Minimum-Cost Spanning Tree

1. Minimum-Cost Spanning Tree CS 110: Data Structures and Algorithms First Semester, 2010-2011

2. Minimum-Cost Spanning Tree • Given a weighted graph G, determine a spanning tree with minimum total edge cost • Recall: spanning subgraph means all vertices are included, tree means the graph is connected and has no cycles • Useful in applications that ensure connectivity of nodes of optimal cost

3. Kruskal’s Algorithm • Solves the minimum-cost spanning tree problem • Strategy: repeatedly select the lowest-cost edge as long as it does not form a cycle with previously selected edges • Stop when n-1 edges have been selected (n is the number of vertices)

4. Kruskal’s Algorithm • Use a priority queue of edges to facilitate selection of lowest-edge cost (just disregard edges that form a cycle) • Time complexityO( m log m )  O( m log n )

5. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

6. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

7. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

8. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

9. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

10. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

11. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

12. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

13. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

14. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

15. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

16. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

17. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

18. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

19. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

20. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

21. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

22. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

23. Kruskal’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342

24. Pseudo-Code: Kruskal function Kruskal( Graph g ) n <-- number of vertices in g for each vertex v in g define an elementary cluster C(v) <-- {v} E <-- all edges in G Es <-- sort(E) T <-- null // will contain edges of MCST i <-- 0 while T has edges fewer than n-1 (u, v) <-- Es[i] Let C(v) be the cluster containing v 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 i = i + 1 return tree T

25. Prim’s Algorithm • Start at a specific vertex • Choose the edge of minimum cost which is incident on the vertex being considered • Add the new vertex on which the previously chosen edge is incident • Repeat until the MCST is found • Unlike Kruskal’s, make sure that a tree is always build as the algorithm progresses

26. Prim’s Algorithm • Start at a specific vertex • Choose the edge of minimum cost which is incident on the vertex being considered • Add the new vertex on which the previously chosen edge is incident • Repeat until the MCST is found • Unlike Kruskal’s, make sure that a tree is always build as the algorithm progresses

27. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

28. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

29. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

30. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

31. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

32. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

33. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

34. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

35. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

36. Prim’s Algorithm 8 7 b c d 4 2 9 4 14 a i e 11 7 6 8 10 h g f 1 2

37. Prim’s Algorithm 8 7 b c d 4 2 9 4 a i e h g f 1 2

38. Pseudo-Code: Prim function Prim( Graph g ) select any vertex v of g D[v] <-- 0 for each vertex u != v D[u] <-- infinity Initialize T <-- null Initalize a priority queue Q with an item ( (u, null), D[u] ) for each vertex u, where (u, null) is the element and D[u] is the key while Q is not empty (u, e) <-- Q.removeMin() Add vertex u and edge e to T for each vertex z adjacent to u such that z is in Q if w( (u,z) ) < D[z] D[z] <-- w( (u,z) ) Change to (z, (u,z)) the element of vertex z in Q Change to D[z] the key of vertex z in Q return the tree T