1. CS200: Algorithm Analysis

2. GREEDY CHOICE PROPERTY
A locally optimal greedy solution => a globally optimal solution.
Problem : Minimum Weight Spanning Tree (MST)
Revolves around the idea of spanning trees.
Undirected, connected graph G = (V,E).
Weight function w : E–>R.

3. MST

4. Example MST

5. Example MST

6. Another MST Example

7. w(T) = S(w(u,v) is minimized).
What we want is a MST T :
w(T) = S(w(u,v) is minimized).
(u,v) in T
1. Optimal substructure: optimal tree has optimal sub-trees.

8. Optimal Substructure

9. Optimal Substructure

10. Optimal Substructure

11. Optimal Substructure

12. Claim: T1 is MST of G1 = (V1,E1), the sub-graph of G with vertices in T1. (V1= vertices in T1, E1 = {(u,v) in E : (u,v) in V1}).
T2 is MST of G2 (description similar to above)

13. Proof of Optimal Substructure

14. Proof of Optimal Substructure

15. Proof of Optimal Substructure

16. Hallmark of Greedy Algorithms

17. Hallmark of Greedy Algorithms

18. Proof of Theorem

19. Proof of Theorem

20. Proof of Theorem

21. Proof of Theorem

22. Proof of Theorem