Graphs

Paths

Circuits

Euler

Last Chapter

Discrete MathTraveling 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…

Lecture Circuit

Pick 5 of the cities below 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.

Create a Graph

Find the cost for your flights.

You should have 10 flights to look up.

Write the cost on each edge.

Go to Southwest Airlines

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.

Hamilton Paths and Circuits

Hamilton v. Euler

Hamilton v. Euler

Euler Circuit Euler Path Hamilton

CircuitHamilton

Path

(a) Yes No Yes Yes

(b) No Yes No Yes

(c) No No Yes Yes

(d) Yes No No Yes

(e) No Yes No No

(f) No No No 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.

What we learn from this table

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

#1

Find the weight of edge BD. Find a Hamilton circuit that starts with BD

and give its weight. Find a Hamilton circuit that starts withDB and give its weight.

#13

Page 226 #2,5,6,9,10,

14,15, and 16

Problems

When every vertex is connected by an edge

Complete Graphs

The notation for a complete graph

The number of edges is equal to

Complete Graph

◦ 1225 edges

How many edges

Use a factorial!!!

Hamilton Circuits

How many paths

Given that

#17 (a)

(9!+11!)10 !

#19 (a)

20!

#21 (a)

A super computer can generate one billion Hamilton circuits per second.

Estimate the number of years it will take for the computer to generate Hamilton circuits.

#23 (a)

If the number of edges in is x, and the number edges in is y, what is the value of ?

#25 (c)

Find the value of N when:◦ has 45 edges

#27 (b)

Page 229 #17-28

Problems

Sites: The vertices on the graph

Costs: The weight of the edges

Tour: A Hamilton Circuit

Optimal Tour: Hamilton Circuit of least weight

TSP vocab

Try to find the optimal route

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.

Simple Strategies

1. Make a list of all possible routes2. Calculate all the tours3. Choose the one with the smallest number

Brute-Force Algorithm

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

Brute-Force

If a computer can calculate one quadrillion tours a second (1,000,000,000,000,000). It will take the computer seconds to calculate until we hit

Super Computer

n Time

20 2 mins

21 40 mins

22 14 hours

23 13 days

24 10 months

25 20 years

26 500 years

27 13,000 years

28 350,000 years

29 9.8 million years

30 284 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

Nearest-Neighbor

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

Nearest-Neighbor

Relative Error

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?

Approximate Algorithms

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

Repetitive Nearest-Neighbor

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

Cheapest Link

A C E B D A

Chapter 6 #30, 31, 35, 36,

38, 41, 42, 44, 47, 48

Problems