Minimum Spanning Tree

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Minimum Spanning Tree

• Clayton Andrews
• COT4810
• 1/31/08

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Firstly, a graph

• A graph is a series of nodes connected with edges.

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Types of Graphs

• Connected
• All nodes can be reached by all other nodes by
  following some series of edges
• All nodes are reachable from all other nodes
• Disconnected
• There exists a node that is not reachable from
  some other node by any series of edges
• One or more nodes are not reachable from one or
  more other nodes

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Types of Edges
A
B

• Undirected
• An edge goes to and from a node
• An edge can be taken both directions
• Edges show connections
• Directed
• Edges can specifically go to or from nodes
• Edges can still be bidirectional
• Edges can show a one way trip, visibility, etc..

C
A
B
C
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Types of Edges 2

• Unweighted
• Edges have no values
• Edges just show some sort of connection
• These edges can show neighbours, possible paths,
  etc..
• Weighted
• Edges have a value attached to them
• These values could be distance, cost, time, etc...

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What is a spanning tree?

• A spanning tree is a sub-graph of another graph
  that connects all the nodes of the graph without
  forming a cycle(a loop)
• The graph must be connected and be undirected

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What is a minimum spanning tree?

• If a graph has weights for its edges, than a
  minimum spanning tree is the spanning tree that
  gives the lowest total sum of the edges.
• Basically, it connects all of the nodes together
  for the least total weight

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Why should we care?

• Why would anyone want to care about these trees
  that connect a series of nodes together?
• A minimum spanning tree can be used for several
  things
• Identifying Clusters
• Determining a central point of operations

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For Example

• Determining the location of roads
• Where they should be placed
• What cities they should pass through

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For Example

• Railroad Tracks
• Where is a good central hub
• What would cost us the least and be the quickest

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For Example

• Airports and Flight paths
• Where airports should be
• What path a flight should take

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Finding a minimum spanning tree

• There are several ways to find a minimum spanning
  tree, but I am going to discuss the two most
  common methods
• Prim's algorithmKruskal's algorithm

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Prim's Algorithm

• Prim's Algorithm starts by making a set with a
  single node inside
• It then looks for the shortest edge that does not
  form a cycle and adds its node to a set
• It keeps going until all nodes are consumed

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Prim's Algorithm

• Step 1 Pick an arbitrary starting node and put
  it into an empty set, X
• Step 2 Look at the shortest edge coming from the
  set X
• Step 3 Add the edge and add the node to the set
• Step 4 Remove all edges that connect the set to
  itself
• Step 5 If all nodes have not been added to set
  X, then repeat Steps 2 - 4 until all nodes have
  been added to the set X

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I chose D as my starting point
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The edge to A was the shortestfrom D so I
added it to the set
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From the set A, Dthe edge to F is shortest
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The edge to B was the shortest edge from the set
A, D, F so I added B to the set. The edge BD
connects the set to itself, so it was removed
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From the set A, B, D, F the edge to E is the
shortest. E is added to the set and two edges
are removed
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From the set A, B, D, E, F the edge to C is the
shortest. So, C is added to the set and an edge
is removed
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Finally, from the set A, B, C, D, E, F the edge
to G is the shortest so add G and remove an edge
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Now since all nodes are in the set, this is a
minimum spanning tree
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Kruskal's Algorithm

• Kruskal's algorithm works by selecting the
  shortest edges and adds them if they do not form
  a cycle
• If an edge forms a cycle, it removes it from the
  list of edges and keeps going
• It keeps going as long as all edges have not been
  considered

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Kruskal's Algorithm

• Step 1 Put every node into a set by itself
• Step 2 Choose the shortest edge that connects
  two of these sets, and merge these two sets into
  one
• Step 3 Remove all edges that connect the set to
  itself
• Step 4 If all edges have not been removed repeat
  steps 2 3 until there are no more remaining
  edges

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All nodes are now a setA, B, C, D, F,
G
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The edge from set A to set D is tied for
shortest with the edge from C to E, so just
choose one, I chose A D to make A, D
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Now the edge from C to E is the shortest, so
they combine to become C, E
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The shortest edge is now from A, D to F so
they combine to become A, D, F
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The shortest edge is from A, D, F to B so
combine to A, B, D, F. The edge from B to D
connects the set to itself, it is removed
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The shortest edge is fromA, B, D, F to C, E
so combine and remove the three edges that
connect the new set to itself
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Finally, the smallest edge is the edge to G from
A,B,C,D,E,F, so combine these two sets and
remove the edge that connects the new set to
itself
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Since all edges have been added or removed, there
are no more edges to consider, this is a minimum
spanning tree.
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Both are fine

• Both algorithms have the same time complexity, so
  it is usually up to the programmer to decide
• Multiple minimum spanning trees can exist for a
  given graph, so these algorithms may produce
  different results for the same graph

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Summary

• Minimum Spanning Trees are an efficient way to
  connect all of the nodes of a graph
• They have several real world applications
• There are 2 very common, very straight-forward
  solutions

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References

• Wikipedia(www.wikipedia.com)?
• All graph images
• All minimum spanning tree images
• Information related to minimum spanning tree
• http//citeseer.ist.psu.edu/nesetril00otakar.html
• Information related to algorithms
• http//www.mapsofworld.com/
• Map of Florida Railroads
• http//www.this-town.com/
• Map of Florida Highways
• http//www.airchive.com/
• Map of Airline Flight Paths

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Homework

• Name 2 algorithms used to find the minimum
  spanning tree of a graph.
• What is another real world example of a use of a
  minimum spanning tree?