Prim's AlgorithmIt is a greedy algorithm. It starts with an empty spanning tree. The idea is to maintain two sets of vertices:
At every step, it considers all the edges and picks the minimum weight edge. After picking the edge, it moves the other endpoint of edge to set containing MST. Steps for finding MST using Prim's Algorithm:
MSTPRIM (G, w, r) 1. for each u ∈ V [G] 2. do key [u] ← ∞ 3. π [u] ← NIL 4. key [r] ← 0 5. Q ← V [G] 6. While Q ? ∅ 7. do u ← EXTRACT  MIN (Q) 8. for each v ∈ Adj [u] 9. do if v ∈ Q and w (u, v) < key [v] 10. then π [v] ← u 11. key [v] ← w (u, v) Example: Generate minimum cost spanning tree for the following graph using Prim's algorithm. Solution: In Prim's algorithm, first we initialize the priority Queue Q. to contain all the vertices and the key of each vertex to ∞ except for the root, whose key is set to 0. Suppose 0 vertex is the root, i.e., r. By EXTRACT  MIN (Q) procure, now u = r and Adj [u] = {5, 1}. Removing u from set Q and adds it to set V  Q of vertices in the tree. Now, update the key and π fields of every vertex v adjacent to u but not in a tree. Now by EXTRACT_MIN (Q) Removes 5 because key [5] = 10 which is minimum so u = 5. Now remove 4 because key [4] = 25 which is minimum, so u =4 Update key value of key [3] as 22 and key [6] as 24. And the parent of 3, 6 as 4. Now remove 3 because key [3] = 22 is minimum so u =3. Now in Q, key [2] = 12, key [6] = 18, key [1] = 28 and parent of 2 and 6 is 3. Now by EXTRACT_MIN (Q) Removes 2, because key [2] = 12 is minimum. So update key value of key [1] as 16 and its parent as 2. Now by EXTRACT_MIN (Q) Removes 1 because key [1] = 16 is minimum. Update key value of 6 as 14 and its parent as 1. Now all the vertices have been spanned, Using above the table we get Minimum Spanning Tree. Thus the final spanning Tree is Total Cost = 10 + 25 + 22 + 12 + 16 + 14 = 99
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