1. Greedy Algorithm (17.4/16.4) Greedy Algorithm (GA)
always makes choice which is the best at the moment
does not look ahead: local
very powerful in practice: simplest and fastest
in general, does not return the optimal solution
Example: convex polygon minimum triangulation
convex = each diagonal is completely inside
triangulation = cut into triangles
minimum = the sum of all edges is minimized
Homework (extra grade):
does greedy always give optimum solution?
how far can greedy be from optimum?

2. Minimum Spanning Tree (24.1/23.1)
MST (G) =
tree: acyclic subgraph of G
spanning: spans (connects) all the vertices of G
has the minimum total weight = sum of edge weights
G is matroid  GA gives maximum spanning tree
how GA can find minimum spanning tree?
Greedy edge selection
Kruskal’s: between two connected components
Prim’s: from connected component to new vertex

3. Prim’s Algorithm (24.2/23.2) Priority queue  binary heap
Q  V(G)
for each u  Q do key(u)  
key(r)  0; (r)  Nil;
while Q   do
u  Extract-Min(Q)
for each v adjacent to u do //total 2|E| times
if v  Q and w(u,v) < key(v) //membership bit
then (v)  u; key(v)  w(u,v) //Decrease-Key
Priority queue  binary heap
Extract-Min, Decrease-Key = O(log |V|)
Total run-time = O(|E|log |V|)

4. 2 1 6 3 4 uExtract-Min-Key(Q) u = 1; for each  adjacent to u do 8 512 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 8 3 12 11 6 4 20 3 10 8 1 7

5. 2 1 6 3 4 uExtract-Min-Key(Q) u = 1; for each  adjacent to u do 8 512 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 8 3 12 11 6 4 20 3 10 8 1 7

6. 2 1 6 3 4 uExtract-Min-Key(Q) u = 2; for each  adjacent to u do 8 512 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 8 3 12 11 6 4 20 3 10 8 1 7

7. 2 1 6 3 4 uExtract-Min-Key(Q) u = 2; for each  adjacent to u do 8 512 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 8 3 12 11 6 4 20 3 10 8 1 7

8. 2 1 6 3 4 uExtract-Min-Key(Q) u = 4; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 4;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 8 3 12 11 6 4 20 3 10 8 1 7

9. 2 1 6 3 4 uExtract-Min-Key(Q) u = 4; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 4;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 8 3 12 11 6 4 20 3 10 8 1 7

10. 2 1 6 3 4 uExtract-Min-Key(Q) u = 6; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 6;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 3 12 8 4 11 20 6 3 10 8 1 7

11. 2 1 6 3 4 uExtract-Min-Key(Q) u = 6; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 6;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 12 9 14 1 10 3 12 8 4 11 20 6 3 10 8 1 7

12. 2 1 6 3 4 uExtract-Min-Key(Q) u = 11; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 11;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 14 12 9 1 10 3 12 8 4 11 20 6 3 8 10 1 7

13. 2 1 6 3 4 uExtract-Min-Key(Q) u = 11; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 11;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 6 5 22 11 7 12 14 12 9 1 10 3 12 8 4 11 20 6 3 8 10 1 7

14. 2 1 6 3 4 uExtract-Min-Key(Q) u = 12; for each  adjacent to u do 8 517 2 13 5 10 9 8 6 22 5 11 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

15. 2 1 6 3 4 uExtract-Min-Key(Q) u = 12; for each  adjacent to u do 8 517 2 13 5 10 9 8 6 22 5 11 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

16. 2 1 6 3 4 uExtract-Min-Key(Q) u = 10; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 10;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 22 5 11 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

17. 2 1 6 3 4 uExtract-Min-Key(Q) u = 10; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 10;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 22 5 11 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

18. 2 1 6 3 4 uExtract-Min-Key(Q) u = 7; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 7;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 5 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

19. 2 1 6 3 4 uExtract-Min-Key(Q) u = 7; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 7;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 5 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

20. 2 1 6 3 4 uExtract-Min-Key(Q) u = 3; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 3;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 5 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

21. 2 1 6 3 4 uExtract-Min-Key(Q) u = 3; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 3;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 5 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

22. 2 1 6 3 4 uExtract-Min-Key(Q) u = 9; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 9;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 5 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

23. 2 1 6 3 4 uExtract-Min-Key(Q) u = 9; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 9;
for each  adjacent to u do 2 8 17 3 2 13 5 10 9 8 6 5 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

24. 2 1 6 3 4 uExtract-Min-Key(Q) u = 5; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 5;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

25. 2 1 6 3 4 uExtract-Min-Key(Q) u = 5; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 5;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

26. 2 1 6 3 4 uExtract-Min-Key(Q) u = 8; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 8;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

27. 2 1 6 3 4 uExtract-Min-Key(Q) u = 8; for each  adjacent to u do 8 512
uExtract-Min-Key(Q)
u = 8;
for each  adjacent to u do 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

28. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

29. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

30. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

31. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

32. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

33. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 22 11 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

34. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

35. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 12 9 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

36. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

37. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

38. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

39. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

40. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

41. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

42. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

43. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

44. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

45. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

46. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

47. 4 12 2 8 3 17 2 13 5 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

48. 4 12 2 8 3 17 2 5 13 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

49. 4 12 2 8 3 17 2 5 13 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7

50. 4 12 2 8 3 17 2 5 13 10 9 8 5 6 11 22 7 12 9 12 14 1 10 12 3 8 4 11 20 6 3 8 10 1 7