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Minimum Spanning Trees

Minimum Spanning Trees. Text Read Weiss, § 9.5 Prim’s Algorithm Weiss §9.5.1 Similar to Dijkstra’s Algorithm Kruskal’s Algorithm Weiss §9.5.2 Focuses on edges, rather than nodes. Definition.

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Minimum Spanning Trees

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  1. Minimum Spanning Trees Text • Read Weiss, §9.5 Prim’s Algorithm • Weiss §9.5.1 • Similar to Dijkstra’s Algorithm Kruskal’s Algorithm • Weiss §9.5.2 • Focuses on edges, rather than nodes

  2. Definition • A Minimum Spanning Tree (MST) is a subgraph of an undirected graph such that the subgraph spans (includes) all nodes, is connected, is acyclic, and has minimum total edge weight

  3. Algorithm Characteristics • Both Prim’s and Kruskal’s Algorithms work with undirected graphs • Both work with weighted and unweighted graphs but are more interesting when edges are weighted • Both are greedy algorithms that produce optimal solutions

  4. Prim’s Algorithm • Similar to Dijkstra’s Algorithm except that dv records edge weights, not path lengths

  5. Walk-Through 2 Initialize array 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  6. 2 Start with any node, say D 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  7. Update distances of adjacent, unselected nodes 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  8. Select node with minimum distance 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  9. Update distances of adjacent, unselected nodes 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  10. Select node with minimum distance 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  11. Update distances of adjacent, unselected nodes 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  12. Select node with minimum distance 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  13. Update distances of adjacent, unselected nodes 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  14. Select node with minimum distance 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  15. Update distances of adjacent, unselected nodes 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7 Table entries unchanged

  16. Select node with minimum distance 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  17. Update distances of adjacent, unselected nodes 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  18. Select node with minimum distance 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  19. Update distances of adjacent, unselected nodes 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7 Table entries unchanged

  20. Select node with minimum distance 2 3 F C 10 A 7 3 4 8 18 4 B D 9 10 H 25 2 3 G E 7

  21. Cost of Minimum Spanning Tree =  dv = 21 2 3 F C A 3 4 4 B D H 2 3 G E Done

  22. Kruskal’s Algorithm • Work with edges, rather than nodes • Two steps: • Sort edges by increasing edge weight • Select the first |V| – 1 edges that do not generate a cycle

  23. Walk-Through Consider an undirected, weight graph 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  24. Sort the edges by increasing edge weight 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  25. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  26. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  27. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3 Accepting edge (E,G) would create a cycle

  28. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  29. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  30. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  31. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  32. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  33. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  34. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  35. Select first |V|–1 edges which do not generate a cycle 3 F C 10 A 4 3 4 8 6 5 B D 4 4 H 1 2 3 G E 3

  36. Select first |V|–1 edges which do not generate a cycle 3 F C A 3 4 5 B D H 1 } 2 3 G E not considered Done Total Cost = dv = 21

  37. Detecting Cycles • Use Disjoint Sets (Chapter 8)

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