Spanning Trees

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• Spanning Trees

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Trees

• Definition A tree is a connected undirected
  graph with no simple circuits.
• Since a tree cannot have a simple circuit, a tree
  cannot contain multiple edges or loops.
• Therefore, any tree must be a simple graph.
• Theorem An undirected graph is a tree if and
  only if there is a unique simple path between any
  of its vertices.

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Trees

• Example Are the following graphs trees?

Yes.
No.
No.
Yes.
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Applications of Trees

• There are numerous important applications of
  trees, only three of which we will discuss today
• Network optimization with minimum spanning
  trees
• Problem solving with backtracking in decision
  trees
• Shortest Path Trees - Minimum Spanning Trees

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Spanning Trees

• Definition Let G be a simple graph. A spanning
  tree of G is a subgraph of G that is a tree
  containing every vertex of G.
• Note A spanning tree of G (V, E) is a
  connected graph on V with a minimum number of
  edges (V - 1).
• Example Find a minimum spanning tree for the
  following network

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Spanning Trees

• The complete graph

The spanning trees of this graph connect all
vertices with a minimum number of edges.
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Spanning Trees

• Example for a spanning tree

Since there are 6 vertices, 5 edges are
sufficient to connect all of them.
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Spanning Trees

• Definition A minimum spanning tree in a
  connected weighted graph is a spanning tree that
  has the smallest possible sum of weights of its
  edges.
• How can we find a minimum spanning tree?

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Spanning Trees

• Prims Algorithm
• Begin by choosing any edge with smallest weight
  and putting it into the spanning tree,
• successively add to the tree edges of minimum
  weight that are incident to a vertex already in
  the tree and not forming a simple circuit
  with those edges already in the tree,
• stop when (n 1) edges have been added.

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Spanning Trees

• Kruskals Algorithm
• Kruskals algorithm is identical to Prims
  algorithm, except that it does not demand new
  edges to be incident to a vertex already in the
  tree.
• Both algorithms are guaranteed to produce a
  minimum spanning tree of a connected weighted
  graph.

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Summary of Network Algorithms

• Topological Sort
• Unweighted Shortest Path
• Dijkstras Shortest Path
• Maximum Flow
• Prims
• Kruskals

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Topological Sort
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Unweighted Shortest Path
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Weighted Shortest Path(Dijkstras Algorithm)
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Maximal Flow Problem
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Prims AlgorithmShortest Path Tree
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Kruskals AlgorithmShortest Path Tree
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Questions??
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