1. MST – KRUSKAL UNIT IV

2. Disjoint-Set Union Problem Want a data structure to support disjoint sets – Collection of disjoint sets S = {S i }, S i ∩ S j =  Need to support following operations: – MakeSet(x): S = S U {{x}} – Union(S i, S j ): S = S - {S i, S j } U {S i U S j } – FindSet(X): return S i  S such that x  S i

3. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); }

4. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

5. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

6. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

7. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1? 5 13 17 25 14 8 21 Run the algorithm:

8. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

9. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2? 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

10. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

11. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5? 13 17 25 14 8 21 Run the algorithm:

12. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

13. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8? 21 Run the algorithm:

14. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

15. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9? 1 5 13 17 25 14 8 21 Run the algorithm:

16. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

17. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13? 17 25 14 8 21 Run the algorithm:

18. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

19. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14? 8 21 Run the algorithm:

20. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

21. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17? 25 14 8 21 Run the algorithm:

22. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19? 9 1 5 13 17 25 14 8 21 Run the algorithm:

23. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21? Run the algorithm:

24. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25? 14 8 21 Run the algorithm:

25. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

26. Kruskal’s Algorithm Kruskal() { T =  ; for each v  V MakeSet(v); sort E by increasing edge weight w for each (u,v)  E (in sorted order) if FindSet(u)  FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); } 2 19 9 1 5 13 17 25 14 8 21 Run the algorithm:

27. MST

28. MST - KRUSKAL KRUSKALKRUSKAL

29. PRIMPRIM

30. MST

31. MST - KRUSKAL

32. MST - PRIM

33. MST – KRUSKAL

34. M S T – K R U S K A L

35. TRY MST(PRIM/KRUSKAL)