1. Minimum Spanning Trees
Text
Read Weiss, §9.5
Prim’s Algorithm
Weiss §9.5.1
Similar to Dijkstra’s Algorithm
Kruskal’s Algorithm
Weiss §9.5.2
Focuses on edges, rather than nodes

2. Definition
A Minimum Spanning Tree (MST) is a subgraph of an undirected graph such that the subgraph spans (includes) all nodes, is connected, is acyclic, and has minimum total edge weight

3. Algorithm Characteristics
Both Prim’s and Kruskal’s Algorithms work with undirected graphs
Both work with weighted and unweighted graphs but are more interesting when edges are weighted
Both are greedy algorithms that produce optimal solutions

4. Prim’s Algorithm
Similar to Dijkstra’s Algorithm except that dv records edge weights, not path lengths

5. Walk-Through F C A B D H G E Initialize array K dv pv A F   B C D E2
Initialize array 3 F C K dv pv A F   B C D E G H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

6. F C A B D H G E Start with any node, say D K dv pv A B C D T  E F G H2
Start with any node, say D 3 F C K dv pv A B C D T  E F G H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

7. Update distances of adjacent, unselected nodes2 3 F C K dv pv A B C 3 D T  E 25 F 18 G 2 H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

8. Select node with minimum distance2 3 F C K dv pv A B C 3 D T  E 25 F 18 G 2 H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

9. Update distances of adjacent, unselected nodes2 3 F C K dv pv A B C 3 D T  E 7 G F 18 2 H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

10. Select node with minimum distance2 3 F C K dv pv A B C T 3 D  E 7 G F 18 2 H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

11. Update distances of adjacent, unselected nodes2 3 F C K dv pv A B 4 C T 3 D  E 7 G F 2 H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

12. Select node with minimum distance2 3 F C K dv pv A B 4 C T 3 D  E 7 G F 2 H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

13. Update distances of adjacent, unselected nodes2 3 F C K dv pv A 10 F B 4 C T 3 D  E 2 G H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

14. Select node with minimum distance2 3 F C K dv pv A 10 F B 4 C T 3 D  E 2 G H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

15. F C A B D H G E Update distances of adjacent, unselected nodes2 3 F C K dv pv A 10 F B 4 C T 3 D  E 2 G H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7
Table entries unchanged

16. Select node with minimum distance2 3 F C K dv pv A 10 F B 4 C T 3 D  E 2 G H 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

17. Update distances of adjacent, unselected nodes2 3 F C K dv pv A 4 H B C T 3 D  E 2 F G 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

18. Select node with minimum distance2 3 F C K dv pv A T 4 H B C 3 D  E 2 F G 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

19. F C A B D H G E Update distances of adjacent, unselected nodes2 3 F C K dv pv A T 4 H B C 3 D  E 2 F G 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7
Table entries unchanged

20. Select node with minimum distance2 3 F C K dv pv A T 4 H B C 3 D  E 2 F G 10 A 7 3 8 4 18 4 B D 9 H 10 25 2 3 G E 7

21. Cost of Minimum Spanning Tree =  dv = 213 F C K dv pv A T 4 H B C 3 D  E 2 F G A 3 4 4 B D H 2 3 G E
Done

22. Kruskal’s Algorithm Work with edges, rather than nodes Two steps:
Sort edges by increasing edge weight
Select the first |V| – 1 edges that do not generate a cycle

23. Walk-Through F C A B D H G E Consider an undirected, weight graph 3 104 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

24. F C A B D H G E Sort the edges by increasing edge weight edge dv (D,E)3 F C
edge dv
(D,E) 1
(D,G) 2
(E,G) 3
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

25. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

26. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

27. F C A B D H G E Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3
Accepting edge (E,G) would create a cycle

28. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

29. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

30. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

31. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

32. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

33. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

34. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

35. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5 
(D,F) 6
(A,B) 8
(A,F) 10 10 A 4 3 8 4 6 5 B D 4 H 4 1 2 3 G E 3

36. Select first |V|–1 edges which do not generate a cycle3 F C
edge dv
(D,E) 1 
(D,G) 2
(E,G) 3 
(C,D)
(G,H)
(C,F)
(B,C) 4
edge dv
(B,E) 4 
(B,F)
(B,H)
(A,H) 5 
(D,F) 6
(A,B) 8
(A,F) 10 A 3 4 5 B D H 1 2 } 3 G E
not
considered
Done
Total Cost =  dv = 21

37. Detecting Cycles
Use Disjoint Sets (Chapter 8)