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Minimum Spanning Tree Algorithm Assignment Help


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What is Minimum Spanning Tree Algorithm?


1) The weight of a sub graph is equal to the sum of the weights of its edges.

2) A minimum spanning tree for a weighted graph is a spanning tree with minimum weight.

Minimum Spanning Tree Algorithm Assignment HelpHere the graph is connected. Graph has more than one minimum spanning tree?

Minimum Spanning Tree Algorithm Assignment Help


1) Given a connected undirected graph G = (V, E), a spanning tree is an acyclic subset of edges T that connects all the vertices together.

2) Assuming each edge (u, v) of G has a numeric weight w(u, v), the cost of a spanning tree T is the sum of the weights of the edges in T.

sum of the weights of the edges in T

Generic-MST algorithm


Two main algorithms for computing MST

1)  Kruskal

2)  Prism,  both greedy algorithm


Generic-MST(G, w)
A=emptyset
while A does not form a spanning tree do
find an edge (u, v) that is sae for A
A= A U {(u, v)}
return A


1) Kruskal's algorithm


1) Add edges to ‘A’ increasing sequence of the weights.

2)  If the edge being thought of introduces a cycle, skit it.

3) or else add to ‘A’.


2) Prim's algorithm:


1) It starts with a tree, ‘T’, containing of the starting vertex, ‘y’.

2) Next, it adds the shortest edge emanating from y that connects T to the rest of the graph.

3) It then moves to the added vertex and repeats the process.


consider a graph G = ( V, E );
let T be a tree consisting of only the starting vertex x;
while ( T has fewer than IVI vertices )
{
find a smallest edge connecting T to ( G - T);
add it to T;
}


The prim algorithm main idea


select a vertex to be a tree-node
while (there are non-tree vertices){
if there isno edge connecting a tree node with a non-tree node
return "no spanning tree"

select an edge of minimum weight
between a tree node and a non tree node

add the selected edge and its new vertex to the tree
}
return tree


Example of Prim’s algorithm: vertices A, B, C, D, E, F,G, H, I and its cost is 3, 5, 4, 2, 4, 5, 7, 8 6, 8. Find total weight.

Example of Prim’s algorithm

Solution: In this graph all vertices are connected to each other and have the different cost. So the solution of this graph is:

Edge

Cost

A, B

3

A, E

4

E, D

2

D, C

4

E, F

5


Total weight of tree: 18



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