The document discusses minimum spanning trees and Kruskal's algorithm. It begins by defining minimum spanning trees and the cut property. It then explains Kruskal's algorithm, which finds a minimum spanning tree by iteratively adding the lightest edge that does not form a cycle. It uses disjoint sets to determine if an edge connects two vertices in different sets, and thus would not form a cycle.

1. POSCAT Seminar 10 :
Graph 3
yougatup @ POSCAT
1

2. Topic
 Topic today
− Minimum Spanning Tree
• Cut Property
• Prim Algorithm
• Kruskal Algorithm
POSCAT Seminar 1-2
16 July 2014
yougatup

3. Spanning Tree
 Definition
A tree which “spans” whole vertices
i.e. a tree with n vertices
POSCAT Seminar 1-3
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

4. Spanning Tree
 Definition
A tree which “spans” whole vertices
i.e. a tree with n vertices
POSCAT Seminar 1-4
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

5. Minimum Spanning Tree
 Problem
Find a spanning tree with minimum cost
POSCAT Seminar 1-5
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

6. Minimum Spanning Tree
 Problem
Find a spanning tree with minimum cost
POSCAT Seminar 1-6
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
Is it MST ?

7. Minimum Spanning Tree
 Problem
Find a spanning tree with minimum cost
POSCAT Seminar 1-7
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
Is it MST ?

8. Cut Property
 Theorem
Suppose edge X are part of a MST of 𝐺 = (𝑉, 𝐸). Pick any subset of
nodes S for which X does not cross between S and 𝑉S, and let 𝑒
be the lightest edge across this partition. Then 𝑋 ∪ {𝑒} is part of
some MST
POSCAT Seminar 1-8
16 July 2014
yougatup
Proof ?

9. Cut Property
 Theorem
Suppose X is a set of edges which are part of a MST of 𝐺 = (𝑉, 𝐸).
Pick any subset of nodes S for which X does not cross between S
and 𝑉S, and let 𝑒 be the lightest edge across this partition. Then
𝑋 ∪ {𝑒} is part of some MST
POSCAT Seminar 1-9
16 July 2014
yougatup
Assume 𝑒 ∉ 𝑇. Then we can construct
a different MST 𝑇′ containing 𝑋 ∪ 𝑒
by altering T slightly.
Compare the cost(𝑇′) and cost (𝑇)

10. Cut Property
 Theorem
Suppose X is a set of edges which are part of a MST of 𝐺 = (𝑉, 𝐸).
Pick any subset of nodes S for which X does not cross between S
and 𝑉S, and let 𝑒 be the lightest edge across this partition. Then
𝑋 ∪ {𝑒} is part of some MST
POSCAT Seminar 1-10
16 July 2014
yougatup
by cut property, we can derive
beautiful greedy algorithm !

11. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-11
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

12. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-12
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

13. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-13
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

14. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-14
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

15. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-15
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

16. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-16
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

17. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-17
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
?

18. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-18
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
?
It make a cycle !
∴ we can’t select it

19. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-19
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

20. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-20
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

21. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-21
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

22. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-22
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

23. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-23
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
Done ! we have no remaining edges

24. Kruskal Algorithm
 Approach
For each iteration, choose minimum edge which doesn’t make
a cycle. Then it will make a MST by cut property.
POSCAT Seminar 1-24
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
Done ! we have no remaining edges

25. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?
POSCAT Seminar 1-25
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

26. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-26
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

27. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
If a edge connects two vertices with different group, it will never
make a cycle.
POSCAT Seminar 1-27
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

28. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
If not, it will make a cycle !
POSCAT Seminar 1-28
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3

29. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-29
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1 2 3 4
5 6 7 8

30. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-30
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1 2 3 4
5 6 7 8

31. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-31
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1 23 4
5 6 7 8

32. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-32
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1 23 4
5 6 7 8

33. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-33
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3 4
5 6 7 8

34. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-34
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3 4
5 6 7 8

35. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-35
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3 4
5
6
7 8

36. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-36
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3 4
5
6
7 8

37. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-37
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4
5
6
7 8

38. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-38
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4
5
6
7 8

39. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-39
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4
5
6
7 8

40. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-40
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4
5
6
7 8
No !!

41. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-41
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4
5
6
7 8

42. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-42
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4 5
6 7
8

43. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-43
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4 5
6 7
8

44. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-44
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4 5
6 7
8

45. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-45
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4 5
6 7
8

46. Kruskal Algorithm
 Question
How can we determine whether adding a edge makes a cycle or
not ?  by using Disjoint Set !
POSCAT Seminar 1-46
16 July 2014
yougatup
1 2
3
4
6
5
7
8
3
1 1
2 3
1 2
9
3
1
2
3
4 5
6 7
8

47. Kruskal Algorithm
 Implementation
− We have to sort the set of edges via their costs  with STL
− Union & Find
 Quite simple !
POSCAT Seminar 1-47
16 July 2014
yougatup

48. Kruskal Algorithm
 Implementation
− We have to sort the set of edges via their costs  with STL
− Union & Find
 Quite simple !
Analysis ?
POSCAT Seminar 1-48
16 July 2014
yougatup

49. Kruskal Algorithm
 Implementation
− We have to sort the set of edges via their costs  with STL
− Union & Find
 Quite simple !
Analysis ? Sorting takes O(𝐸 log 𝐸)
Each iteration takes almost O(1)
because we use Union & Find
POSCAT Seminar 1-49
16 July 2014
yougatup

50. Kruskal Algorithm
 Implementation
− We have to sort the set of edges via their costs  with STL
− Union & Find
 Quite simple !
Analysis ? Sorting takes O(𝐸 log 𝐸)
Each iteration takes almost O(1)
because we use Union & Find  O(𝑬 𝐥𝐨𝐠 𝑬 )
POSCAT Seminar 1-50
16 July 2014
yougatup