville  Chicago  Indianapolis  Minneapolis/St. Paul  Detroit  Cleveland  Pittsburgh  Charlotte

 Create a graph so that all of the cities are connected.  Make it easy to read for yourself.

 Find the cost for your flights.  You should have 10 flights to look up.  Write the cost on each edge.

 How many different circuits do you make?  Which circuit would you use?  What was its cost?  How did you find this circuit? What was your process for choosing flights?

 Hamilton Path- A path that crosses ever VERTEX once and only once  Hamilton Circuit- A Circuit that crosses ever VERTEX once and ends in the place it started.

Euler CircuitEuler PathHamilton CircuitHamilton Path (a)YesNoYes (b)NoYesNoYes (c)No Yes (d)YesNo Yes (e)NoYesNo (f)No

 If there is a Hamilton Circuit, then there is a Hamilton Path (Drop the last edge that creates the circuit).  There is no connection between Euler and Hamilton  There is no easy theorem to see if there is a Hamilton Circuit or Path.

 Find 3 different Hamilton circuits  Find a Hamilton path that starts at A and ends at B  Find a Hamilton path that starts at D and ends at F

 Find the weight of edge BD.  Find a Hamilton circuit that starts with BD and give its weight.  Find a Hamilton circuit that starts with DB and give its weight.

Page 226 #2,5,6,9,10, 14,15, and 16

 When every vertex is connected by an edge

20!

Page 229 #17-28

 Sites: The vertices on the graph  Costs: The weight of the edges  Tour: A Hamilton Circuit  Optimal Tour: Hamilton Circuit of least weight

 Exhaustive Search: Make a list of all possible routes. The previous example has 24 possible.  Go Cheap: Go to the cheapest city. Continue by going through the cheapest routes possible.

1. Make a list of all possible routes 2. Calculate all the tours 3. Choose the one with the smallest number

 This will ALWAYS get you the optimal tour.  Problem: it is an INEFFICIENT ALGORITHM. This means that it can take way to long to find your solution. ◦ Even with computers

nTime 202 mins 2140 mins 2214 hours 2313 days 2410 months 2520 years years 2713,000 years 28350,000 years million years million years

 Start at the starting vertex  Go to the “nearest neighbor” (edge with the lowest amount)  Continue through all the points  End at your starting vertex

 This is not optimal because it might not give us our optimal route  But this is effective because it is proportional to the size of the graph ◦ 10 vertices= 10 steps ◦ 30 vertices= 30 steps

 Using Algorithms to get close to the optimal but might not be the optimal.  Ask yourself: Is a 12.49% relative error good or would we have to be closer?

 Just like Nearest-Neighbor, but you create a circuit for all the vertices to be your starting point.

 A C E D B A ◦ 773  B C A E D B ◦ 722  C A E D B C ◦ 722  D B C A E D ◦ 722  E C A D B E ◦ 741

 Find the cheapest edge and mark it  Pick the next smallest edge  Keep picking the smallest edges ◦ Do not close the circuit ◦ Do not have 3 edges go to the same vertex  Close the circuit to finish

Chapter 6 #30, 31, 35, 36, 38, 41, 42, 44, 47, 48