MST
0 likes•93 views
There are two algorithms for finding the minimum spanning tree of a graph: Prim's algorithm and Kruskal's algorithm. Prim's algorithm starts with one vertex and adds the lowest cost edge that connects it to another vertex at each step. Kruskal's algorithm sorts the edges by cost and adds edges one by one if they do not form cycles. Both algorithms produce a spanning tree with the minimum total cost.
MST
- 1. Spanning Subgraph A subgraph of a graph, which contains all the vertices of the graph. Spanning Tree A spanning tree is a subgraph (spanning subgraph) of a graph, which contains all the vertices of the graph and contains no cycle. Minimum Cost Spanning Tree It is a spanning tree whose cost is minimum. There are two well-known algorithms to build minimum cost spanning tree. One is Prim’s algorithm and another is Kruskal’s algorithm. Prim's Algorithm Prim's algorithm is a minimum spanning tree algorithm that takes a graph as input and finds the subset of the edges of that graph which form a tree that includes every vertex has the minimum sum of weights among all the trees that can be formed from the graph The steps for implementing Prim's algorithm are as follows: 1. Initialize the minimum spanning tree with a vertex chosen at random. 2. Find all the edges that connect the tree to new vertices, find the minimum and add it to the tree 3. Keep repeating step 2 until we get a minimum spanning tree Example Construct the minimum spanning tree (MST) for the given graph using Prim’s Algorithm-
- 2. Solution: Step-01 Step-02 Step-03 Step-04 Step-05 Step-06 Now, Cost of Minimum Spanning Tree = Sum of all edge weights = 10 + 25 + 22 + 12 + 16 + 14 = 99 units Prim's Algorithm pseudocode T = ∅; U = { 1 }; while (U ≠ V) let (u, v) be the lowest cost edge such that u ∈ U and v ∈ V - U; T = T ∪ {(u, v)} U = U ∪ {v}
- 3. Kruskal's Algorithm Kruskal's algorithm is a minimum spanning tree algorithm that takes a graph as input and finds the subset of the edges of that graph which form a tree that includes every vertex has the minimum sum of weights among all the trees that can be formed from the graph The steps for implementing Kruskal's algorithm are as follows: 1. Sort all the edges from low weight to high 2. Take the edge with the lowest weight and add it to the spanning tree. If adding the edge created a cycle, then reject this edge. 3. Keep adding edges until we reach all vertices. Example Construct the minimum spanning tree (MST) for the given graph using Kruskal’s Algorithm- Solution Step-01 Step-02
- 4. Step-03 Step-04 Step-05 Step-06 Weight of the MST = Sum of all edge weights = 10 + 25 + 22 + 12 + 16 + 14 = 99 units Kruskal Algorithm Pseudocode KRUSKAL(G): A = ∅ For each vertex v ∈ G.V: MAKE-SET(v) For each edge (u, v) ∈ G.E ordered by increasing order by weight(u, v): if FIND-SET(u) ≠ FIND-SET(v): A = A ∪ {(u, v)} UNION(u, v) return A