1. Chapter 2: Business Efficiency Lesson Plan
Visiting Vertices-Graph Theory Problem
Hamiltonian Circuits
Vacation Planning Problem
Minimum Cost-Hamiltonian Circuit
Method of Trees
Fundamental Principle of Counting
Traveling Salesman Problem
Helping Traveling Salesman
Nearest Neighbor and Sorted Edges Algorithms
Minimum-Cost Spanning Trees
Kruskal’s Algorithm
Prim’s Algorithm

2. Chapter 2: Business Efficiency Business Efficiency
Visiting Vertices
In some graph theory problems, it is only necessary to visit specific locations (using the travel routes, or streets available).
Problem: Find an efficient route along distinct edges of a graph that visits each vertex only once in a simple circuit.
Applications:
Salesman visiting particular cities
Inspecting traffic signals
Delivering mail to drop-off boxes
Pharmaceutical representative visiting doctors

3. Chapter 2: Business Efficiency Vocabulary Review

4. Remember….Chapter 1: Urban Services Finding Euler Circuits
Is there an Euler Circuit?
Does it have even valence? (Yes)
Is the graph connected? (Yes)
Euler circuit exists if both “yes.”
Create (Find) an Euler Circuit
Pick a point to start (if none has been given to you).
Number the edges in order of travel, showing the direction with arrows.
Cover every edge only once, and end at the same vertex where you started.

5. Chapter 2: Business Efficiency Hamiltonian Circuit
A tour that starts and ends at the same vertex (circuit definition).
Visits each vertex once. (Vertices cannot be reused or revisited.)
Circuits can start at any location.
Use wiggly edges to show the circuit.
Starting at vertex A, the tour can be written as ABDGIHFECA, or starting at E, it would be EFHIGDBACE.
A different circuit visiting each vertex once and only once would be CDBIGFEHAC (starting at vertex C).

6. Chapter 2: Business Efficiency Hamiltonian Circuit vs. Euler Circuits
Hamiltonian vs. Euler Circuits
Similarities
Both forbid re-use.
Hamiltonian do not reuse vertices.
Euler do not reuse edges.
Differences
Hamiltonian is a circuit of vertices.
Euler is a circuit of edges.
Euler graphs are easy to spot (connectedness and even valence).
Hamiltonian circuits are NOT as easy to determine upon inspection.
(some classes of graphs have them, so do not)
Hamiltonian circuit – A tour (showed by wiggly edges) that starts at a vertex of a graph and visits each vertex once and only once, returning to where it started.
Euler circuit – A circuit that traverses each edge of a graph exactly once and starts and stops at the same point.

7. Chapter 2: Business Efficiency Hamiltonian Circuit
Graphs that will NEVER have a Hamiltonian Circuit
A family of graphs that never has a Hamiltonian circuit
Constructed with vertices on two parallel vertical columns with one column having more vertices than the other.

8. Task: Find a Hamiltonian circuit for this graph starting at vertex A

9. Solution: One solution…others???

10. Task:
Does this graph have a Hamiltonian circuit?

11. Solution:
No...cannot avoid visiting vertices C and D more than one time to reach all vertices

12. Chapter 2: Business Efficiency Hamiltonian Circuits
Vacation–Planning Problem
Hamiltonian circuit concept is used to find the best route that minimizes the total distance traveled to visit friends in different cities (assume less mileage  less gas  minimizes costs)
Hamiltonian circuit with weighted edges
Edges of the graph are given weights, or in this case mileage or distance between cities.
As you travel from vertex to vertex, add the numbers (mileage in this case).
Each Hamiltonian circuit will produce a particular sum.
Road mileage between four cities

13. Chapter 2: Business Efficiency Hamiltonian Circuit
Minimum-Cost Hamiltonian Circuit
A Hamiltonian circuit with the lowest possible sum of the weights of its edges.
Algorithm (step-by-step process) for Solving This Problem
Generate all possible Hamiltonian tours (starting with Chicago).
Add up the distances on the edges of each tour.
Choose the tour of minimum distance.
Algorithm – A step-by-step description of how to solve a problem.

14. Chapter 2: Business Efficiency Hamiltonian Circuits
Method of Trees
For the first step of the algorithm, disregard distances
Begin by selecting a starting vertex, say Chicago, and making a tree-diagram showing the next possible locations.
At each stage down, there will be one less choice (3, 2,then 1 choice).
In this example, the method of trees generated six different paths, all starting and ending with Chicago.
*However, only three are unique circuits….why?
Method of trees for vacation-planning problem

15. Chapter 2: Business Efficiency Hamiltonian Circuits
Minimum-Cost Hamiltonian Circuit: Vacation-Planning Example
Method of tree used to find all tours (for four cities: three unique paths). On the graph, the unique paths are drawn with wiggly lines.
Add up distance on edges of each unique tour. The smallest sum would give us the minimal distance, which is the minimum cost.
Choose the tour of minimum distance. The smallest sum would give us the minimal distance, which is the minimum cost.
The three Hamiltonian circuits’ sums of the tours

16. Task: Use the method of trees to find all possible Hamiltonian circuits for this graph starting at A.

17. Solution: Next three are reversals
First three branches yield three distinct Hamiltonian circuits
ABCDA
ABDCA
Next three are reversals
ADCBA
ACDBA
ADBCA

18. Chapter 2: Business Efficiency Hamiltonian Circuit
Principle of Counting for Hamiltonian Circuits
For a complete graph of n vertices, there are (n - 1)! possible routes.
Half of these routes are repeats, the result is:
Possible unique Hamiltonian circuits are
(n - 1)! / 2
Complete graph – A graph in which every pair of vertices is joined by an edge.

19. Chapter 2: Business Efficiency Complete Graph
Complete graph – A graph in which every pair of vertices is joined by an edge.

20. Task: Determine how many Hamiltonian circuits there are for a complete graph with 10 vertices.

21. Solution:

22. Chapter 2: Business Efficiency Traveling Salesman Problem
Traveling Salesman Problem (TSP)
Difficult to solve Hamiltonian circuits when the number of vertices in a complete graph increases (n becomes very large).
This problem originated from a salesman determining his trip that minimizes costs (less mileage) as he visits the cities in a sales territory, starting and ending the trip in the same city.

23. Chapter 2: Business Efficiency Traveling Salesman Problem
Traveling Salesman Problem (TSP)
Many applications today:
Lobster Fisherman needs to pick up nets
Bus picking up campers and then returning them
Picking up records from ATM machines
Driving service dropping people from the airport
You running your errands…thank you Jared Macke!

24. Chapter 2: Business Efficiency Hamiltonian Circuit
Fundamental Principle of Counting
If there are a ways of choosing one thing, b ways of choosing a second after the first is chosen, c ways of choosing a third after the second is chosen…, and so on…, and z ways of choosing the last item after the earlier choices, then the total number of choice patterns is a × b × c × … × z.
This means that…if Jack has 3 pairs of pants, 4 shirts and 4 hats, then he has 3 x 4 x 4 = 48 outfits!

25. Task: To encourage her son to try new things, a mother offers to take him for a dish of ice cream with a topping once a week, for as many weeks he does not get the same choice as on a previous occasion If the store offers 12 flavors and six toppings, how many weeks will she have to do this if her son never picks either of the two types of chocolate ice cream or the three types of nut topping?

26. Solution: (12-2) x (6-3) = 10 x 3 = 30 weeks

27. Task: A large corporation has found that it has “outgrown” its current code system for routing interoffice mail. Current system places a code of three ordered, distinct nonzero digits on the mail New system calls for the use of two ordered capital letters Is it an improvement?

28. Task: Current system 9 x 8 x 7 = 504 New system calls for the use of two ordered capital letters 26 x 26 = 676 Is it an improvement? Yes! 676 – 504 = 172 more codes

29. Chapter 2: Business Efficiency Traveling Salesman Problem

30. Chapter 2: Business Efficiency Traveling Salesman Problem
How can the TSP be solved?
Computer program can find optimal route (not always practical).
Heuristic methods can be used to find a “fast” answer, but does not guarantee that it is always the optimal answer.
Nearest-Neighbor algorithm
Sorted-Edges algorithm
Review…
What is an algorithm?
What is our goal?

31. Nearest-Neighbor Algorithm
Chapter 2: Business Efficiency Traveling Salesman Problem: Nearest-Neighbor Algorithm
Nearest-Neighbor Algorithm
Starting from the home city, first visit the nearest city, then visit the nearest city that has not already been visited. We return to the start city when no other choice is available.

32. Chapter 2: Business Efficiency Demonstrate the Nearest-Neighbor Algorithm *Start in Seattle*

33. Chapter 2: Business Efficiency Nearest-Neighbor Algorithm
Chapter 2: Business Efficiency Nearest-Neighbor Algorithm *Start at vertex A*

34. Sorted Edges Algorithm
Chapter 2: Business Efficiency Traveling Salesman Problem: Sorted Edges Algorithm
Sorted Edges Algorithm
Start by sorting or arranging the edges of the complete graph in order of increasing cost or distance. Then at each stage we can select that edge of least cost that (1) never requires that three used edges meet at a vertex and that (2) never closes up a circular tour that doesn’t include all the vertices.

35. Chapter 2: Business Efficiency Demonstrate the Sorted Edges Algorithm
Chapter 2: Business Efficiency Demonstrate the Sorted Edges Algorithm *Start???*

36. Chapter 2: Business Efficiency Sorted Edges Algorithm

37. Chapter 2: Business Efficiency Minimum-Cost Spanning Trees
What is the cost to construct underground
power cables to service among five cities?
The diagram shows the cost to build the connection from each vertex to all other vertices (connected graph).
Cities are linked in order of increasing costs to make the connection.
The cost of redirecting the service may be small compared to adding another link.
Costs (in millions of dollars) of installing service among five cities

38. Chapter 2: Business Efficiency Minimum-Cost Spanning Trees
A tree
consists of one piece
contains no circuits.
A spanning tree
a tree that connects all vertices of a graph with each other
no redundancy.

39. Task: With A as the original graph, which of the graphs shown in bold represent trees and/or spanning trees?

40. Chapter 2: Business Efficiency Minimum-Cost Spanning Trees
graph theory optimization problem that links all the vertices together, in order of increasing costs, to form a “tree.”
The cost of the tree is the sum of the weights on the edges.

41. Chapter 2: Business Efficiency Spanning Trees
Two spanning trees for graph A are shown in B and C.

42. Chapter 2: Business Efficiency Minimum-Cost Spanning Trees
Kruskal’s Algorithm — Developed by Joseph Kruskal (AT&T research).
Goal: Create a tree that links all the vertices together and achieves the lowest cost to create.
Add links in order of cheapest cost according to the rules:
No circuit is created (no loops).
If a circuit (or loop) is created by adding the next largest link, eliminate this largest (most expensive link)—it is not needed.
Every vertex must be included in the final tree.

43. Chapter 2: Business Efficiency Minimum-Cost Spanning Trees
Greedy Algorithms
Extend a tree by including the cheapest outgoing edge
Add the cheapest edge that joins disjoint components
Delete the most expensive edge that does not disconnect the graph

44. Task: Using this graph, find the MSP by applying Prim’s Algorithm beginning at vertex A
Extend a tree by including the cheapest outgoing edge

45. Task: Using this graph, find the MSP by applying Kruskal’s Algorithm
Add the cheapest edge that joins disconnected components

46. Task: Using this graph, find the MSP by applying the Reverse-Delete Algorithm.
Delete the most expensive edge that does not disconnect the graph

47. Task: Using this graph, apply Kruskal’s Algorithm

48. Chapter 2: Business Efficiency Critical Path Analysis
Most often, scheduling jobs consists of complicated tasks that cannot be done in a random order.
Due to a pre-defined order of tasks, the entire job may not be done any sooner than the longest path of dependent tasks.

49. Chapter 2: Business Efficiency Critical Path Analysis
Order-Requirement Digraph
A directed graph (digraph) that shows which tasks precede other tasks among the collection of tasks making up a job.

50. Chapter 2: Business Efficiency Order Requirement Graph True or False
T1 must be done before T2
T2 must be done before T4
T1 must be done before T7
T2 need not be done before T5
T3 need not be done before T5

51. Chapter 2: Business Efficiency Critical Path Analysis
The longest path in an order-requirement digraph.
The length is measured in terms of summing the task times of the tasks making up the path.
An order-requirement digraph, tasks A – E with task times in the circles
Critical Path is BE = = 52 min.

52. Chapter 2: Business Efficiency Find the Critical Path for this order-requirement digraph

53. Chapter 2: Business Efficiency What is the length of the earliest completion time for this order-requirement digraph?