1. Minimum- Spanning Trees
1. Concrete example: computer connection
2. Definition of a Minimum- Spanning Tree
3. The Crucial Fact about Minimum- Spanning Trees
4. Algorithms to find Minimum- Spanning Trees
- Kruskal‘s Algorithm
- Prim‘s Algorithm
- Barůvka‘s Algorithm
Minimum-Spanning Trees

2. Minimum-Spanning Trees
Concrete example
Imagine: You wish to connect all the computers in an
office building using the least amount of cable
a weighted graph problem !!
Each vertex in a graph G represents a computer
Each edge represents the amount of cable needed to
connect all computers
Minimum-Spanning Trees

3. Minimum-Spanning Tree
We are interested in:
Finding a tree T that contains all the vertices
of a graph G spanning tree
and has the least total weight over all
such trees minimum-spanning tree
(MST)
Minimum-Spanning Tree

4. The Crucial Fact about MST
min-weight
“bridge“ edge
Crucial Fact

5. The Crucial Fact about MST The basis of the following algorithms-
The basis of the following algorithms
Proposition: Let G = (V,E) be a weighted graph, and let and
be two disjoint nonempty sets such that
Furthermore, let e be an edge with minimum weight from
among those with one vertex in and the other in
There is a minimum- spanning tree T that has e as one of
its edges.
Crucial Fact

6. The Crucial Fact about MST The basis of the following algorithms-
The basis of the following algorithms
Justification: There is no minimum- spanning tree that has e as one of
of ist edges The addition of e must create a cycle.
There exists an edge f (one endpoint in the other in ).
Choose: By removing f from , a
spanning tree is created, whose total weight is no more than
before A new MSTcontaining e
Contradiction!!!
There is a MST containing e after all!!!
Crucial Fact

7. MST-Algorithms
Input: A weighted connected graph G = (V,E) with n vertices
and m edges
Output: A minimum- spanning tree T
MST-Algorithms

8. Kruskal‘s Algorithm Each vertex is in its own cluster
2. Take the edge e with the smallest weight
- if e connects two vertices in different clusters,
then e is added to the MST and the two clusters,
which are connected by e, are merged into a single cluster
- if e connects two vertices, which are already in the same
cluster, ignore it
3. Continue until n-1 edges were selected
Kruskal's Algorithm

9. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Kruskal's Algorithm

10. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Kruskal's Algorithm

11. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Kruskal's Algorithm

12. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Kruskal's Algorithm

13. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Kruskal's Algorithm

14. 5 A B 4 6 2 2 D
cycle!! C 3 1 2 3 E F 4
Kruskal's Algorithm

15. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Kruskal's Algorithm

16. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Kruskal's Algorithm

17. minimum- spanning treeB 2 2 D C 1 2 3 E F
Kruskal's Algorithm

18. The correctness of Kruskal‘s Algorithm
Crucial Fact about MSTs
Running time: O ( m log n )
By implementing queue Q as a heap, Q could be initialized in O ( m ) time and a vertex could be extracted in each iteration in O ( log n ) time
Kruskal's Algorithm

19. Code Fragment
Input: A weighted connected graph G with n vertices and m edges
Output: A minimum-spanning tree T for G
for each vertex v in G do
Define a cluster C(v)  {v}.
Initialize a priority queue Q to contain all edges in G, using weights as keys.
T  
while Q   do
Extract (and remove) from Q an edge (v,u) with smallest weight.
Let C(v) be the cluster containing v, and let C(u) be the cluster containing u.
if C(v)  C(u) then
Add edge (v,u) to T.
Merge C(v) and C(u) into one cluster, that is, union C(v) and C(u).
return tree T
Kruskal's Algorithm

20. Prim‘s Algorithm All vertices are marked as not visited
2. Any vertex v you like is chosen as starting vertex and
is marked as visited (define a cluster C)
The smallest- weighted edge e = (v,u), which connects
one vertex v inside the cluster C with another vertex u outside
of C, is chosen and is added to the MST.
4. The process is repeated until a spanning tree is formed
Prim's Algorithm

21. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Prim's Algorithm

22. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Prim's Algorithm

23. We could delete these edges because of Dijkstra‘s label D[u] for each vertex outside of the cluster5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Prim's Algorithm

24. A B 2 2 D C 3 1 2 3 E F 4
Prim's Algorithm

25. A B 2 2 D C 3 1 2 3 E F
Prim's Algorithm

26. A B 2 2 D C 3 1 2 3 E F
Prim's Algorithm

27. A B 2 2 D C 1 2 3 E F
Prim's Algorithm

28. A B 2 2 D C 1 2 3 E F
Prim's Algorithm

29. minimum- spanning treeB 2 2 D C 1 2 3 E F
Prim's Algorithm

30. The correctness of Prim‘s Algorithm
Crucial Fact about MSTs
Running time: O ( m log n )
By implementing queue Q as a heap, Q could be initialized in O ( m ) time and a vertex could be extracted in each iteration in O ( log n ) time
Prim's Algorithm

31. Barůvka‘s Algorithm
For all vertices search the edge with the smallest weight
of this vertex and mark these edges
Search connected vertices (clusters) and replace them by
a “new“ vertex (cluster)
Remove the cycles and, if two vertices are connected by
more than one edge, delete all edges except the “cheapest“
Baruvka's Algorithm

32. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Baruvka's Algorithm

33. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Baruvka's Algorithm

34. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Baruvka's Algorithm

35. 5 A B 4 6 2 2 D C 3 1 2 3 E F 4
Baruvka's Algorithm

36. A B 2 2 D C 1 2 3 E F
Baruvka's Algorithm

37. minimum- spanning treeB 2 2 D C 1 2 3 E F
Baruvka's Algorithm

38. The correctness of Barůvka‘s Algorithm
Crucial Fact about MSTs
Running time: O ( m log n )
The number of edges is at least reduced by half in each step.
Number of steps: O ( log n )
Baruvka's Algorithm

39. Comparison
Kruskal‘s,
Prim‘s,
and Borůvka‘s
algorithm
Comparison

40. Comparison
Although each of the above algorithms has the same worth-case running time, each one achieves this running time using different data structures and different approaches to build the MST.
there is no clear winner among these three algorithms
Comparison