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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=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) 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’.
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; } |
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.
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