MANAGEMENT SCIENCE

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MANAGEMENT SCIENCE
NETWORKS
(Spanning trees Kruskals Algorithm)
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Creating a network (an application)

• The city council of Geometry (which, of course is
  in the country of Mathland) has decided to build
  an underground utility network to service the
  citys 5 major tourist attractions.
• After doing some research, they create a diagram
  depicting possible tunnel routes.
• Their goal is to come up with a system that would
  allow service to each site, but would require the
  fewest miles of tunneling.

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Geometry, MathlandMajor Tourist Attractions
Gardens of Galileo
The Euclidean Plain
Fibonaccis Fountain
Eulers Arch
Point Pythagorus
The management problem that occurs here is
trying to come up with a solution that covers the
least number of miles.
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Draw an appropriate graph...
Suppose that the engineers involved in the
project find that tunnels in any of the following
places would be feasible.
To complete the application, you must determine
which of these locations should be used to create
the lowest cost utility network.
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Label the graph...
The mileage between the attractions is
calculated...
6
4
2
6.25
6
2.50
3.50
7
1.50
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Examine some possibilitiesLook at subgraphs...
Subgraph 1

• What is the COST (total mileage) of the network
  described by this subgraph?
• 19.50 miles
• Can you see any way that money could be saved?

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More possibilitiesLook at another subgraph...

• If we take out some redundant edges, we could
  reduce the total mileage.
• We could save 6 miles by removing GE, and simply
  allowing a route through F.
• Similarly, removing edge PA would cut 1.50 miles
  off of the total.
• New mileage 12

Subgraph 2
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NEW VOCABULARY...

• This type of graph is called a SPANNING TREE of
  the original graph

• A TREE is any connected graph that contains NO
  CIRCUITS
• A spanning tree is a connected subgraph that
  contains all the vertices of the original graph,
  but has no circuits

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A tree or not a tree . . .
. . . that is the question.
Not a tree
A tree
NOT a tree
A tree must be connected and cannot have a
circuit.
A tree
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Minimum Cost Spanning Trees

• The OPTIMAL solution for our application will be
  the spanning tree of lowest cost.

• KRUSKALS ALGORITHM is the method we will use to
  find the optimal solution.
• Unlike the algorithms in the Hamilton circuit
  chapter, this algorithm guarantees the BEST
  solution.

We will see that THIS is NOT the optimal solution!
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Minimum Cost Spanning Trees

• Spanning trees of a graph with N vertices must
  have N-1 edges
• Spanning Trees will contain NO CIRCUITS
• Kruskals Algorithm will give the LOWEST COST
  spanning tree (often called the MINIMUM spanning
  tree or the OPTIMAL spanning tree.)

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KRUSKALS ALGORITHM

• The algorithm is very much like the CHEAPEST LINK
  algorithm we used for Hamilton circuit problems.
• The graph must be connected.
• BEGIN by listing the edges in order of increasing
  weight (MILEAGE)

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KRUSKALS ALGORITHM

• As in the Cheapest Link Algorithm, simply use
  edges of lowest weight UNLESS they break a rule
  for TREES
• The only rule Dont create a circuit.

Continue until all the vertices are connected.

• This spanning trees weight is 10 MILES

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The BEST solution...

• Remember that, originally, the problem was to
  look at a way to create a network of tunnels that
  will be used to build a utility network for the 5
  tourist attractions.
• The city councils GOAL was to do this so that
  the mileage of tunneling was a minimum.
• This would, most likely, also minimize the cost
  of building the network.
• For this type of problem, a CIRCUIT was not
  necessary.
• And the algorithm used to find the solution is an
  OPTIMAL ALGORITHM
• It guarantees the lowest cost solution!

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One more example . . .
Find a minimum spanning tree.
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Is the graph connected?
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Start with the edge of lowest weight.
Use either edge of weight 8.
But only one!
With this graph, we end up using the longest
edge.
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Suggested Problems for Chapter 7
Chapter 7 1, 3, 9, 11, 19, 21, 23