Finding lowcost Hamilton circuits

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Finding low-cost Hamilton circuits
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Repetitive Nearest Neighbor

• Since the Hamilton circuit we find using the
  Nearest Neighbor algorithm depends on the
  starting vertex, we can repeat the process from
  different vertices and find different circuits.
• This algorithm is called the Repetitive Nearest
  Neighbor.
• We use the Nearest Neighbor algorithm starting at
  each vertex and compare the results, choosing the
  cheapest one.

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• Chip is responsible for marketing the new Brutus
  clothing line in Atlanta, Baltimore, Detroit, and
  East St. Louis, as well as his hometown of
  Columbus.
• He needs to plan a trip to each of these cities
  to meet with clients and see how sales are going.
  
• He wants to find the cheapest way to visit all
  the cities.

The cost of one-way travel to each city is
represented in the graph to the right.
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Repetitive Nearest Neighbor Algorithm

• We use the Nearest Neighbor Algorithm from each
  of the vertices.
• Lets start first with vertex A.
• Copy the vertices and begin the process.

• Start with vertex A and find the nearest vertex
  (the one connected by the cheapest edge).
• Add that edge to the graph and continue.

• Remember the rule
• Dont close the circuit until you get to all the
  vertices.

The total weight of this circuit is 125 170
125 150 160 730
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Repetitive Nearest Neighbor Algorithm

• Now repeat the process starting at vertex B.

• Previous result starting at A 730

• Remember the rule
• Dont close the circuit until you get to all the
  vertices.

The total weight of this circuit is 140 125
135 125 230 745
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Repetitive Nearest Neighbor Algorithm

• Now repeat the process starting at vertex C.

• Previous results
• Starting at A 730
• Starting at B 745

• Remember the rule
• Dont close the circuit until you get to all the
  vertices.

The total weight of this circuit is 125 140
160 125 180 730
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Repetitive Nearest Neighbor Algorithm

• We complete this process from each vertex and
  compare the results.
• From A 730
• From B 745
• From C 730
• From D 755
• From E 755

Our best result is the circuit we found starting
at A or at C. We can choose either of these and
rewrite it so we start at C, (Chips home town).
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Whats the difference?

• The CHEAPEST LINK algorithm creates a Hamilton
  circuit subgraph where
• The edges are chosen by increasing weight.
• No initial vertex is chosen
• There is a unique cheapest link subgraph (up to
  edges of equal weight)

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Whats the difference?

• The NEAREST NEIGHBOR algorithm creates a Hamilton
  circuit subgraph where
• The weight of the Hamilton circuit is dependent
  on the initial point chosen.
• There are many nearest neighbor subgraphs,
  depending on the vertex you initially choose.
• This algorithm can be repeated in order to
  compare weights of different subgraphs

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REMEMBER!!!...

• These algorithms do not guarantee an OPTIMAL
  SOLUTION.
• They are to help you QUICKLY find a Hamilton
  circuit subgraph that has a number of associated
  Hamilton circuits.
• Your response to an application may also require
  that you include an appropriate path or circuit.
• Algorithms are especially useful for LARGE
  APPLICATIONS.
• The same algorithm may produce a number of
  different subgraphs depending on the initial
  point chosen.
• Different algorithms do not necessarily produce
  the same solutions.

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Suggested Problems for Chapter 6

• Chapter 6 1, 7, 11, 17, 19,
• 25, 27, 29, 35, 39, 41,
• 43, 59