# 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. Here the graph is connected. Graph has more than one minimum spanning tree?  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. Generic-MST algorithm

Two main algorithms for computing MST

1)  Kruskal

2)  Prism,  both greedy algorithm

 Generic-MST(G, w)A=emptysetwhile A does not form a spanning tree dofind an edge (u, v) that is sae for AA= 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-nodewhile (there are non-tree vertices){if there isno edge connecting a tree node with a non-tree nodereturn "no spanning tree"select an edge of minimum weightbetween a tree node and a non tree nodeadd 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. 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