1. Minimum- Spanning Trees1. Concrete example: computer connection2. Definition of a Minimum- Spanning Tree3. The Crucial Fact about Minimum- Spanning Trees4. Algorithms to find Minimum- Spanning Trees - Kruskal‘s Algorithm - Prim‘s Algorithm - Barůvka‘s Algorithm Minimum-Spanning Trees
2. Concrete exampleImagine: 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 verticesof a graph G spanning treeand has the least total weight over allsuch trees minimum-spanning tree (MST) w(T ) = ∑ w((v, u )) ( v ,u )∈T Minimum-Spanning Tree
4. The Crucial Fact about MST e V 1 min-weight V 2 “bridge“ edge Crucial Fact
5. The Crucial Fact about MST - The basis of the following algorithmsProposition: Let G = (V,E) be a weighted graph, and let V1 and V2 be two disjoint nonempty sets such that V = V1 ∪ V2 . Furthermore, let e be an edge with minimum weight from among those with one vertex in V1 and the other in V2 . 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 algorithmsJustification: 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 V1 the other V2in ). w(e) ≤ w( f ) T ∪ {e} Choose: . By removing f from ,a spanning tree is created, whose total weight is no more than before. A new MSTcontaining e Contradiction!!! Crucial Fact There is a MST containing e after all!!!
7. MST-AlgorithmsInput: A weighted connected graph G = (V,E) with n vertices and m edgesOutput: A minimum- spanning tree T w(T ) = ∑ w((v, u)) ( v ,u )∈T MST-Algorithms
8. Kruskal‘s Algorithm1. Each vertex is in its own cluster2. 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 it3. Continue until n-1 edges were selected Kruskals Algorithm
9. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Kruskals Algorithm
10. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Kruskals Algorithm
11. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Kruskals Algorithm
12. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Kruskals Algorithm
13. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Kruskals Algorithm
14. 5 A B 4 6 2 2 D 3 cycle!!C 3 1 2 E F 4 Kruskals Algorithm
15. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Kruskals Algorithm
16. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Kruskals Algorithm
17. minimum- spanning tree A B 2 2 DC 3 1 2 E F Kruskals Algorithm
18. The correctness of Kruskal‘s Algorithm Crucial Fact about MSTsRunning 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 Kruskals Algorithm
19. Code Fragment Input: A weighted connected graph G with n vertices and m edges Output: A minimum-spanning tree T for Gfor 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 Kruskals Algorithm
20. Prim‘s Algorithm1. All vertices are marked as not visited2. 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 Prims Algorithm
21. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Prims Algorithm
22. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Prims Algorithm
23. We could delete these edges because ofDijkstra‘s label D[u] for each vertexoutside of the cluster 5 A B 4 6 2 2 D 3 C 3 1 2 E F 4 Prims Algorithm
24. A B 2 2 D 3C 3 1 2 E F 4 Prims Algorithm
25. A B 2 2 D 3C 3 1 2 E F Prims Algorithm
26. A B 2 2 D 3C 3 1 2 E F Prims Algorithm
27. A B 2 2 DC 3 1 2 E F Prims Algorithm
28. A B 2 2 DC 3 1 2 E F Prims Algorithm
29. minimum- spanning tree A B 2 2 D C 3 1 2 E F Prims Algorithm
30. The correctness of Prim‘s Algorithm Crucial Fact about MSTsRunning 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 Prims Algorithm
31. Barůvka‘s Algorithm1. For all vertices search the edge with the smallest weight of this vertex and mark these edges4. Search connected vertices (clusters) and replace them by a “new“ vertex Ci (cluster)7. Remove the cycles and, if two vertices are connected by more than one edge, delete all edges except the “cheapest“ Baruvkas Algorithm
32. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Baruvkas Algorithm
33. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Baruvkas Algorithm
34. 5 A B 4 6 2 Ci 2 D 3C 3 1 2 E F 4 Baruvkas Algorithm
35. 5 A B 4 6 2 2 D 3C 3 1 2 E F 4 Baruvkas Algorithm
36. A B 2 2 DC 3 1 2 E F Baruvkas Algorithm
37. minimum- spanning tree A B 2 2 DC 3 1 2 E F Baruvkas 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 ) Baruvkas Algorithm
39. ComparisonKruskal‘s, Prim‘s, and Borůvka‘s algorithm Comparison
40. ComparisonAlthough each of the above algorithms has the same worth-caserunning time, each one achieves this running time using different datastructures and different approaches to build the MST. there is no clear winner among these three algorithms Comparison
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