1. Prim’s algorithm
IDEA: Maintain V – A as a priority queue Q. Key each vertex in Q with the weight of the least-weight edge connecting it to a vertex in A.
Q  V
key[v]  ¥ for all v Î V
key[s]  0 for some arbitrary s Î V
while Q ¹ 
do u  EXTRACT-MIN(Q)
for each v Î Adj[u]
do if v Î Q and w(u, v) < key[v]
then key[v]  w(u, v) ⊳ DECREASE-KEY
p[v]  u
At the end, {(v, p[v])} forms the MST.

2. Example of Prim’s algorithm
Î A
Î V – A 6 12 5 9 14 7 15 8 3 10

3. Example of Prim’s algorithm
Î A
Î V – A 6 12 5 9 14 7 15 8 3 10

4. Example of Prim’s algorithm
Î A
Î V – A 6 12 5 9 7 14 7 15 8 15 3 10 10

5. Example of Prim’s algorithm
Î A
Î V – A 6 12 5 9 7 14 7 15 8 15 3 10 10

6. Example of Prim’s algorithm12
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 15 3 10 10

7. Example of Prim’s algorithm12
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 15 3 10 10

8. Example of Prim’s algorithm6
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 14 15 3 10 8

9. Example of Prim’s algorithm6
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 14 15 3 10 8

10. Example of Prim’s algorithm6
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 14 15 3 10 8

11. Example of Prim’s algorithm6
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 3 15 3 10 8

12. Example of Prim’s algorithm6
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 3 15 3 10 8

13. Example of Prim’s algorithm6
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 3 15 3 10 8

14. Example of Prim’s algorithm6
Î A
Î V – A 6 12 5 9 5 7 9 14 7 15 8 3 15 3 10 8

15. Analysis of Prim Time = Q(V)·TEXTRACT-MIN + Q(E)·TDECREASE-KEY Q  V
key[v]  ¥ for all v Î V
key[s]  0 for some arbitrary s Î V
while Q ¹ 
do u  EXTRACT-MIN(Q)
for each v Î Adj[u]
do if v Î Q and w(u, v) < key[v]
then key[v]  w(u, v)
p[v]  u
Q(V) total
|V | times
degree(u)
times
Handshaking Lemma  Q(E) implicit DECREASE-KEY’s.
Time = Q(V)·TEXTRACT-MIN + Q(E)·TDECREASE-KEY

16. Analysis of Prim (continued)
Time = Q(V)·TEXTRACT-MIN + Q(E)·TDECREASE-KEY Q
TEXTRACT-MIN
TDECREASE-KEY
Total
array
O(V)
O(1)
O(V2)
binary heap
O(lg V)
O(E lg V)
Fibonacci heap
O(lg V)
amortized
O(1)
O(E + V lg V)
worst case

17. MST algorithms Kruskal’s algorithm (see CLRS):
Uses the disjoint-set data structure (Lecture 20).
Running time = O(E lg V).
Best to date:
Karger, Klein, and Tarjan [1993].
Randomized algorithm.
O(V + E) expected time.