1. f d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2 Prim’s Algorithm – an Example edge candidates choosen

2. f d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2

3. f d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2

4. f d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2

5. f d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2

6. f d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2 Total weight of the MST: 14

7. Prim’s Algorithm - Description MST-Prim(G, w) choose an arbitrary r  V(G) Initialize T as a tree consisting of vertex r only. while T has < n vertices do let (u, v) be the lightest edge with u  V(T) and v  V(G) – V(T) T  T  { (u, v) } return T Correctness follows a previous corollary: Let A be a subset of E that is included in some MST of G, and C be a connected component in the forest G =. If (u, v) is a light edge connecting C to some other component in G, then (u, v) is safe for A.

8. Implementation At every iteration, maintain the following information for each vertex v in G: cost(v) is the weight of the lightest edge connecting v to T. closest(v) is the vertex y in T such that cost(v) = w(v, y). The next edge to add is (u, closest(u)). Here u is the vertex with minimum cost in V(G) – V(T). u closest(u) T cost(u)

9. Updating Closest Vertex u v z After adding (u, closest(u)), scan every neighbor v of u: set closest(v) = u if cost(v) decreases, update cost(v) if necessary. MST v u z new cost

10. Priority Queue Implementation MST-Prim(G, w) choose r  V(G) for each u  V(G) do cost(u)   cost(r)  0 Q  V(G) // Build a priority queue keyed by cost closest(r)  NIL while Q  { } do do u  Extract-Min(Q) for each v  Adj(u) do if v  Q and w(u, v) < cost(v) then closest(v)  u cost(v)  w(u, v) // Decrease-Key

11. Running Time Analysis Queue opertions: Build-Queue creates the initial queue Extract-Min extracts the vertex of minimum cost Decrease-Key decreases cost(v) for v in Q, if necessary Heap  (V) O(lg V) O(lg V) O(E lg V) #operations 1 V  2E Array  (V) O(V) O(1) O(V ) 2 Q Build-Queue Extract-Min Decrease-Key Time Prim’s