Both Prim's and Kruskal's algorithms can be used to find the minimum spanning tree (MST) of a weighted, undirected graph. Prim's algorithm grows the MST from an initial node by repeatedly adding the lowest-cost edge that connects to the MST. Kruskal's algorithm sorts the edges by weight and builds the MST by adding edges in order as long as they do not form cycles.

1. Minimum Spanning Trees
Prim’s Algorithm
• Similar to Dijkstra’s Algorithm
Kruskal’s Algorithm
• 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
Initialize array
K dv pv
A F ∞ −
B F ∞ −
C F ∞ −
D F ∞ −
E F ∞ −
F F ∞ −
G F ∞ −
H F ∞ −
4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
2

6. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Start with any node, say D
K dv pv
A
B
C
D T 0 −
E
F
G
H
2

7. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of
adjacent, unselected nodes
K dv pv
A
B
C 3 D
D T 0 −
E 25 D
F 18 D
G 2 D
H
2

8. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with
minimum distance
K dv pv
A
B
C 3 D
D T 0 −
E 25 D
F 18 D
G T 2 D
H
2

9. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of
adjacent, unselected nodes
K dv pv
A
B
C 3 D
D T 0 −
E 7 G
F 18 D
G T 2 D
H 3 G
2

10. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with
minimum distance
K dv pv
A
B
C T 3 D
D T 0 −
E 7 G
F 18 D
G T 2 D
H 3 G
2

11. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of
adjacent, unselected nodes
K dv pv
A
B 4 C
C T 3 D
D T 0 −
E 7 G
F 3 C
G T 2 D
H 3 G
2

12. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with
minimum distance
K dv pv
A
B 4 C
C T 3 D
D T 0 −
E 7 G
F T 3 C
G T 2 D
H 3 G
2

13. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of
adjacent, unselected nodes
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 −
E 2 F
F T 3 C
G T 2 D
H 3 G
2

14. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with
minimum distance
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H 3 G
2

15. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of
adjacent, unselected nodes
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H 3 G
2
Table entries unchanged

16. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with
minimum distance
K dv pv
A 10 F
B 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

17. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of
adjacent, unselected nodes
K dv pv
A 4 H
B 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

18. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with
minimum distance
K dv pv
A T 4 H
B 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

19. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Update distances of
adjacent, unselected nodes
K dv pv
A T 4 H
B 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
Table entries unchanged

20. 4
25
A
H
B
F
E
D
C
G 7
2
10
18
3
4
3
7
8
9
3
10
Select node with
minimum distance
K dv pv
A T 4 H
B T 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2

21. 4
A
H
B
F
E
D
C
G
2
3
4
3
3
Cost of Minimum
Spanning Tree = Σ dv = 21
K dv pv
A T 4 H
B T 4 C
C T 3 D
D T 0 −
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
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
Consider an undirected, weight graph
5
1
A
H
B
F
E
D
C
G 3
2
4
6
3
4
3
4
8
4
3
10

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

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

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

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

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

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

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

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

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

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

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

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

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