Chapter 2 Methods For Finding A Hamiltonian Circuit Complete Graph

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Chapter 2Methods For Finding A Hamiltonian
Circuit(Complete Graph)

• I. Brute Force Method
• II. Nearest Neighbor Algorithm
• III. Sorted Edges Algorithm

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Credits

• COMAP, For All Practical Purposes, 5th ed.

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I. Brute Force Method

• Use Fundamental Counting Principle to obtain
  number of possible paths, K
• Find to obtain total of
  different paths
• Pick vertex create tree diagram (delete
  duplicate paths)
• Add distances of each different path, choose path
  with smallest total
• Always gives optimal solution

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Using graph on left, draw the tree diagram
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Add up distances. Pick shortest path
1985
2040
1877
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II. Nearest Neighbor Algorithm

• Select a starting vertex
• Select the edge with the shortest path to the
  next vertex
• Continue sequentially in this manner until you
  reach the original vertex and you have covered
  all vertices in the graph.
• Remember If your selection will create a
  subcircuit, select an alternate shortest edge to
  the next vertex.

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We will begin with vertex CFollow the path to
the nearest vertexRemember you cant form
subcircuit
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III. Sorted Edges Algorithm

• Sort the distances in increasing order 1,2,3,
• Select the edge with distance 1
• Select the next edge with distance 2 (unless that
  edge leads to a subcircuit or to a vertex with 3
  edges)then skip edge with distance 2 and select
  edge with distance 3
• Continue until you return to starting point

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Sort graphs edges 300,349,425,541,562,774Begin
with shortest edge, continue sequentiallyCant
form subcircuit, or 3 edges into a vertex
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Traveling Salesman Problem

• Find a minimum-cost Hamiltonian circuit in a
  complete graph where each edge has been assigned
  a cost or weight.
• Can use any of the three methods described
  previously.