Minimum Spanning Trees

1. Minimum Spanning Trees Jeff Chastine

2. Minimum Spanning Trees • Create a Tree that connect all vertices • If is weight between u and v, then: • Our job: minimize w(T) • We can use a greedy approach Jeff Chastine

3. General Proof for MSTs • General idea for loop invariant: • “Prior to each iteration, A is a subset of a MST” • Do this by finding safe edges that don’t violate this property • Other terms • A cut is a partitioning of V • A crossing edge crosses a cut • A light edge is a minimally weighted edge crossing the cut Jeff Chastine

4. A Tale of Two Algorithms • Pretty simple… • Kruskal’s: forms a forest • Prim’s: forms a single tree • Both: • Sort edges by weight • Add edges by increasing order if connects a new vertex Jeff Chastine

5. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

6. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

7. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

8. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

9. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

10. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

11. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

12. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

13. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

14. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

15. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

16. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

17. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

18. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

19. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

20. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

21. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

22. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

23. 8 7 b c d 4 9 2 Kruskal’s 11 a i 4 14 e 7 6 10 8 h g f 1 2 Note: the algorithms still run on the remaining edges! 8 7 b c d 4 9 2 11 a i 4 14 e Prim’s 7 6 10 8 h g f 1 2 Jeff Chastine

24. 7 b c d 4 9 2 Kruskal’s a i 4 e 8 h g f 1 2 Why do we get different shapes? 8 7 b c d 4 9 2 a i 4 e Prim’s h g f 1 2 Jeff Chastine

25. Summary • Minimum Spanning Trees • Calculate smallest weighted tree to connect all vertices • Use greedy algorithms • Prim’s algorithm (single tree) • Kruskal’s algorithm (forest) • Must sort the edges Jeff Chastine