Minimum- Spanning Trees

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. 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. 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 e min-weight “bridge“ edge Crucial Fact

5. The Crucial Fact about MST - 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 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 tree A B 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 cluster 5 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 tree A B 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 tree A B 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