Minimum Spanning Tree

it is the tree of a graph with minimum total weight
Examples:

connect all computers in a building with the least cable
 connect locations on a map with the least amound of road
Minimum Spanning Tree
:
Let V’ and V’’ be partitions of the vertices of graph G.
Let (v’, v’’) be an edge across the partitions such that v’ is in V’ and v’’ is in V’’, then
Then the edge with minimum weight across the partition is in the minimum spanning
tree
Example:
=>
minimum spanning tree contains edge with weight 8.
Prim Jarnick Algorithm
Input:
A weighted graph G
Output:
A Minimum Spanning Tree (MST) T of G
pick a vertex v of G
T = {v} //add initial vertex to MST
D[v] = 0 //Distance from v to the MST
E[v] = {} //edge that links v to MST
D[u] = infinity for all u != v
Q = {all vertices in G}
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while Q is not empty
choose the minimum D[u] such that u is in Q
remove u from Q
add E[u] to T
for each vertex z adjacent to u
if weight(u, z) < D[z]
D[z] = weight (u, z)
E[z] = (u, z)
end if
end loop
end loop
Ex:
Step 1:
Choose any vertex
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 Fall '08
 Staff
 Graph Theory, Data Structures, Glossary of graph theory, minimum weight

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